CMSA

Entanglement and its key role in quantum information
2022-06-30 16:23 - 17:23 (America/New_York) | MA
Abstract: Entanglement is a type of correlation found in composite quantum systems, connected with various non-classical phenomena. Currently, entanglement plays a key role in quantum information applications such as quantum computing, quantum communication, and quantum sensing. In this talk the concept of entanglement will be introduced along with various methods that have been proposed to detect and quantify it. The fundamental role of entanglement in both quantum theory and quantum technology will also be discussed.
Bio: Spyros Tserkis is a postdoctoral researcher at Harvard University, working on quantum information theory. Before joining Harvard in Fall 2021, he was a postdoctoral researcher at MIT and the Australian National University. He received his PhD from the University of Queensland.

Some new algorithms in statistical genomics
2022-06-23 09:00 - 10:00 | MA
Abstract: The statistical analysis of genomic data has incubated many innovations for computational method development. This talk will discuss some simple algorithms that may be useful in analyzing such data. Examples include algorithms for efficient resampling-based hypothesis testing, minimizing the sum of truncated convex functions, and fitting equality-constrained lasso problems. These algorithms have the potential to be used in other applications beyond statistical genomics.
Bio: Hui Jiang is an Associate Professor in the Department of Biostatistics at the University of Michigan. He received his Ph.D. in Computational and Mathematical Engineering from Stanford University. Before joining the University of Michigan, he was a postdoc in the Department of Statistics and Stanford Genome Technology Center at Stanford University. He is interested in developing statistical and computational methods for analyzing large-scale biological data generated using modern high-throughput technologies.

Joint BHI/CMSA Conference on Flat Holography
2022-06-21 09:00 - 2022-06-24 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On June 21–24, 2022, the Harvard Black Hole Initiative and the CMSA hosted the Joint BHI/CMSA Conference on Flat Holography (and related topics).
The recent discovery of infinitely-many soft symmetries for all quantum theories of gravity in asymptotically flat space has provided a promising starting point for a bottom-up construction of a holographic dual for the real world. Recent developments have brought together previously disparate studies of soft theorems, asymptotic symmetries, twistor theory, asymptotically flat black holes and their microscopic duals, self-dual gravity, and celestial scattering amplitudes, and link directly to AdS/CFT.
The conference was held in room G10 of the CMSA, 20 Garden Street, Cambridge, MA.
Organizers:

Daniel Kapec, CMSA
Andrew Strominger, BHI
Shing-Tung Yau, Harvard & Tsinghua

Confirmed Speakers:

Nima Arkani-Hamed, IAS
Shamik Banerjee, Bhubaneswar, Inst. Phys.
Miguel Campiglia, Republica U., Montevido
Geoffrey Compere, Brussels
Laura Donnay, Vienna
Netta Engelhardt, MIT
Laurent Freidel, Perimeter
Alex Lupsasca, Princeton
Juan Maldacena, IAS
Lionel Mason, Oxford
Natalie Paquette, U. Washington
Sabrina Pasterski, Princeton/Perimeter
Andrea Puhm, Ecole Polytechnique
Ana-Maria Raclariu, Perimeter
Marcus Spradlin, Brown
Tomasz Taylor, Northeastern
Herman Verlinde, Princeton
Anastasia Volovich, Brown
Bin Zhu, Northeastern

Short talks by: Gonçalo Araujo-Regado (Cambridge), Adam Ball (Harvard), Eduardo Casali (Harvard), Jordan Cotler (Harvard), Erin Crawley (Harvard), Stéphane Detournay (Brussels), Alfredo Guevara (Harvard), Temple He (UC Davis), Elizabeth Himwich (Harvard), Yangrui Hu (Brown), Daniel Kapec (Harvard), Rifath Khan (Cambridge), Albert Law (Harvard), Luke Lippstreu (Brown), Noah Miller (Harvard), Sruthi Narayanan (Harvard), Lecheng Ren (Brown), Francisco Rojas (UAI), Romain Ruzziconi (Vienna), Andrew Strominger (Harvard), Adam Tropper (Harvard), Tianli Wang (Harvard), Walker Melton (Harvard)

Schedule
Monday, June 20, 2022

Arrival

7:00–9:00 pm
Welcome Reception at Andy’s residence

Tuesday, June 21, 2022

9:00–9:30 am
Breakfast
light breakfast provided

Morning Session
Chair: Dan Kapec

9:30–10:00 am
Herman Verlinde
Title: Comments on Celestial Dynamics

10:00–10:30 am
Juan Maldacena
Title: What happens when you spend too much time looking at supersymmetric
black holes?

10:30–11:00
Coffee break

11:00–11:30 am
Miguel Campiglia
Title: Asymptotic symmetries and loop corrections to soft theorems

11:30–12:00 pm
Geoffrey Compere
Title: Metric reconstruction from $Lw_{1+infty}$ multipoles
Abstract: The most general vacuum solution to Einstein’s field equations with no incoming radiation can be constructed perturbatively from two infinite sets of canonical multipole moments, which are found to be exchanged under gravitational electric-magnetic duality at the non-linear level. We demonstrate that in non-radiative regions such spacetimes are completely determined by a set of conserved celestial charges, which uniquely label transitions among non-radiative regions caused by radiative processes. The algebra of the conserved celestial charges is derived from the real $Lw_{1+infty}$ algebra. The celestial charges are expressed in terms of multipole moments, which allows to holographically reconstruct the metric in de Donder, Newman-Unti or Bondi gauge outside of sources.

12:00–2:00 pm
Lunch break

Afternoon Session
Chair: Eduardo Casali

2:00–2:30 pm
Natalie Paquette
Title: New thoughts on old gauge amplitudes

2:30–3:00 pm
Lionel Mason
Title: An open sigma model for celestial gravity
Abstract: A global twistor construction for conformally self-dual split signature metrics on $S2times S2$ was developed 15 years ago by Claude LeBrun and the speaker. This encodes the conformal metric into the location of a finite deformation of the real twistor space inside the flat complex twistor space, $mathbb{CP}3$. This talk adapts the construction to construct global SD Einstein metrics from conformal boundary data and perturbations around the self-dual sector. The construction entails determining a family of holomorphic discs in $mathbb{CP}3$ whose boundaries lie on the deformed real slice and the (chiral) sigma model controls these discs in the Einstein case and provides amplitude formulae.

3:00–3:30 pm
Coffee break

3:30–4:30 pm
Short Talks
Daniel Kapec: Soft Scalars and the Geometry of the Space of Celestial CFTs
Albert Law: Soft Scalars and the Geometry of the Space of Celestial CFTs
Sruthi Narayanan: Soft Scalars and the Geometry of the Space of Celestial CFTs
Stéphane Detournay: Non-conformal symmetries and near-extremal black holes
Francisco Rojas: Celestial string amplitudes beyond tree level
Temple He: An effective description of energy transport from holography

4:30–5:00 pm
Nima Arkani-Hamed
(Dual) surfacehedra and flow particles know about strings

Wednesday, June 22, 2022

9:00–9:30 am
Breakfast
light breakfast provided

Morning Session
Chair: Alfredo Guevara

9:30–10:00 am
Laurent Freidel
Title: Higher spin symmetry in gravity
Abstract: In this talk, I will review how the gravitational conservation laws at infinity reveal a tower of symmetry charges in an asymptotically flat spacetime.
I will show how the conservation laws, at spacelike infinity, give a tower of soft theorems that connect to the ones revealed by celestial holography.
I’ll present the expression for the symmetry charges in the radiative phase space, which opens the way to reveal the structure of the algebra beyond the positive helicity sector. Then, if time permits I’ll browse through many questions that these results raise:
such as the nature of the spacetime symmetry these charges represent, the nature of the relationship with multipole moments, and the insights their presence provides for quantum gravity.

10:00–10:30 am
Ana-Maria Raclariu
Title: Eikonal approximation in celestial CFT

10:30–11:00 am
Coffee break

11:00–11:30 am
Anastasia Volovich
Title: Effective Field Theories with Celestial Duals

11:30–12:00 pm
Marcus Spradlin
Title: Loop level gluon OPE’s in celestial holography

12:00–2:00 pm
Lunch break

Afternoon Session
Chair: Chiara Toldo

2:00–2:30 pm
Netta Engelhardt
Title: Wormholes from entanglement: true or false?

2:30–3:00 pm
Short Talks
Luke Lippstreu: Loop corrections to the OPE of celestial gluons
Yangrui Hu: Light transforms of celestial amplitudes
Lecheng Ren: All-order OPE expansion of celestial gluon and graviton primaries from MHV amplitudes

3:00–3:30 pm
Coffee break

3:30–4:30 pm
Short Talks
Noah Miller: C Metric Thermodynamics
Erin Crawley: Kleinian black holes
Rifath Khan: Cauchy Slice Holography: A New AdS/CFT Dictionary
Gonçalo Araujo-Regado: Cauchy Slice Holography: A New AdS/CFT Dictionary
Tianli Wang: Soft Theorem in the BFSS Matrix Model
Adam Tropper: Soft Theorem in the BFSS Matrix Model

7:00–9:00 pm
Banquet
Maharaja Restaurant, 57 JFK Street, Cambridge, MA

Thursday, June 23, 2022

9:00–9:30 am
Breakfast
light breakfast provided

Morning Session
Chair: Jordan Cotler

9:30–10:00 am
Laura Donnay
Title: A Carrollian road to flat space holography

10:00–10:30 am
Andrea Puhm
Title: Celestial wave scattering on Kerr-Schild backgrounds

10:30–11:00 am
Coffee break

11:00–11:30 am
Sabrina Pasterski
Title: Mining Celestial Symmetries
Abstract: The aim of this talk is to delve into the common thread that ties together recent work with H. Verlinde, L. Donnay, A. Puhm, and S. Banerjee exploring, explaining, and exploiting the symmetries encoded in the conformally soft sector.
Come prepared to debate the central charge, loop corrections, contour prescriptions, and orders of limits!

11:30–12:00 pm
Shamik Banerjee
Title: Virasoro and other symmetries in CCFT
Abstract: In this talk I will briefly describe my ongoing work with Sabrina Pasterski. In this work we revisit the standard construction of the celestial stress tensor as a shadow of the subleading conformally soft graviton. In its original formulation, we find that there is an obstruction to reproducing the expected $TT$ OPE in the double soft limit. This obstruction is related to the existence of the $SL_2$ current algebra symmetry of the CCFT. We propose a modification to the definition of the stress tensor which circumvents this obstruction and also discuss its implications for the existence of other current algebra (w_{1+infty}) symmetries in CCFT.

12:00–2:00 pm
Lunch break

Afternoon Session
Chair: Albert Law

2:00–2:30 pm
Tomasz Taylor
Title: Celestial Yang-Mills amplitudes and D=4 conformal blocks

2:30–3:00 pm
Bin Zhu
Title: Single-valued correlators and Banerjee-Ghosh equations
Abstract: Low-point celestial amplitudes are plagued with singularities resulting from spacetime translation. We consider a marginal deformation of the celestial CFT which is realized by coupling Yang-Mills theory to a background dilaton field, with the (complex) dilaton source localized on the celestial sphere. This picture emerges from the physical interpretation of the solutions of the system of differential equations discovered by Banerjee and Ghosh. We show that the solutions can be written as Mellin transforms of the amplitudes evaluated in such a dilaton background. The resultant three-gluon and four-gluon amplitudes are single-valued functions of celestial coordinates enjoying crossing symmetry and all other properties expected from standard CFT correlators.

3:00–3:30 pm
Coffee break

3:30–4:00 pm
Alex Lupsasca
Title: Holography of the Photon Ring

4:00–5:30 pm
Short Talks
Elizabeth Himwich: Celestial OPEs and w(1+infinity) symmetry of massless and massive amplitudes
Adam Ball: Perturbatively exact $w_{1+infty}$ asymptotic symmetry of quantum self-dual gravity
Romain Ruzziconi: A Carrollian Perspective on Celestial Holography
Jordan Cotler: Soft Gravitons in 3D
Alfredo Guevara: Comments on w_1+inf
Andrew Strominger: Top-down celestial holograms
Eduardo Casali: Celestial amplitudes as AdS-Witten diagrams
Walker Melton: Top-down celestial holograms

Friday, June 24, 2022

9:00–9:30 am
Breakfast

9:30–12:30 pm
Open Discussion

12:30–2:30 pm
Lunch provided at the BHI

Departure

Surface hopping algorithms for non-adiabatic quantum systems
2022-06-16 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Interdisciplinary Science Seminar

Speaker: Jianfeng Lu, Duke UniversityTitle: Surface hopping algorithms for non-adiabatic quantum systems
Abstract: Surface hopping algorithm is widely used in chemistry for mixed quantum-classical dynamics. In this talk, we will discuss some of our recent works in mathematical understanding and algorithm development for surface hopping methods. These methods are based on stochastic approximations of semiclassical path-integral representation to the solution of multi-level Schrodinger equations; such methodology also extends to other high-dimensional transport systems.

Symposium on Foundations of Responsible Computing (FORC)
2022-06-06 09:00 - 2022-06-08 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On June 6-8, 2022, the CMSA hosted the 3rd annual Symposium on Foundations of Responsible Computing (FORC).
The Symposium on Foundations of Responsible Computing (FORC) is a forum for mathematical research in computation and society writ large. The Symposium aims to catalyze the formation of a community supportive of the application of theoretical computer science, statistics, economics and other relevant analytical fields to problems of pressing and anticipated societal concern.
Organizers: Cynthia Dwork, Harvard SEAS | Omer Reingold, Stanford | Elisa Celis, Yale
Schedule
June 6, 2022

9:15 am–10:15 am
Opening Remarks
Keynote Speaker: Caroline Nobo, Yale University
Title: From Theory to Impact: Why Better Data Systems are Necessary for Criminal Legal Reform
Abstract: This talk will dive into the messy, archaic, and siloed world of local criminal justice data in America. We will start with a 30,000 foot discussion about the current state of criminal legal data systems, then transition to the challenges of this broken paradigm, and conclude with a call to measure new things – and to measure them better! This talk will leave you with an understanding of criminal justice data infrastructure and transparency in the US, and will discuss how expensive case management software and other technology are built on outdated normative values which impede efforts to reform the system. The result is an infuriating paradox: an abundance of tech products built without theoretical grounding, in a space rich with research and evidence.

10:15 am–10:45 am
Coffee Break

10:45 am–12:15 pm
Paper Session 1
Session Chair: Ruth Urner

Georgy Noarov, University of Pennsylvania
Title: Online Minimax Multiobjective Optimization
Abstract: We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. The learner’s objective is to minimize the maximum cumulative loss over all coordinates. We give a simple algorithm that lets the learner do almost as well as if she knew the adversary’s actions in advance. We demonstrate the power of our framework by using it to (re)derive optimal bounds and efficient algorithms across a variety of domains, ranging from multicalibration to a large set of no-regret algorithms, to a variant of Blackwell’s approachability theorem for polytopes with fast convergence rates. As a new application, we show how to “(multi)calibeat” an arbitrary collection of forecasters — achieving an exponentially improved dependence on the number of models we are competing against, compared to prior work.

Matthew Eichhorn, Cornell University
Title: Mind your Ps and Qs: Allocation with Priorities and Quotas
Abstract: In many settings, such as university admissions, the rationing of medical supplies, and the assignment of public housing, decision-makers use normative criteria (ethical, financial, legal, etc.) to justify who gets an allocation. These criteria can often be translated into quotas for the number of units available to particular demographics and priorities over agents who qualify in each demographic. Each agent may qualify in multiple categories at different priority levels, so many allocations may conform to a given set of quotas and priorities. Which of these allocations should be chosen? In this talk, I’ll formalize this reserve allocation problem and motivate Pareto efficiency as a natural desideratum. I’ll present an algorithm to locate efficient allocations that conform to the quota and priority constraints. This algorithm relies on beautiful techniques from integer and linear programming, and it is both faster and more straightforward than existing techniques in this space. Moreover, its clean formulation allows for further refinement, such as the secondary optimization of some heuristics for fairness.

Haewon Jeong, Harvard University
Title: Fairness without Imputation: A Decision Tree Approach for Fair Prediction with Missing Values
Abstract: We investigate the fairness concerns of training a machine learning model using data with missing values. Even though there are a number of fairness intervention methods in the literature, most of them require a complete training set as input. In practice, data can have missing values, and data missing patterns can depend on group attributes (e.g. gender or race). Simply applying off-the-shelf fair learning algorithms to an imputed dataset may lead to an unfair model. In this paper, we first theoretically analyze different sources of discrimination risks when training with an imputed dataset. Then, we propose an integrated approach based on decision trees that does not require a separate process of imputation and learning. Instead, we train a tree with missing incorporated as attribute (MIA), which does not require explicit imputation, and we optimize a fairness-regularized objective function. We demonstrate that our approach outperforms existing fairness intervention methods applied to an imputed dataset, through several experiments on real-world datasets.

Emily Diana, University of Pennsylvania
Title: Multiaccurate Proxies for Downstream Fairness
Abstract: We study the problem of training a model that must obey demographic fairness conditions when the sensitive features are not available at training time — in other words, how can we train a model to be fair by race when we don’t have data about race? We adopt a fairness pipeline perspective, in which an “upstream” learner that does have access to the sensitive features will learn a proxy model for these features from the other attributes. The goal of the proxy is to allow a general “downstream” learner — with minimal assumptions on their prediction task — to be able to use the proxy to train a model that is fair with respect to the true sensitive features. We show that obeying multiaccuracy constraints with respect to the downstream model class suffices for this purpose, provide sample- and oracle efficient-algorithms and generalization bounds for learning such proxies, and conduct an experimental evaluation. In general, multiaccuracy is much easier to satisfy than classification accuracy, and can be satisfied even when the sensitive features are hard to predict.

12:15 pm–1:45 pm
Lunch Break

1:45–3:15 pm
Paper Session 2
Session Chair: Guy Rothblum

Elbert Du, Harvard University
Title: Improved Generalization Guarantees in Restricted Data Models
Abstract: Differential privacy is known to protect against threats to validity incurred due to adaptive, or exploratory, data analysis — even when the analyst adversarially searches for a statistical estimate that diverges from the true value of the quantity of interest on the underlying population. The cost of this protection is the accuracy loss incurred by differential privacy. In this work, inspired by standard models in the genomics literature, we consider data models in which individuals are represented by a sequence of attributes with the property that where distant attributes are only weakly correlated. We show that, under this assumption, it is possible to “re-use” privacy budget on different portions of the data, significantly improving accuracy without increasing the risk of overfitting.

Ruth Urner, York University
Title: Robustness Should not be at Odds with Accuracy
Abstract: The phenomenon of adversarial examples in deep learning models has caused substantial concern over their reliability and trustworthiness: in many instances an imperceptible perturbation can falsely flip a neural network’s prediction. Applied research in this area has mostly focused on developing novel adversarial attack strategies or building better defenses against such. It has repeatedly been pointed out that adversarial robustness may be in conflict with requirements for high accuracy. In this work, we take a more principled look at modeling the phenomenon of adversarial examples. We argue that deciding whether a model’s label change under a small perturbation is justified, should be done in compliance with the underlying data-generating process. Through a series of formal constructions, systematically analyzing the the relation between standard Bayes classifiers and robust-Bayes classifiers, we make the case for adversarial robustness as a locally adaptive measure. We propose a novel way defining such a locally adaptive robust loss, show that it has a natural empirical counterpart, and develop resulting algorithmic guidance in form of data-informed adaptive robustness radius. We prove that our adaptive robust data-augmentation maintains consistency of 1-nearest neighbor classification under deterministic labels and thereby argue that robustness should not be at odds with accuracy.

Sushant Agarwal, University of Waterloo
Title: Towards the Unification and Robustness of Perturbation and Gradient Based Explanations
Abstract: As machine learning black boxes are increasingly being deployed in critical domains such as healthcare and criminal justice, there has been a growing emphasis on developing techniques for explaining these black boxes in a post hoc manner. In this work, we analyze two popular post hoc interpretation techniques: SmoothGrad which is a gradient based method, and a variant of LIME which is a perturbation based method. More specifically, we derive explicit closed form expressions for the explanations output by these two methods and show that they both converge to the same explanation in expectation, i.e., when the number of perturbed samples used by these methods is large. We then leverage this connection to establish other desirable properties, such as robustness and linearity, for these techniques. We also derive finite sample complexity bounds for the number of perturbations required for these methods to converge to their expected explanation. Finally, we empirically validate our theory using extensive experimentation on both synthetic and real world datasets.

Tijana Zrnic, University of California, Berkeley
Title: Regret Minimization with Performative Feedback
Abstract: In performative prediction, the deployment of a predictive model triggers a shift in the data distribution. As these shifts are typically unknown ahead of time, the learner needs to deploy a model to get feedback about the distribution it induces. We study the problem of finding near-optimal models under performativity while maintaining low regret. On the surface, this problem might seem equivalent to a bandit problem. However, it exhibits a fundamentally richer feedback structure that we refer to as performative feedback: after every deployment, the learner receives samples from the shifted distribution rather than only bandit feedback about the reward. Our main contribution is regret bounds that scale only with the complexity of the distribution shifts and not that of the reward function. The key algorithmic idea is careful exploration of the distribution shifts that informs a novel construction of confidence bounds on the risk of unexplored models. The construction only relies on smoothness of the shifts and does not assume convexity. More broadly, our work establishes a conceptual approach for leveraging tools from the bandits literature for the purpose of regret minimization with performative feedback.

3:15 pm–3:45 pm
Coffee Break

3:45 pm–5:00 pm
Panel Discussion
Title: What is Responsible Computing?
Panelists: Jiahao Chen, Cynthia Dwork, Kobbi Nissim, Ruth Urner
Moderator: Elisa Celis

June 7, 2022

9:15 am–10:15 am
Keynote Speaker: Isaac Kohane, Harvard Medical School
Title: What’s in a label? The case for and against monolithic group/ethnic/race labeling for machine learning
Abstract: Populations and group labels have been used and abused for thousands of years. The scale at which AI can incorporate such labels into its models and the ways in which such models can be misused are cause for significant concern. I will describe, with examples drawn from experiments in precision medicine, the task dependence of how underserved and oppressed populations can be both harmed and helped by the use of group labels. The source of the labels and the utility models underlying their use will be particularly emphasized.

10:15 am–10:45 am
Coffee Break

10:45 am–12:15 pm
Paper Session 3
Session Chair: Ruth Urner

Rojin Rezvan, University of Texas at Austin
Title: Individually-Fair Auctions for Multi-Slot Sponsored Search
Abstract: We design fair-sponsored search auctions that achieve a near-optimal tradeoff between fairness and quality. Our work builds upon the model and auction design of Chawla and Jagadeesan, who considered the special case of a single slot. We consider sponsored search settings with multiple slots and the standard model of click-through rates that are multiplicatively separable into an advertiser-specific component and a slot-specific component. When similar users have similar advertiser-specific click-through rates, our auctions achieve the same near-optimal tradeoff between fairness and quality. When similar users can have different advertiser-specific preferences, we show that a preference-based fairness guarantee holds. Finally, we provide a computationally efficient algorithm for computing payments for our auctions as well as those in previous work, resolving another open direction from Chawla and Jagadeesan.

Judy Hanwen Shen, Stanford
Title: Leximax Approximations and Representative Cohort Selection
Abstract: Finding a representative cohort from a broad pool of candidates is a goal that arises in many contexts such as choosing governing committees and consumer panels. While there are many ways to define the degree to which a cohort represents a population, a very appealing solution concept is lexicographic maximality (leximax) which offers a natural (pareto-optimal like) interpretation that the utility of no population can be increased without decreasing the utility of a population that is already worse off. However, finding a leximax solution can be highly dependent on small variations in the utility of certain groups. In this work, we explore new notions of approximate leximax solutions with three distinct motivations: better algorithmic efficiency, exploiting significant utility improvements, and robustness to noise. Among other definitional contributions, we give a new notion of an approximate leximax that satisfies a similarly appealing semantic interpretation and relate it to algorithmically-feasible approximate leximax notions. When group utilities are linear over cohort candidates, we give an efficient polynomial-time algorithm for finding a leximax distribution over cohort candidates in the exact as well as in the approximate setting. Furthermore, we show that finding an integer solution to leximax cohort selection with linear utilities is NP-Hard.

Jiayuan Ye,
National University of Singapore
Title: Differentially Private Learning Needs Hidden State (or Much Faster Convergence)
Abstract: Differential privacy analysis of randomized learning algorithms typically relies on composition theorems, where the implicit assumption is that the internal state of the iterative algorithm is revealed to the adversary. However, by assuming hidden states for DP algorithms (when only the last-iterate is observable), recent works prove a converging privacy bound for noisy gradient descent (on strongly convex smooth loss function) that is significantly smaller than composition bounds after a few epochs. In this talk, we extend this hidden-state analysis to various stochastic minibatch gradient descent schemes (such as under “shuffle and partition” and “sample without replacement”), by deriving novel bounds for the privacy amplification by random post-processing and subsampling. We prove that, in these settings, our privacy bound is much smaller than composition for training with a large number of iterations (which is the case for learning from high-dimensional data). Our converging privacy analysis, thus, shows that differentially private learning, with a tight bound, needs hidden state privacy analysis or a fast convergence. To complement our theoretical results, we present experiments for training classification models on MNIST, FMNIST and CIFAR-10 datasets, and observe a better accuracy given fixed privacy budgets, under the hidden-state analysis.

Mahbod Majid, University of Waterloo
Title: Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism
Abstract: We give the first polynomial-time algorithm to estimate the mean of a d-variate probability distribution from O(d) independent samples (up to logarithmic factors) subject to pure differential privacy.
Our main technique is a new approach to use the powerful Sum of Squares method (SoS) to design differentially private algorithms. SoS proofs to algorithms is a key theme in numerous recent works in high-dimensional algorithmic statistics – estimators which apparently require exponential running time but whose analysis can be captured by low-degree Sum of Squares proofs can be automatically turned into polynomial-time algorithms with the same provable guarantees. We demonstrate a similar proofs to private algorithms phenomenon: instances of the workhorse exponential mechanism which apparently require exponential time but which can be analyzed with low-degree SoS proofs can be automatically turned into polynomial-time differentially private algorithms. We prove a meta-theorem capturing this phenomenon, which we expect to be of broad use in private algorithm design.

12:15 pm–1:45 pm
Lunch Break

1:45–3:15 pm
Paper Session 4
Session Chair: Kunal Talwar

Kunal Talwar,
Apple
Title: Differential Secrecy for Distributed Data and Applications to Robust Differentially Secure Vector Summation
Abstract: Computing the noisy sum of real-valued vectors is an important primitive in differentially private learning and statistics. In private federated learning applications, these vectors are held by client devices, leading to a distributed summation problem. Standard Secure Multiparty Computation (SMC) protocols for this problem are susceptible to poisoning attacks, where a client may have a large influence on the sum, without being detected.
In this work, we propose a poisoning-robust private summation protocol in the multiple-server setting, recently studied in PRIO. We present a protocol for vector summation that verifies that the Euclidean norm of each contribution is approximately bounded. We show that by relaxing the security constraint in SMC to a differential privacy like guarantee, one can improve over PRIO in terms of communication requirements as well as the client-side computation. Unlike SMC algorithms that inevitably cast integers to elements of a large finite field, our algorithms work over integers/reals, which may allow for additional efficiencies.

Giuseppe Vietri, University of Minnesota
Title: Improved Regret for Differentially Private Exploration in Linear MDP
Abstract: We study privacy-preserving exploration in sequential decision-making for environments that rely on sensitive data such as medical records. In particular, we focus on solving the problem of reinforcement learning (RL) subject to the constraint of (joint) differential privacy in the linear MDP setting, where both dynamics and rewards are given by linear functions. Prior work on this problem due to Luyo et al. (2021) achieves a regret rate that has a dependence of O(K^{3/5}) on the number of episodes K. We provide a private algorithm with an improved regret rate with an optimal dependence of O(K^{1/2}) on the number of episodes. The key recipe for our stronger regret guarantee is the adaptivity in the policy update schedule, in which an update only occurs when sufficient changes in the data are detected. As a result, our algorithm benefits from low switching cost and only performs O(log(K)) updates, which greatly reduces the amount of privacy noise. Finally, in the most prevalent privacy regimes where the privacy parameter ? is a constant, our algorithm incurs negligible privacy cost — in comparison with the existing non-private regret bounds, the additional regret due to privacy appears in lower-order terms.

Mingxun Zhou,
Carnegie Mellon University
Title: The Power of the Differentially Oblivious Shuffle in Distributed Privacy MechanismsAbstract: The shuffle model has been extensively investigated in the distributed differential privacy (DP) literature. For a class of useful computational tasks, the shuffle model allows us to achieve privacy-utility tradeoff similar to those in the central model, while shifting the trust from a central data curator to a “trusted shuffle” which can be implemented through either trusted hardware or cryptography. Very recently, several works explored cryptographic instantiations of a new type of shuffle with relaxed security, called differentially oblivious (DO) shuffles. These works demonstrate that by relaxing the shuffler’s security from simulation-style secrecy to differential privacy, we can achieve asymptotical efficiency improvements. A natural question arises, can we replace the shuffler in distributed DP mechanisms with a DO-shuffle while retaining a similar privacy-utility tradeoff?
In this paper, we prove an optimal privacy amplification theorem by composing any locally differentially private (LDP) mechanism with a DO-shuffler, achieving parameters that tightly match the shuffle model. Moreover, we explore multi-message protocols in the DO-shuffle model, and construct mechanisms for the real summation and histograph problems. Our error bounds approximate the best known results in the multi-message shuffle-model up to sub-logarithmic factors. Our results also suggest that just like in the shuffle model, allowing each client to send multiple messages is fundamentally more powerful than restricting to a single message.

Badih Ghazi,
Google Research
Title: Differentially Private Ad Conversion Measurement
Abstract: In this work, we study conversion measurement, a central functionality in the digital advertising space, where an advertiser seeks to estimate advertiser site conversions attributed to ad impressions that users have interacted with on various publisher sites. We consider differential privacy (DP), a notion that has gained in popularity due to its strong and rigorous guarantees, and suggest a formal framework for DP conversion measurement, uncovering a subtle interplay between attribution and privacy. We define the notion of an operationally valid configuration of the attribution logic, DP adjacency relation, privacy
budget scope and enforcement point, and provide, for a natural space of configurations, a complete characterization.

3:15 pm–3:45 pm
Coffee Break

3:45 pm–5:00 pm
Open Poster Session

June 8, 2022

9:15 am–10:15 am
Keynote Speaker: Nuria Oliver, Data-Pop Alliance
Title: Data Science against COVID-19
Abstract: In my talk, I will describe the work that I have been doing since March 2020, leading a multi-disciplinary team of 20+ volunteer scientists working very closely with the Presidency of the Valencian Government in Spain on 4 large areas: (1) human mobility modeling; (2) computational epidemiological models (both metapopulation, individual and LSTM-based models); (3) predictive models; and (4) citizen surveys via the COVID19impactsurvey with over 600,000 answers worldwide.
I will describe the results that we have produced in each of these areas, including winning the 500K XPRIZE Pandemic Response Challenge and best paper award at ECML-PKDD 2021. I will share the lessons learned in this very special initiative of collaboration between the civil society at large (through the survey), the scientific community (through the Expert Group) and a public administration (through the Commissioner at the Presidency level). WIRED magazine just published an article describing our story.

10:15 am–10:45 am
Coffee Break

10:45 am–12:15 pm
Paper Session 5
Session Chair: Kunal Talwar

Shengyuan Hu, Carnegie Mellon University
Title: Private Multi-Task Learning: Formulation and Applications to Federated Learning
Abstract: Many problems in machine learning rely on multi-task learning (MTL), in which the goal is to solve multiple related machine learning tasks simultaneously. MTL is particularly relevant for privacy-sensitive applications in areas such as healthcare, finance, and IoT computing, where sensitive data from multiple, varied sources are shared for the purpose of learning. In this work, we formalize notions of task-level privacy for MTL via joint differential privacy (JDP), a relaxation of differential privacy for mechanism design and distributed optimization. We then propose an algorithm for mean-regularized MTL, an objective commonly used for applications in personalized federated learning, subject to JDP. We analyze our objective and solver, providing certifiable guarantees on both privacy and utility. Empirically, our method allows for improved privacy/utility trade-offs relative to global baselines across common federated learning benchmarks

Christina Yu,
Cornell University
Title: Sequential Fair Allocation: Achieving the Optimal Envy-Efficiency Tradeoff Curve
Abstract: We consider the problem of dividing limited resources to individuals arriving over T rounds with a goal of achieving fairness across individuals. In general there may be multiple resources and multiple types of individuals with different utilities. A standard definition of `fairness’ requires an allocation to simultaneously satisfy envy-freeness and Pareto efficiency. However, in the online sequential setting, the social planner must decide on a current allocation before the downstream demand is realized, such that no policy can guarantee these desiderata simultaneously with probability 1, requiring a modified metric of measuring fairness for online policies. We show that in the online setting, the two desired properties (envy-freeness and efficiency) are in direct contention, in that any algorithm achieving additive counterfactual envy-freeness up to L_T necessarily suffers an efficiency loss of at least 1 / L_T. We complement this uncertainty principle with a simple algorithm, HopeGuardrail, which allocates resources based on an adaptive threshold policy and is able to achieve any fairness-efficiency point on this frontier. Our result is the first to provide guarantees for fair online resource allocation with high probability for multiple resource and multiple type settings. In simulation results, our algorithm provides allocations close to the optimal fair solution in hindsight, motivating its use in practical applications as the algorithm is able to adapt to any desired fairness efficiency trade-off.

Hedyeh Beyhaghi, Carnegie Mellon University
Title: On classification of strategic agents who can both game and improve
Abstract: In this work, we consider classification of agents who can both game and improve. For example, people wishing to get a loan may be able to take some actions that increase their perceived credit-worthiness and others that also increase their true credit-worthiness. A decision-maker would like to define a classification rule with few false-positives (does not give out many bad loans) while yielding many true positives (giving out many good loans), which includes encouraging agents to improve to become true positives if possible. We consider two models for this problem, a general discrete model and a linear model, and prove algorithmic, learning, and hardness results for each. For the general discrete model, we give an efficient algorithm for the problem of maximizing the number of true positives subject to no false positives, and show how to extend this to a partial-information learning setting. We also show hardness for the problem of maximizing the number of true positives subject to a nonzero bound on the number of false positives, and that this hardness holds even for a finite-point version of our linear model. We also show that maximizing the number of true positives subject to no false positive is NP-hard in our full linear model. We additionally provide an algorithm that determines whether there exists a linear classifier that classifies all agents accurately and causes all improvable agents to become qualified, and give additional results for low-dimensional data.

Keegan Harris, Carnegie Mellon University
Title: Bayesian Persuasion for Algorithmic Recourse
Abstract: When subjected to automated decision-making, decision subjects may strategically modify their observable features in ways they believe will maximize their chances of receiving a favorable decision. In many practical situations, the underlying assessment rule is deliberately kept secret to avoid gaming and maintain competitive advantage. The resulting opacity forces the decision subjects to rely on incomplete information when making strategic feature modifications. We capture such settings as a game of Bayesian persuasion, in which the decision maker offers a form of recourse to the decision subject by providing them with an action recommendation (or signal) to incentivize them to modify their features in desirable ways. We show that when using persuasion, both the decision maker and decision subject are never worse off in expectation, while the decision maker can be significantly better off. While the decision maker’s problem of finding the optimal Bayesian incentive-compatible (BIC) signaling policy takes the form of optimization over infinitely-many variables, we show that this optimization can be cast as a linear program over finitely-many regions of the space of possible assessment rules. While this reformulation simplifies the problem dramatically, solving the linear program requires reasoning about exponentially-many variables, even under relatively simple settings. Motivated by this observation, we provide a polynomial-time approximation scheme that recovers a near-optimal signaling policy. Finally, our numerical simulations on semi-synthetic data empirically illustrate the benefits of using persuasion in the algorithmic recourse setting.

12:15 pm–1:45 pm
Lunch Break

1:45–3:15 pm
Paper Session 6
Session Chair: Elisa Celis

Mark Bun, Boston University
Title: Controlling Privacy Loss in Sampling Schemes: An Analysis of Stratified and Cluster Sampling
Abstract: Sampling schemes are fundamental tools in statistics, survey design, and algorithm design. A fundamental result in differential privacy is that a differentially private mechanism run on a simple random sample of a population provides stronger privacy guarantees than the same algorithm run on the entire population. However, in practice, sampling designs are often more complex than the simple, data-independent sampling schemes that are addressed in prior work. In this work, we extend the study of privacy amplification results to more complex, data-dependent sampling schemes. We find that not only do these sampling schemes often fail to amplify privacy, they can actually result in privacy degradation. We analyze the privacy implications of the pervasive cluster sampling and stratified sampling paradigms, as well as provide some insight into the study of more general sampling designs.

Samson Zhou, Carnegie Mellon University
Title: Private Data Stream Analysis for Universal Symmetric Norm Estimation
Abstract: We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include L_p norms, k-support norms, top-k norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the “heavy” coordinates in important levels and releases approximate level sizes for the “light” coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits (1+alpha)-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.

Aloni Cohen, University of Chicago
Title: Attacks on Deidentification’s Defenses
Abstract: Quasi-identifier-based deidentification techniques (QI-deidentification) are widely used in practice, including k-anonymity, ?-diversity, and t-closeness. We present three new attacks on QI-deidentification: two theoretical attacks and one practical attack on a real dataset. In contrast to prior work, our theoretical attacks work even if every attribute is a quasi-identifier. Hence, they apply to k-anonymity, ?-diversity, t-closeness, and most other QI-deidentification techniques.
First, we introduce a new class of privacy attacks called downcoding attacks, and prove that every QI-deidentification scheme is vulnerable to downcoding attacks if it is minimal and hierarchical. Second, we convert the downcoding attacks into powerful predicate singling-out (PSO) attacks, which were recently proposed as a way to demonstrate that a privacy mechanism fails to legally anonymize under Europe’s General Data Protection Regulation. Third, we use LinkedIn.com to reidentify 3 students in a k-anonymized dataset published by EdX (and show thousands are potentially vulnerable), undermining EdX’s claimed compliance with the Family Educational Rights and Privacy Act.
The significance of this work is both scientific and political. Our theoretical attacks demonstrate that QI-deidentification may offer no protection even if every attribute is treated as a quasi-identifier. Our practical attack demonstrates that even deidentification experts acting in accordance with strict privacy regulations fail to prevent real-world reidentification. Together, they rebut a foundational tenet of QI-deidentification and challenge the actual arguments made to justify the continued use of k-anonymity and other QI-deidentification techniques.

Steven Wu,
Carnegie Mellon University
Title: Fully Adaptive Composition in Differential Privacy
Abstract: Composition is a key feature of differential privacy. Well-known advanced composition theorems allow one to query a private database quadratically more times than basic privacy composition would permit. However, these results require that the privacy parameters of all algorithms be fixed before interacting with the data. To address this, Rogers et al. introduced fully adaptive composition, wherein both algorithms and their privacy parameters can be selected adaptively. The authors introduce two probabilistic objects to measure privacy in adaptive composition: privacy filters, which provide differential privacy guarantees for composed interactions, and privacy odometers, time-uniform bounds on privacy loss. There are substantial gaps between advanced composition and existing filters and odometers. First, existing filters place stronger assumptions on the algorithms being composed. Second, these odometers and filters suffer from large constants, making them impractical. We construct filters that match the tightness of advanced composition, including constants, despite allowing for adaptively chosen privacy parameters. We also construct several general families of odometers. These odometers can match the tightness of advanced composition at an arbitrary, preselected point in time, or at all points in time simultaneously, up to a doubly-logarithmic factor. We obtain our results by leveraging recent advances in time-uniform martingale concentration. In sum, we show that fully adaptive privacy is obtainable at almost no loss, and conjecture that our results are essentially not improvable (even in constants) in general.

3:15 pm–3:45 pm
FORC Reception

3:45 pm–5:00 pm
Social Hour

Fast Point Transformer
2022-06-02 16:13 - 17:13 | MA
Abstract: The recent success of neural networks enables a better interpretation of 3D point clouds, but processing a large-scale 3D scene remains a challenging problem. Most current approaches divide a large-scale scene into small regions and combine the local predictions together. However, this scheme inevitably involves additional stages for pre- and post-processing and may also degrade the final output due to predictions in a local perspective. This talk introduces Fast Point Transformer that consists of a new lightweight self-attention layer. Our approach encodes continuous 3D coordinates, and the voxel hashing-based architecture boosts computational efficiency. The proposed method is demonstrated with 3D semantic segmentation and 3D detection. The accuracy of our approach is competitive to the best voxel-based method, and our network achieves 129 times faster inference time than the state-of-the-art, Point Transformer, with a reasonable accuracy trade-off in 3D semantic segmentation on S3DIS dataset.
Bio: Jaesik Park is an Assistant Professor at POSTECH. He received his Bachelor’s degree from Hanyang University in 2009, and he received his Master’s degree and Ph.D. degree from KAIST in 2011 and 2015, respectively. Before joining POSTECH, He worked at Intel as a research scientist, where he co-created the Open3D library. His research interests include image synthesis, scene understanding, and 3D reconstruction. He serves as a program committee at prestigious computer vision conferences, such as Area Chair for ICCV, CVPR, and ECCV.

Extinction and coexistence for reaction-diffusion systems on metric graphs
2022-05-26 09:00 - 10:00 | MA
Abstract: In spatial population genetics, it is important to understand the probability of extinction in multi-species interactions such as growing bacterial colonies, cancer tumor evolution and human migration. This is because extinction probabilities are instrumental in determining the probability of coexistence and the genealogies of populations. A key challenge is the complication due to spatial effect and different sources of stochasticity. In this talk, I will discuss about methods to compute the probability of extinction and other long-time behaviors for stochastic reaction-diffusion equations on metric graphs that flexibly parametrizes the underlying space. Based on recent joint work with Adrian Gonzalez-Casanova and Yifan (Johnny) Yang.

Oblique Lessons from the W Mass Measurement at CDF II
2022-05-25 10:30 - 12:00 | Hybrid
Speaker: Seth Koren (University of Chicago)
Title: Baryon Minus Lepton Number BF Theory for the Cosmological Lithium Problem
Abstract: The cosmological lithium problem—that the observed primordial abundance is lower than theoretical expectations by order one—is perhaps the most statistically significant anomaly of SM+ ΛCDM, and has resisted decades of attempts by cosmologists, nuclear physicists, and astronomers alike to root out systematics. We upgrade a discrete subgroup of the anomaly-free global symmetry of the SM to an infrared gauge symmetry, and UV complete this at a scale Λ as the familiar U(1)_{B-N_cL} Abelian Higgs theory. The early universe phase transition forms cosmic strings which are charged under the emergent higher-form symmetry of the baryon minus lepton BF theory. These topological defects catalyze interactions which turn N_g baryons into N_g leptons at strong scale rates in an analogue of the Callan-Rubakov effect, where N_g=3 is the number of SM generations. We write down a model in which baryon minus lepton strings superconduct bosonic global baryon plus lepton number currents and catalyze solely 3p^+ → 3e^+. We suggest that such cosmic strings have disintegrated O(1) of the lithium nuclei formed during Big Bang Nucleosynthesis and estimate the rate, with our benchmark model finding Λ ~ 10^8 GeV gives the right number density of strings.

2022 Summer Introduction to Mathematical Research
2022-05-21 09:00 - 2022-06-12 17:00 | Virtual
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.

The geometry of conditional independence models with hidden variables
2022-05-19 09:00 - 10:00 | MA
Abstract: Conditional independence (CI) is an important tool instatistical modeling, as, for example, it gives a statistical interpretation to graphical models. In general, given a list of dependencies among random variables, it is difficult to say which constraints are implied by them. Moreover, it is important to know what constraints on the random variables are caused by hidden variables. On the other hand, such constraints are corresponding to some determinantal conditions on the tensor of joint probabilities of the observed random variables. Hence, the inference question in statistics relates to understanding the algebraic and geometric properties of determinantal varieties such as their irreducible decompositions or determining their defining equations. I will explain some recent progress that arises by uncovering the link to point configurations in matroid theory and incidence geometry. This connection, in particular, leads to effective computational approaches for (1) giving a decomposition for each CI variety; (2) identifying each component in the decomposition as a matroid variety; (3) determining whether the variety has a real point or equivalently there is a statistical model satisfying a given collection of dependencies. The talk is based on joint works with Oliver Clarke, Kevin Grace, and Harshit Motwani.
The papers are available on arxiv: https://arxiv.org/pdf/2011.02450
and https://arxiv.org/pdf/2103.16550.pdf

Boundary conditions and LSM anomalies of conformal field theories in 1+1 dimensions
2022-05-18 16:00 - 17:30 | Virtual
Speaker: Linhao Li (ISSP, U Tokyo)
Title: Boundary conditions and LSM anomalies of conformal field theories in 1+1 dimensions
Abstract: In this talk, we will study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+1 dimensions. For a given CFT with a global symmetry, we consider symmetric gapping potentials which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we will systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries. As an application, we propose 1d LSM theorem involving magnetic space group symmetries on the lattice. The extended LSM theorems apply to systems with a broader class of spin interactions, such as Dzyaloshinskii-Moriya interactions and chiral three-spin interactions.

The Generalized Landau Paradigm (a review of generalized symmetries in condensed matter)
2022-05-18 16:00 - 17:30 | Virtual
Abstract: Recent advances in our understanding of symmetry in quantum many-body systems offer the possibility of a generalized Landau paradigm that encompasses all equilibrium phases of matter. This talk will be an elementary review of some of these developments, based on: https://arxiv.org/abs/2204.03045

Statistical Mechanics of Mutilated Sheets and Shells
2022-05-18 09:30 - 10:30 | MA
Speaker: David Nelson, Harvard University
Title: Statistical Mechanics of Mutilated Sheets and Shells
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure.

Hypergraph Matchings Avoiding Forbidden Submatchings
2022-05-17 09:30 - 10:30 | MA
Abstract: In 1973, Erdős conjectured the existence of high girth (n,3,2)-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős’ conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős’ conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth (n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n,q,r)-Steiner systems. If time permits, I will explain some of the other applications of our main results such as to rainbow matchings. This is joint work with Luke Postle.

SMaSH: Symposium for Mathematical Sciences at Harvard
2022-05-17 09:00 - 18:00 | Science and Engineering Complex (SEC), 150 Western Ave, Allston, MA 02134, MA
SMaSH: Symposium for Mathematical Sciences at Harvard
On Tuesday, May 17, 2022, from 9:00 am – 5:30 pm, the Harvard John A Paulson School of Engineering and Applied Sciences (SEAS) and the Harvard Center of Mathematical Sciences and Applications (CMSA) held a Symposium for Mathematical Sciences for the mathematical sciences community at Harvard.
Organizing Committee

Michael Brenner, Applied Mathematics (SEAS)
Michael Desai, Organismic and Evolutionary Biology (FAS)
Sam Gershman, Psychology (FAS)
Michael Hopkins, Mathematics (FAS)
Gary King, Government (FAS)
Peter Koellner, Philosophy (FAS)
Scott Kominers, Economics (FAS) & Entrepreneurial Management (HBS)
Xihong Lin, Biostatistics (HSPH) & Statistics (FAS)
Yue Lu, Electrical Engineering (SEAS)
Susan Murphy, Statistics (FAS) & Computer Science (SEAS)
Lisa Randall, Physics (SEAS)
Eugene Shakhnovich, Chemistry (FAS)
Salil Vadhan, Computer Science (SEAS)
Horng-Tzer Yau, Mathematics (FAS)

This event was held in-person at the Science and Engineering Complex (SEC) at 150 Western Ave, Allston, MA 02134, and streamed on Zoom.
Harvard graduate students and postdocs presented Poster Sessions.

Venue: Science and Engineering Complex (SEC)

Speakers

Anurag Anshu, Computer Science (SEAS)
Morgane Austern, Statistics (FAS)
Demba Ba, Electrical Engineering & Bioengineering (SEAS)
Michael Brenner, Applied Mathematics (SEAS)
Rui Duan, Biostatistics (HSPH)
Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Kosuke Imai, Government & Statistics (FAS)
Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Seth Neel, Technology & Operations Management (HBS)
Melanie Matchett Wood, Mathematics (FAS)

Schedule PDF
Schedule

9:00–9:30 am
Coffee and Breakfast
West Atrium (ground floor of the SEC)

9:30–10:30 am
Faculty Talks
Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumKosuke Imai, Government & Statistics (FAS): Use of Simulation Algorithms for Legislative Redistricting Analysis and EvaluationYannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS): The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization

10:30–11:00 am
Coffee Break
West Atrium (ground floor of the SEC)

11:00–12:00 pm
Faculty Talks
Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumSeth Neel, Technology & Operations Management (HBS): “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy LawDemba Ba, Electrical Engineering & Bioengineering (SEAS): Geometry, AI, and the Brain

12:00–1:00 pm
Lunch Break
Engineering Yard Tent

1:00–2:30 pm
Faculty Talks
Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMelanie Matchett Wood, Mathematics (FAS): Understanding distributions of algebraic structures through their momentsMorgane Austern, Statistics (FAS): Limit theorems for structured random objectsAnurag Anshu, Computer Science (SEAS): Operator-valued polynomial approximations and their use.

2:30–3:00 pm
Coffee Break
West Atrium (ground floor of the SEC)

3:00–4:30 pm
Faculty Talks
Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMichael Brenner, Applied Mathematics (SEAS): Towards living synthetic materialsRui Duan, Biostatistics (HSPH): Federated and transfer learning for healthcare data integrationSham M. Kakade, Computer Science (SEAS) & Statistics (FAS): What is the Statistical Complexity of Reinforcement Learning?

4:30–5:30 pm
Reception with Jazz musicians
& Poster Session
Engineering Yard Tent

Faculty Talks

Speaker
Title / Abstract / Bio

Anurag Anshu, Computer Science (SEAS)
Title: Operator-valued polynomial approximations and their use.
Abstract: Approximation of complicated functions with low degree polynomials is an indispensable tool in mathematics. This becomes particularly relevant in computer science, where the complexity of interesting functions is often captured by the degree of the approximating polynomials. This talk concerns the approximation of operator-valued functions (such as the exponential of a hermitian matrix, or the intersection of two projectors) with low-degree operator-valued polynomials. We will highlight the challenges that arise in achieving similarly good approximations as real-valued functions, as well as recent methods to overcome them. We will discuss applications to the ground states in physics and quantum complexity theory: correlation lengths, area laws and concentration bounds.
Bio: Anurag Anshu is an Assistant Professor of computer science at Harvard University. He spends a lot of time exploring the rich structure of quantum many-body systems from the viewpoint of quantum complexity theory, quantum learning theory and quantum information theory. He held postdoctoral positions at University of California, Berkeley and University of Waterloo and received his PhD from National University of Singapore, focusing on quantum communication complexity.

Morgane Austern, Statistics (FAS)
Title: Limit theorems for structured random objects
Abstract: Statistical inference relies on numerous tools from probability theory to study the properties of estimators. Some of the most central ones are the central limit theorem and the free central limit theorem. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting. We consider distributionally “structured’ and dependent random object i.e random objects whose distribution is invariant under the action of an amenable group. We show, under mild moment and mixing conditions, a series of universal second and third order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network and information theory.
Bio: Morgane Austern is an assistant professor in the Statistics Department of Harvard University. Broadly, she is interested in developing probability tools for modern machine learning and in establishing the properties of learning algorithms in structured and dependent data contexts. She graduated with a PhD in statistics from Columbia University in 2019 where she worked in collaboration with Peter Orbanz and Arian Maleki on limit theorems for dependent and structured data. She was a postdoctoral researcher at Microsoft Research New England from 2019 to 2021.

Demba Ba, Electrical Engineering & Bioengineering (SEAS)
Title: Geometry, AI, and the Brain
Abstract: A large body of experiments suggests that neural computations reflect, in some sense, the geometry of “the world”. How do artificial and neural systems learn representations of “the world” that reflect its geometry? How, for instance, do we, as humans, learn representations of objects, e.g. fruits, that reflect the geometry of object space? Developing artificial systems that can capture/understand the geometry of the data they process may enable them to learn representations useful in many different contexts and tasks. My talk will describe an artificial neural-network architecture that, starting from a simple union-of-manifold model of data comprising objects from different categories, mimics some aspects of how primates learn, organize, and retrieve concepts, in a manner that respects the geometry of object space.
Bio: Demba Ba serves as an Associate Professor of electrical engineering and bioengineering in Harvard University’s School of Engineering and Applied Sciences, where he directs the CRISP group. Recently, he has taken a keen interest in the connection between artificial neural networks and sparse signal processing. His group leverages this connection to solve data-driven unsupervised learning problems in neuroscience, to understand the principles of hierarchical representations of sensory signals in the brain, and to develop explainable AI. In 2016, he received a Research Fellowship in Neuroscience from the Alfred P. Sloan Foundation. In 2021, Harvard’s Faculty of Arts and Sciences awarded him the Roslyn Abramson award for outstanding undergraduate teaching.

Michael Brenner, Applied Mathematics (SEAS)
Title: Towards living synthetic materials
Abstract: Biological materials are much more complicated and functional than synthetic ones. Over the past several years we have been trying to figure out why. A sensible hypothesis is that biological materials are programmable. But we are very far from being able to program materials we create with this level of sophistication. I will discuss our (largely unsuccessful) efforts to bridge this gap, though as of today I’m somewhat optimistic that we are arriving at a set of theoretical models that is rich enough to produce relevant emergent behavior.
Bio: I’ve been at Harvard for a long time. My favorite part of Harvard is the students.

Rui Duan, Biostatistics (HSPH)
Title: Federated and transfer learning for healthcare data integration
Abstract: The growth of availability and variety of healthcare data sources has provided unique opportunities for data integration and evidence synthesis, which can potentially accelerate knowledge discovery and improve clinical decision-making. However, many practical and technical challenges, such as data privacy, high dimensionality, and heterogeneity across different datasets, remain to be addressed. In this talk, I will introduce several methods for the effective and efficient integration of multiple healthcare datasets in order to train statistical or machine learning models with improved generalizability and transferability. Specifically, we develop communication-efficient federated learning algorithms for jointly analyzing multiple datasets without the need of sharing patient-level data, as well as transfer learning approaches that leverage shared knowledge learned across multiple datasets to improve the performance of statistical models in target populations of interest. We will discuss both the theoretical properties and examples of implementation of our methods in real-world research networks and data consortia.
Bio: Rui Duan is an Assistant Professor of Biostatistics at the Harvard T.H. Chan School of Public Health. She received her Ph.D. in Biostatistics in May 2020 from the University of Pennsylvania. Her research interests focus on developing statistical, machine learning, and informatics tools for (1) efficient data integration in biomedical research, (2) understanding and accounting for the heterogeneity of biomedical data, and improving the generalizability and transferability of models across populations (3) advancing precision medicine research on rare diseases and underrepresented populations.

Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Title: The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
Abstract: We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of n additive bidders whose values for m heterogeneous items are drawn independently. For any such instance and any ε > 0, we show that it is possible to learn an ε-Bayesian Incentive Compatible auction whose expected revenue is within ε of the optimal ε-BIC auction from only polynomially many samples.
Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.
Joint work with S. Matthew Weinberg
Bio: Yannai A. Gonczarowski is an Assistant Professor of Economics and of Computer Science at Harvard University—the first faculty member at Harvard to have been appointed to both of these departments. Interested in both economic theory and theoretical computer science, Yannai explores computer-science-inspired economics: he harnesses approaches, aesthetics, and techniques traditionally originating in computer science to derive economically meaningful insights. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. Yannai’s doctoral dissertation was recognized with several awards, including the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. For the design and implementation of the National Matching System for Gap-Year Programs in Israel, he was awarded the Best Paper Award at MATCH-UP’19 and the inaugural INFORMS AMD Michael H. Rothkopf Junior Researcher Paper Prize (first place) for 2020. Yannai is also the recipient of the inaugural ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.

Kosuke Imai, Government & Statistics (FAS)
Title: Use of Simulation Algorithms for Legislative Redistricting Analysis and Evaluation
Abstract: After the 2020 Census, many states have been redrawing the boundaries of Congressional and state legislative districts. To evaluate the partisan and racial bias of redistricting plans, scholars have developed Monte Carlo simulation algorithms. The idea is to generate a representative sample of redistricting plans under a specified set of criteria and conduct a statistical hypothesis test by comparing a proposed plan with these simulated plans. I will give a brief overview of these redistricting simulation algorithms and discuss how they are used in real-world court cases.
Bio: Kosuke Imai is Professor in the Department of Government and Department of Statistics at Harvard University. Before moving to Harvard in 2018, Imai taught at Princeton University for 15 years where he was the founding director of the Program in Statistics and Machine Learning. Imai specializes in the development of statistical methods and machine learning algorithms and their applications to social science research. His areas of expertise include causal inference, computational social science, program evaluation, and survey methodology.

Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Title: What is the Statistical Complexity of Reinforcement Learning?
Abstract: This talk will highlight much of the recent progress on the following fundamental question in the theory of reinforcement learning: what (representational or structural) conditions govern our ability to generalize and avoid the curse of dimensionality? With regards to supervised learning, these questions are reasonably well understood, both practically and theoretically: practically, we have overwhelming evidence on the value of representational learning (say through modern deep networks) as a means for sample efficient learning, and, theoretically, there are well-known complexity measures (e.g. the VC dimension and Rademacher complexity) that govern the statistical complexity of learning. Providing an analogous theory for reinforcement learning is far more challenging, where even characterizing structural conditions which support sample efficient generalization has been far less well understood, until recently.
This talk will survey recent advances towards characterizing when generalization is possible in RL, focusing on both necessary and sufficient conditions. In particular, we will introduce a new complexity measure, the Decision-Estimation Coefficient, that is proven to be necessary (and, essentially, sufficient) for sample-efficient interactive learning.
Bio: Sham Kakade is a professor at Harvard University and a co-director of the Kempner Institute for the Study of Artificial and Natural Intelligence. He works on the mathematical foundations of machine learning and AI. Sham’s thesis helped lay the statistical foundations of reinforcement learning. With his collaborators, his additional contributions include foundational results on: policy gradient methods in reinforcement learning; regret bounds for linear bandit and Gaussian process bandit models; the tensor and spectral methods for latent variable models; and a number of convergence analyses for convex and non-convex algorithms. He is the recipient of the ICML Test of Time Award, the IBM Pat Goldberg best paper award, and INFORMS Revenue Management and Pricing Prize. He has been program chair for COLT 2011.
Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He completed his Ph.D. with Peter Dayan in computational neuroscience at the Gatsby Computational Neuroscience Unit. He was a postdoc with Michael Kearns at the University of Pennsylvania.

Seth Neel, Technology & Operations Management (HBS)
Title: “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy Law
Abstract: Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks. In a series of papers we help formalize a rigorous notion of data-deletion and propose algorithms to efficiently delete user data from trained models with provable guarantees in both convex and non-convex settings.
Bio: Seth Neel is a first-year Assistant Professor in the TOM Unit at Harvard Business School, and Co-PI of the SAFR ML Lab in the D3 Institute, which develops methodology to incorporate privacy and fairness guarantees into techniques for machine learning and data analysis, while balancing other critical considerations like accuracy, efficiency, and interpretability. He obtained his Ph.D. from the University of Pennsylvania in 2020 where he was an NSF graduate fellow. His work has focused primarily on differential privacy, notions of fairness in a variety of machine learning settings, and adaptive data analysis.

Melanie Matchett Wood, Mathematics (FAS)
Title: Understanding distributions of algebraic structures through their moments
Abstract: A classical tool of probability and analysis is to use the moments (mean, variance, etc.) of a distribution to recognize an unknown distribution of real numbers. In recent work, we are interested in distributions of algebraic structures that can’t be captured in a single number. We will explain one example, the fundamental group, that captures something about the shapes of possibly complicated or high dimensional spaces. We are developing a new theory of the moment problem for random algebraic structures which helps to to identify distributions of such, such as fundamental groups of random three dimensional spaces. This talk is based partly on joint work with Will Sawin.
Bio: Melanie Matchett Wood is a professor of mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Her work spans number theory, algebraic geometry, algebraic topology, additive combinatorics, and probability. Wood has been awarded a CAREER grant, a Sloan Research Fellowship, a Packard Fellowship for Science and Engineering, and the AWM-Microsoft Research Prize in Algebra and Number Theory, and she is a Fellow of the American Mathematical Society. In 2021, Wood received the National Science Foundation’s Alan T. Waterman Award, the nation’s highest honor for early-career scientists and engineers.

Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
2022-05-13 09:30 - 11:00 | MA
Member Seminar
Speaker: Juven Wang
Title: Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
Abstract: ‘t Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a d-dim spacetime are known to be classified by a d+1-dim cobordism group TPd+1(G), whose group generator is a d+1-dim cobordism invariant written as a d+1-dim invertible topological field theory. Deformation class of QFT is recently proposed to be specified by its symmetry G and a d+1-dim invertible topological field theory. Seemly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. We ask which deformation class controls the 4d ungauged or gauged (SU(3)×SU(2)×U(1))/Zq Standard Model (SM) for q=1,2,3,6 with a continuous or discrete (B−L) symmetry and with also a compatible discrete baryon plus lepton Z_{2Nf} B+L symmetry. (The Z_{2Nf} B+L is discrete due to the ABJ anomaly under the BPST instanton.) We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM’s deformation class. While many Grand Unified Theories violating the discrete B+L symmetry suffer from the proton decay, the SM and some versions of Ultra Unification (constrained by Z_{16} class global anomaly that replaces sterile neutrinos with new exotic gapped/gapless topological or conformal sectors) can have a stable proton. Dictated by a Z_2 class global mixed gauge-gravitational anomaly, there can be a gapless deconfined quantum critical region between Georgi-Glashow and Pati-Salam models — the Standard Model and beyond occur as neighbor phases. We will also comment on a new mechanism to give the neutrino mass via topological field theories and topological defects. Work based on arXiv:2112.14765, arXiv:2204.08393, arXiv:2202.13498 and references therein.

Geometric Models for Sets of Probability Measures
2022-05-12 15:38 - 17:38 | MA
Abstract: Many statistical and computational tasks boil down to comparing probability measures expressed as density functions, clouds of data points, or generative models. In this setting, we often are unable to match individual data points but rather need to deduce relationships between entire weighted and unweighted point sets. In this talk, I will summarize our team’s recent efforts to apply geometric techniques to problems in this space, using tools from optimal transport and spectral geometry. Motivated by applications in dataset comparison, time series analysis, and robust learning, our work reveals how to apply geometric reasoning to data expressed as probability measures without sacrificing computational efficiency.

Oblique Lessons from the W Mass Measurement at CDF II
2022-05-12 10:30 - 12:00 | MA
Abstract: The CDF collaboration recently reported a new precise measurement of the W boson mass MW with a central value significantly larger than the SM prediction. We explore the effects of including this new measurement on a fit of the Standard Model (SM) to electroweak precision data. We characterize the tension of this new measurement with the SM and explore potential beyond the SM phenomena within the electroweak sector in terms of the oblique parameters S, T and U. We show that the large MW value can be accommodated in the fit by a large, nonzero value of U, which is difficult to construct in explicit models. Assuming U = 0, the electroweak fit strongly prefers large, positive values of T. Finally, we study how the preferred values of the oblique parameters may be generated in the context of models affecting the electroweak sector at tree- and loop-level. In particular, we demonstrate that the preferred values of T and S can be generated with a real SU(2)L triplet scalar, the humble swino, which can be heavy enough to evade current collider constraints, or by (multiple) species of a singlet-doublet fermion pair. We highlight challenges in constructing other simple models, such as a dark photon, for explaining a large MW value, and several directions for further study.

Cosmology from the vacuum
2022-05-11 10:30 - 12:00 | MA
Abstract: We are familiar with the idea that quantum gravity in AdS can holographically emerge from complex patterns of entanglement, but can the physics of big bang cosmology emerge from a quantum many-body system? In this talk I will argue that standard tools of holography can be used to describe fully non-perturbative microscopic models of cosmology in which a period of accelerated expansion may result from the positive potential energy of time-dependent scalar fields evolving towards a region with negative potential. In these models, the fundamental cosmological constant is negative, and the universe eventually recollapses in a time-reversal symmetric way. The microscopic description naturally selects a special state for the cosmology. In this framework, physics in the cosmological spacetime is dual to the vacuum physics in a static planar asymptotically AdS Lorentzian wormhole spacetime, in the sense that the background spacetimes and observables are related by analytic continuation. The dual spacetime is weakly curved everywhere, so any cosmological observables can be computed in the dual picture via effective field theory without detailed knowledge of the UV completion or the physics near the big bang. Based on 2203.11220 with S. Antonini, P. Simidzija, and M. Van Raamsdonk.

Inflation and light Dark Matter constraints from the Swampland
2022-05-09 13:00 - 14:00 | Virtual
Abstract: I will explore the interplay between Swampland conjectures and models of inflation and light Dark Matter. To that end, I will briefly review the weak gravity conjecture (WGC) and the related Festina Lente (FL) bound. These have implications for light darkly and milli-charged particles and can disfavor a large portion of parameter space. The FL bound also implies strong restrictions on the field content of our universe during inflation and presents an opportunity for inflationary model building. At the same time, it rules out some popular models like chromo-natural inflation and gauge-flation. Finally, I will review another Swampland conjecture related to Stückelberg photon masses and discuss its implications for astro-particle physics.

Conference in Memory of Professor Masatake Kuranishi
2022-05-09 09:00 - 2022-05-12 12:30 | Science and Engineering Complex (SEC), 150 Western Ave, Allston, MA 02134, MA
On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Videos are available on the conference playlist.

Speakers:
Charles Fefferman (Princeton University)
Teng Fei (Rutgers University)
Robert Friedman (Columbia University)
Kenji Fukaya (Simons Center, Stony Brook)
Akito Futaki (Tsinghua University)
Victor Guillemin (Massachusetts Institute of Technology)
Nigel Hitchin (Oxford University)
Blaine Lawson (Stony Brook University)
Yu-Shen Lin (Boston University)
Melissa C.C. Liu (Columbia University)
Takeo Ohsawa (Nagoya University)
Duong H. Phong (Columbia University)
Sebastien Picard (University of British Columbia)
Paul Seidel (Massachusetts Institute of Technology)
Gabor Szekelyhidi (University of Notre Dame)
Claire Voisin (Institut de Mathematiques, Jussieu, France)
Shing-Tung Yau (Harvard University)

Schedule (download pdf)

Monday, May 9, 2022

8:15 am
Light breakfast & coffee/tea

8:45–9:00 am
Opening Remarks

9:00–10:00 am
Kenji Fukaya
Title: Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Video

10:15–11:15 am
Nigel Hitchin (Zoom)
Title: Deformations: A personal perspective
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Video

11:30–12:30 pm
H. Blaine Lawson
Title: Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Gabor Szekelyhidi
Title: Singularities along the Lagrangian mean curvature flow.
Abstract: We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Video

3:30–4:00 pm
Coffee Break

4:00–5:00 pm
Takeo Ohsawa
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Video

Tuesday, May 10, 2022

8:15 am
Light breakfast & coffee/tea

9:00–10:00 am
Charles Fefferman (Zoom)
Title: Interpolation of Data by Smooth Functions
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Video

10:15–11:15 am
Claire Voisin
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Video

11:30– 2:30 pm
Victor Guillemin
Title: Semi-Classical Functions of Isotropic Type
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Teng Fei
Title: Symplectic deformations and the Type IIA flow
Abstract: The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Video

Speakers Banquet

Wednesday, May 11, 2022

8:15 am
Light breakfast & coffee/tea

9:00–10:00 am
Shing-Tung Yau (Zoom)
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Video

10:15–11:15 am
Duong H. Phong
Title: $L^infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^infty$ estimates are particularly important as they are also needed for other estimates. The key $L^infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Video

11:30–2:30 pm
Paul Seidel
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Video

12:30–2:30 pm
Lunch Break

2:30–3:30 pm
Melissa C.C. Liu
Title: Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Video

3:30–4:00 pm
Coffee Break

4:00–5:00 pm
Sebastien Picard
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Video

Thursday, May 12, 2022

8:15 am
Light breakfast & coffee/tea

9:00 am–10:00 am
Akito Futaki (Zoom)
Title: Transverse coupled Kähler-Einstein metrics and volume minimization

Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video

10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Video

11:30 am – 12:30 pm
Robert Friedman
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
Video

12:30 pm
Concluding Remarks

2022 NSF FRG Workshop on Discrete Shapes
2022-05-06 10:00 - 2022-05-08 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On May 6–8, 2022, the CMSA hosted a second NSF FRG Workshop.
This project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology), Computer Science (Scientific Computing and Complexity Theory), and domain sciences, from Data Sciences to Computational Biology.
Scientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time, however, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing, searching, simulating, analyzing, and comparing shapes. These methods and tools can be applied in a wide range of fields, including computer vision, biological imaging, brain mapping, target recognition, and satellite image analysis.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.

Workshop on Discrete Shapes
May 6–8, 2022
Organizers:

David Glickenstein (University of Arizona)
Joel Hass (University of California, Davis)
Patrice Koehl (University of California, Davis)
Feng Luo (Rutgers University, New Brunswick)
Maria Trnkova (University of California, Davis)
Shing-Tung Yau (Harvard)

Speakers:

Miri Ben-Chen (Technion)
Alexander Bobenko (TU Berlin)
John Bowers (James Madison)
Steven Gortler (Harvard)
David Gu (Stony Brook)
Anil Hirani (UIUC)
Yanwen Luo (Rutgers)
Peter Schroeder (Caltech)
Justin Solomon (MIT)
Tianqi Wu (Clark University)

Contributed Talk Speakers:

Oded Stein (MIT)
Bohan Zhou (Dartmouth)

Schedule
Schedule (PDF)
Friday, May 6, 2022

10:00–10:05 am

Welcome Opening

10:05–10:55 am
Anil N. Hirani
Title: Discrete vector bundles with connection
Abstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path, and is the local obstruction to a trivialization. For a bundle with metric, metric compatibility of the discrete connection is equivalent to a Leibniz rule. Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection, and curvature obstructs trivializability. In this talk I will focus on just the first part, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple, Inc.).

11:10–12:00 pm
David Gu
Title: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory
Abstract: Surface quadrilateral meshing plays an important role in many fields. For example, in CAD (computer-aided design), all shapes are represented as Spline surfaces, which requires structured quad-meshing; in CAE (computer-aided engineering), the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes.
In this talk, we introduce our recent discovery: the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety.
This leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities, the metric induced by the quad-mesh can be computed using the discrete Yambe flow, and the meromorphic quartic differential can be constructed, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.

12:00–2:00 pm
Lunch Break

2:00–2:50 pm
Justin Solomon
Title: Geometry Processing with Volumes
Abstract: Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes. Drawing inspiration from simulation, medical imaging, and other application domains, however, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids. In this talk, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes. Specifically, I will cover our recent work on hexahedral remeshing via cuboid decomposition, volumetric correspondence, and minimal surface computation via geometric measure theory.

3:00–3:20 pm
Oded Stein
Title: Optimization for flip-free parametrization
Abstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem, these energies are often nonconvex and unbounded, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).

3:20–4:00 pm
Break

4:00–4:50 pm
John Bowers
Title: Koebe-Andre’ev-Thurston Packings via Flow
Abstract: Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles.
The full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings, which is then used to show that all convex circle polyhedra are infinitesimally rigid, a result of independent interest.

5:00–5:30 pm
Movies
“conform!” & ”Koebe polyhedra”

Saturday, May 7, 2022

9:30–10:20 am
Alexander Bobenko
Title: The Bonnet problem: Is a surface characterized by its metric and curvatures?
Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In particular, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas

10:20–11:00 am
Break

11:00–11:50 am
Miri Ben Chen
Title: Surface Multigrid via Intrinsic Prolongation
Abstract: The solution of a linear system is a required ingredient in many geometry processing applications, and multigrid methods are among the most efficient solution techniques. However, due to the unstructured nature of triangle meshes, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve, and leads to order of magnitude time-efficiency improvement compared to direct solvers.

12:00–2:00 pm
Lunch Break

2:00–2:50 pm
Steven Gortler
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk, I will review what is known about this question.

3:00–3:20 pm
Bohan Zhou
Title: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
Abstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation, it provides a meaningful functional in the image science like image and shape registrations. However, exact computation through LP or PDE is in general not practical in large scale, while the popular entropy-regularized method introduces additional diffusion noise, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger, A Fast Approach to Optimal Transport: the back-and-forth method, Numerische Mathematik, 2020], solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno, we extend from two marginals to multiple marginals, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result, the new proposed method can be used in many fields those require sharp boundaries. If time allows, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries, with applications on sea ice prediction.

3:20–4:00 pm
Break

4:00–4:50 pm
Peter Schroeder
Title: Constrained Willmore Surfaces
Abstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin).

5:00–5:50 pm

Problems and Application discussions

Sunday, May 8, 2022

9:00–9:50 am
Tianqi Wu
Title: Convergence of discrete uniformizations
Abstract: The theory of discrete conformality, based on the notion of vertex scaling, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface, then the related discrete uniformization will converge to the smooth uniformization, with an error bounded linearly by the size of the triangles in the mesh.

10:10–11:00 am
Yanwen Luo
Title: Recent Progress in Spaces of Geodesic Triangulations of Surfaces

Abstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.

11:10–12:00 pm

Problems and Application discussions

12:00–1:00 pm
Movie
“The Discrete Charm of Geometry”

Qianfang: a type-safe and data-driven healthcare system starting from Traditional Chinese Medicine
2022-05-05 15:36 - 17:36 | MA
Abstract: Although everyone talks about AI + healthcare, many people were unaware of the fact that there are two possible outcomes of the collaboration, due to the inherent dissimilarity between the two giant subjects. The first possibility is healthcare-leads, and AI is for building new tools to make steps in healthcare easier, better, more effective or more accurate. The other possibility is AI-leads, and therefore the protocols of healthcare can be redesigned or redefined to make sure that the whole infrastructure and pipelines are ideal for running AI algorithms.
Our system Qianfang belongs to the second category. We have designed a new kind of clinic for the doctors and patients, so that it will be able to collect high quality data for AI algorithms. Interestingly, the clinic is based on Traditional Chinese Medicine (TCM) instead of modern medicine, because we believe that TCM is more suitable for AI algorithms as the starting point.
In this talk, I will elaborate on how we convert TCM knowledge into a modern type-safe large-scale system, the mini-language that we have designed for the doctors and patients, the interpretability of AI decisions, and our feedback loop for collecting data.
Our project is still on-going, not finished yet.Bio: Yang Yuan is now an assistant professor at IIIS, Tsinghua. He finished his undergraduate study at Peking University in 2012. Afterwards, he received his PhD at Cornell University in 2018, advised by Professor Robert Kleinberg. During his PhD, he was a visiting student at MIT/Microsoft New England (2014-2015) and Princeton University (2016 Fall). Before joining Tsinghua, he spent one year at MIT Institute for Foundations of Data Science (MIFODS) as a postdoc researcher. He now works on AI+Healthcare, AI Interpretability and AI system.

The threshold for stacked triangulations
2022-05-03 09:30 - 10:30 | Hybrid
Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y subseteq binom{[n]}3$, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle.
Suppose that every triangle is initially infected independently with probability p=p(n), what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime?
This notion of percolation can be viewed as a simplification of simple-connectivity. Namely, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces.
We ask: for which $p$ does the random simplicial complex Y_2(n,p) contain, for every triple $xyz$, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n].
We consider this problem in every dimension d>=2, and our main result identifies a sharp probability threshold for percolation, showing it is asymptotically (c_d*n)^(-1/d), where c_d is the growth rate of the Fuss–Catalan numbers of order d.
The proof hinges on a second moment argument in the supercritical regime, and on Kalai’s algebraic shifting in the subcritical regime.
Joint work with Eyal Lubetzky.

General Relativity Workshop
2022-05-02 09:00 - 2022-05-05 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Workshop on scalar curvature, minimal surfaces, and initial data sets
Dates: May 2–5, 2022
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138 and via Zoom webinar.
Advanced registration for in-person components is required.
Organizers: Dan Lee (CMSA/CUNY), Martin Lesourd (CMSA/BHI), and Lan-Hsuan Huang (University of Connecticut).
Speakers:

Zhongshan An, University of Connecticut
Paula Burkhardt-Guim, NYU
Hyun Chul Jang, University of Miami
Chao Li, NYU
Christos Mantoulidis, Rice University
Robin Neumayer, Carnegie Mellon University
Andre Neves, University of Chicago
Tristan Ozuch, MIT
Annachiara Piubello, University of Miami
Antoine Song, UC Berkeley
Tin-Yau Tsang, UC Irvine
Ryan Unger, Princeton
Zhizhang Xie, Texas A & M
Xin Zhou, Cornell University
Jonathan Zhu, Princeton University

Schedule
Monday, May 2, 2022

9:30–10:30 am
Hyun Chul Jang
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.

10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.

LUNCH BREAK

1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.

2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.

TEA BREAK

4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.

Tuesday, May 3, 2022

9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.

10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.

LUNCH BREAK

1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.

2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.

TEA BREAK

4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.

Speakers Banquet

Wednesday, May 4, 2022

9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.

10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.

LUNCH BREAK

1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.

2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.

TEA BREAK

4:10–5:10 pm
Xin Zhou
(Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).

May 5, 2022

9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.

10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.

11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.

LUNCH BREAK

1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.

Machine Learning the Gravity Equation for International Trade
2022-04-29 09:30 - 11:00 | Virtual
Member Seminar
Speaker: Sergiy Verstyuk
Title: Machine Learning the Gravity Equation for International Trade
Abstract: We will go through modern deep learning methods and existing approaches to their interpretation. Next, I will describe a graph neural network framework. You will also be introduced to an economic analog of gravity. Finally, we will see how these tools can help understand observed trade flows between 181 countries over 68 years. [Joint work with Michael R. Douglas.]

A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter
2022-04-28 15:35 - 16:35 | MA
Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.

Intersection number and systole on hyperbolic surfaces
2022-04-28 15:33 - 17:33 | MA
Abstract: Let X be a compact hyperbolic surface. We can see that there is a constant C(X) such that the intersection number of the closed geodesics is leq C(X) times the product of their lengths. Consider the optimum constant C(X). In this talk, we describe its asymptotic behavior in terms of systole, length of the shortest closed geodesic on X.

Building active nematic and active polar liquids out of biological machines
2022-04-28 13:00 - 14:30 | Virtual
Speaker: Guillaume Duclos (Brandeis)

Title: Building active nematic and active polar liquids out of biological machines
Abstract: Active matter describes out-of-equilibrium materials composed of motile building blocks that convert free energy into mechanical work. The continuous input of energy at the particle scale liberates these systems from the constraints of thermodynamic equilibrium, leading to emergent collective behaviors not found in passive materials. In this talk, I will describe our recent efforts to build simple active systems composed of purified proteins and identify generic emergent behaviors in active systems. I will first discuss two distinct activity-driven instabilities in suspensions of microtubules and molecular motors. Second, I will describe a new model system for polar fluid whose collective dynamics are driven by the non-equilibrium turnover of actin filaments. Our results illustrate how biomimetic materials can serve as a platform for studying non-equilibrium statistical mechanics, as well as shine light on the physical mechanisms that regulate self-organization in living matter.

Video (Youtube)

Aspects of 4d supersymmetric dynamics and geometry
2022-04-28 10:30 - 12:00 | Virtual
Abstract: We will overview the program of geometrically engineering four dimensional supersymmetric QFTs as compactifications of six dimensional SCFTs. In particular we will discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be better understood in such geometric constructions.
