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1
• Conference

##### Conference on Geometry and Statistics

9:00 am-5:30 pm
03/01/2023-03/01/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.

Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138

This conference will be held in person. Directions and Recommended Lodging

Registration is required.

Register here to attend in-person.

Organizing Committee:
Stephan Huckemann (Georg-August-Universität Göttingen)
Ezra Miller (Duke University)
Zhigang Yao (Harvard CMSA and Committee Chair)

Horng-Tzer Yau (Harvard CMSA)
Shing-Tung Yau (Harvard CMSA)

Speakers:

• Tamara Broderick (MIT)
• David Donoho (Stanford)
• Ian Dryden (Florida International University in Miami)
• David Dunson (Duke)
• Charles Fefferman (Princeton)
• Stefanie Jegelka (MIT)
• Sebastian Kurtek (OSU)
• Lizhen Lin (Notre Dame)
• Steve Marron (U North Carolina)
• Ezra Miller (Duke)
• Hans-Georg Mueller (UC Davis)
• Nicolai Reshetikhin (UC Berkeley)
• Wolfgang Polonik (UC Davis)
• Amit Singer (Princeton)
• Zhigang Yao (Harvard CMSA)
• Bin Yu (Berkeley)

Moderator: Michael Simkin (Harvard CMSA)

#### SCHEDULE

Monday, Feb. 27, 2023 (Eastern Time)

 8:30 am Breakfast 8:45–8:55 am Zhigang Yao Welcome Remarks 8:55–9:00 am Shing-Tung Yau* Remarks Morning Session Chair: Zhigang Yao 9:00–10:00 am David Donoho Title: ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated NoiseAbstract: Truncation of the singular value decomposition is a true scientific workhorse. But where to Truncate?For 55 years the answer, for many scientists, has been to eyeball the scree plot, an approach which still generates hundreds of papers per year.I will describe ScreeNOT, a mathematically solid alternative deriving from the many advances in Random Matrix Theory over those 55 years. Assuming a model of low-rank signal plus possibly correlated noise, and adopting an asymptotic viewpoint with number of rows proportional to the number of columns, we show that ScreeNOT has a surprising oracle property.It typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance – i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure.The talk is based on joint work with Matan Gavish and Elad Romanov, Hebrew University. 10:00–10:10 am Break 10:10–11:10 am Steve Marron Title: Modes of Variation in Non-Euclidean SpacesAbstract: Modes of Variation provide an intuitive means of understanding variation in populations, especially in the case of data objects that naturally lie in non-Euclidean spaces. A variety of useful approaches to finding useful modes of variation are considered in several non-Euclidean contexts, including shapes as data objects, vectors of directional data, amplitude and phase variation and compositional data. 11:10–11:20 am Break 11:20 am–12:20 pm Zhigang Yao Title: Manifold fitting: an invitation to statisticsAbstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y \subset \mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $\widehat{M} \subset \mathbb{R}^D$ to approximate $Y$, and how well can $\widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference:https://arxiv.org/abs/1909.10228 12:20–1:50 pm 12:20 pm Group Photofollowed by Lunch Afternoon Session Chair: Stephan Huckemann 1:50–2:50 pm Bin Yu* Title: Interpreting Deep Neural Networks towards TrustworthinessAbstract: Recent deep learning models have achieved impressive predictive performance by learning complex functions of many variables, often at the cost of interpretability. This lecture first defines interpretable machine learning in general and introduces the agglomerative contextual decomposition (ACD) method to interpret neural networks. Extending ACD to the scientifically meaningful frequency domain, an adaptive wavelet distillation (AWD) interpretation method is developed. AWD is shown to be both outperforming deep neural networks and interpretable in two prediction problems from cosmology and cell biology. Finally, a quality-controlled data science life cycle is advocated for building any model for trustworthy interpretation and introduce a Predictability Computability Stability (PCS) framework for such a data science life cycle. 2:50–3:00 pm Break 3:00-4:00 pm Hans-Georg Mueller Title: Exploration of Random Objects with Depth Profiles and Fréchet RegressionAbstract: Random objects, i.e., random variables that take values in a separable metric space, pose many challenges for statistical analysis, as vector operations are not available in general metric spaces. Examples include random variables that take values in the space of distributions, covariance matrices or surfaces, graph Laplacians to represent networks, trees and in other spaces. The increasing prevalence of samples of random objects has stimulated the development of metric statistics, an emerging collection of statistical tools to characterize, infer and relate samples of random objects. Recent developments include depth profiles, which are useful for the exploration of random objects. The depth profile for any given object is the distribution of distances to all other objects (with P. Dubey, Y. Chen 2022).These distributions can then be subjected to statistical analysis. Their mutual transports lead to notions of transport ranks, quantiles and centrality. Another useful tool is global or local Fréchet regression (with A. Petersen 2019) where random objects are responses and scalars or vectors are predictors and one aims at modeling conditional Fréchet means. Recent theoretical advances for local Fréchet regression provide a basis for object time warping (with Y. Chen 2022). These approaches are illustrated with distributional and other data. 4:00-4:10 pm Break 4:10-5:10 pm Stefanie Jegelka Title: Some benefits of machine learning with invariancesAbstract: In many applications, especially in the sciences, data and tasks have known invariances. Encoding such invariances directly into a machine learning model can improve learning outcomes, while it also poses challenges on efficient model design. In the first part of the talk, we will focus on the invariances relevant to eigenvectors and eigenspaces being inputs to a neural network. Such inputs are important, for instance, for graph representation learning. We will discuss targeted architectures that can universally express functions with the relevant invariances – sign flips and changes of basis – and their theoretical and empirical benefits.Second, we will take a broader, theoretical perspective. Empirically, it is known that encoding invariances into the machine learning model can reduce sample complexity. For the simplified setting of kernel ridge regression or random features, we will discuss new bounds that illustrate two ways in which invariances can reduce sample complexity. Our results hold for learning on manifolds and for invariances to (almost) any group action, and use tools from differential geometry.This is joint work with Derek Lim, Joshua Robinson, Behrooz Tahmasebi, Lingxiao Zhao, Tess Smidt, Suvrit Sra, and Haggai Maron.

Tuesday, Feb. 28, 2023 (Eastern Time)

 8:30-9:00 am Breakfast Morning Session Chair: Zhigang Yao 9:00-10:00 am Charles Fefferman* Title: Lipschitz Selection on Metric SpacesAbstract: The talk concerns the problem of finding a Lipschitz map F from a given metric space X into R^D, subject to the constraint that F(x) must lie in a given compact convex “target” K(x) for each point x in X. Joint work with Pavel Shvartsman and with Bernat Guillen Pegueroles. 10:00-10:10 am Break 10:10-11:10 am David Dunson Title: Inferring manifolds from noisy data using Gaussian processesAbstract: In analyzing complex datasets, it is often of interest to infer lower dimensional structure underlying the higher dimensional observations. As a flexible class of nonlinear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. This article proposes a new methodology for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing application of Gaussian processes for probabilistic manifold reconstruction. In addition to theory justifying the algorithm, we provide simulated and real data examples to illustrate the performance. Joint work with Nan Wu – see https://arxiv.org/abs/2110.07478 11:10-11:20 am Break 11:20 am-12:20 pm Wolfgang Polonik Title: Inference in topological data analysisAbstract: Topological data analysis has seen a huge increase in popularity finding applications in numerous scientific fields. This motivates the importance of developing a deeper understanding of benefits and limitations of such methods. Using this angle, we will present and discuss some recent results on large sample inference in topological data analysis, including bootstrap for Betti numbers and the Euler characteristics process. 12:20–1:50 pm Lunch Afternoon Session Chair: Stephan Huckemann 1:50-2:50 pm Ezra Miller Title: Geometric central limit theorems on non-smooth spacesAbstract: The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think “curvature”) is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural outgrowth, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran. 2:50-3:00 pm Break 3:00-4:00 pm Lizhen Lin Title: Statistical foundations of deep generative modelsAbstract: Deep generative models are probabilistic generative models where the generator is parameterized by a deep neural network. They are popular models for modeling high-dimensional data such as texts, images and speeches, and have achieved impressive empirical success. Despite demonstrated success in empirical performance, theoretical understanding of such models is largely lacking. We investigate statistical properties of deep generative models from a nonparametric distribution estimation viewpoint. In the considered model, data are assumed to be observed in some high-dimensional ambient space but concentrate around some low-dimensional structure such as a lower-dimensional manifold structure. Estimating the distribution supported on this low-dimensional structure is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. We obtain convergence rates with respect to the Wasserstein metric of distribution estimators based on two methods: a sieve MLE based on the perturbed data and a GAN type estimator. Such an analysis provides insights into i) how deep generative models can avoid the curse of dimensionality and outperform classical nonparametric estimates, and ii) how likelihood approaches work for singular distribution estimation, especially in adapting to the intrinsic geometry of the data. 4:00-4:10 pm Break 4:10-5:10 pm Conversation session

Wednesday, March 1, 2023 (Eastern Time)

 8:30-9:00 am Breakfast Morning Session Chair: Ezra Miller 9:00-10:00 am Amit Singer* Title: Heterogeneity analysis in cryo-EM by covariance estimation and manifold learningAbstract: In cryo-EM, the 3-D molecular structure needs to be determined from many noisy 2-D tomographic projection images of randomly oriented and positioned molecules. A key assumption in classical reconstruction procedures for cryo-EM is that the sample consists of identical molecules. However, many molecules of interest exist in more than one conformational state. These structural variations are of great interest to biologists, as they provide insight into the functioning of the molecule. Determining the structural variability from a set of cryo-EM images is known as the heterogeneity problem, widely recognized as one of the most challenging and important computational problem in the field. Due to high level of noise in cryo-EM images, heterogeneity studies typically involve hundreds of thousands of images, sometimes even a few millions. Covariance estimation is one of the earliest methods proposed for heterogeneity analysis in cryo-EM. It relies on computing the covariance of the conformations directly from projection images and extracting the optimal linear subspace of conformations through an eigendecomposition. Unfortunately, the standard formulation is plagued by the exorbitant cost of computing the N^3 x N^3 covariance matrix. In the first part of the talk, we present a new low-rank estimation method that requires computing only a small subset of the columns of the covariance while still providing an approximation for the entire matrix. This scheme allows us to estimate tens of principal components of real datasets in a few minutes at medium resolutions and under 30 minutes at high resolutions. In the second part of the talk, we discuss a manifold learning approach based on the graph Laplacian and the diffusion maps framework for learning the manifold of conformations. If time permits, we will also discuss the potential application of optimal transportation to heterogeneity analysis. Based on joint works with Joakim Andén, Marc Gilles, Amit Halevi, Eugene Katsevich, Joe Kileel, Amit Moscovich, and Nathan Zelesko. 10:00-10:10 am Break 10:10-11:10 am Ian Dryden Title: Statistical shape analysis of molecule dataAbstract: Molecular shape data arise in many applications, for example high dimension low sample size cryo-electron microscopy (cryo-EM) data and large temporal sequences of peptides from molecular dynamics simulations. In both applications it is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principal components analysis (PCA) can be problematic as the data may lie far from in a flat manifold. Principal nested spheres gives a fundamentally different decomposition of data from the usual Euclidean subspace based PCA. Subspaces of successively lower dimension are fitted to the data in a backwards manner with the aim of retaining signal and dispensing with noise at each stage. We adapt the methodology to 3D sub-shape spaces and provide some practical fitting algorithms. The methodology is applied to cryo-EM data of a large sliding clamp multi-protein complex and to cluster analysis of peptides, where different states of the molecules can be identified. Further molecular modeling tasks include resolution matching, where coarse resolution models are back-mapped into high resolution (atomistic) structures. This is joint work with Kwang-Rae Kim, Charles Laughton and Huiling Le. 11:10-11:20 am Break 11:20 am-12:20 pm Tamara Broderick Title: An Automatic Finite-Sample Robustness Metric: Can Dropping a Little Data Change Conclusions?Abstract: One hopes that data analyses will be used to make beneficial decisions regarding people’s health, finances, and well-being. But the data fed to an analysis may systematically differ from the data where these decisions are ultimately applied. For instance, suppose we analyze data in one country and conclude that microcredit is effective at alleviating poverty; based on this analysis, we decide to distribute microcredit in other locations and in future years. We might then ask: can we trust our conclusion to apply under new conditions? If we found that a very small percentage of the original data was instrumental in determining the original conclusion, we might not be confident in the stability of the conclusion under new conditions. So we propose a method to assess the sensitivity of data analyses to the removal of a very small fraction of the data set. Analyzing all possible data subsets of a certain size is computationally prohibitive, so we provide an approximation. We call our resulting method the Approximate Maximum Influence Perturbation. Our approximation is automatically computable, theoretically supported, and works for common estimators. We show that any non-robustness our method finds is conclusive. Empirics demonstrate that while some applications are robust, in others the sign of a treatment effect can be changed by dropping less than 0.1% of the data — even in simple models and even when standard errors are small. 12:20-1:50 pm Lunch Afternoon Session Chair: Ezra Miller 1:50-2:50 pm Nicolai Reshetikhin* Title: Random surfaces in exactly solvable models in statistical mechanics.Abstract: In the first part of the talk I will be an overview of a few models in statistical mechanics where a random variable is a geometric object such as a random surface or a random curve. The second part will be focused on the behavior of such random surfaces in the thermodynamic limit and on the formation of the so-called “limit shapes”. 2:50-3:00 pm Break 3:00-4:00 pm Sebastian Kurtek Title: Robust Persistent Homology Using Elastic Functional Data AnalysisAbstract: Persistence landscapes are functional summaries of persistence diagrams designed to enable analysis of the diagrams using tools from functional data analysis. They comprise a collection of scalar functions such that birth and death times of topological features in persistence diagrams map to extrema of functions and intervals where they are non-zero. As a consequence, variation in persistence diagrams is encoded in both amplitude and phase components of persistence landscapes. Through functional data analysis of persistence landscapes, under an elastic Riemannian metric, we show how meaningful statistical summaries of persistence landscapes (e.g., mean, dominant directions of variation) can be obtained by decoupling their amplitude and phase variations. This decoupling is achieved via optimal alignment, with respect to the elastic metric, of the persistence landscapes. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under geometric, scale and sampling variabilities, the phase function prescribes an optimal rate of increase of the resolution parameter for enhancing the topological signal in a persistence diagram. The proposed approach adds to the statistical analysis of data objects with rich structure compared to past studies. In particular, we focus on two sets of data that have been analyzed in the past, brain artery trees and images of prostate cancer cells, and show that separation of amplitude and phase of persistence landscapes is beneficial in both settings. This is joint work with Dr. James Matuk (Duke University) and Dr. Karthik Bharath (University of Nottingham). 4:00-4:10 pm Break 4:10-5:10 pm Conversation session 5:10-5:20 pm Stephan Huckemann, Ezra Miller, Zhigang Yao Closing Remarks

* Virtual Presentation

2
• Member Seminar

##### New bounds on lattice covering volumes, and nearly uniform covers

12:00 pm-1:00 pm
03/02/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Barak Weiss

Title: New bounds on lattice covering volumes, and nearly uniform covers

Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

• Colloquia

##### The string/black hole transition in anti de Sitter space

4:00 pm-5:00 pm
03/02/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Erez Urbach, Weizmann Institute of Science

Title: The string/black hole transition in anti de Sitter space

Abstract: String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

3
• Quantum Matter

##### Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions

10:00 am
03/03/2023

Quantum Matter Seminar

Speaker: Anna Hasenfratz (University of Colorado)

Title: Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions
Abstract: 4-dimensional gauge-fermion systems exhibit a quantum phase transition from a confining, chirally broken phase to a conformal phase as the number of fermions is increased. While the existence of the conformal phase is well established, very little is known about the nature of the phase transition or the strong coupling phase.
Lattice QCD methods can predict the RG $\beta$ function, but the calculations are often limited by non-physical bulk phase transition that prevent exploring the strong coupling region of the phase diagram. Even the critical flavor number is controversial, estimates vary between $N_f=8$ and 14 for fundamental fermions.
Using an improved lattice actions that include heavy Pauli-Villars (PV) type bosons to reduce ultraviolet fluctuations, I was able to simulate an SU(3) system with 8 fundamental flavors at much stronger renormalized coupling than previously possibly. The numerical results indicate a smooth phase transition from weak coupling to a strongly coupled phase.
I investigate the critical behavior of the transition using finite size scaling. The result of the scaling analysis is not consistent with a first order phase transition, but it is well described by   Berezinsky-Kosterlitz-Thouless or BKT scaling. BKT scaling could imply that the 8-flavor system is the opening of the conformal window, an exciting possibility that warrants further investigations.
The strongly coupled phase appear to be chirally symmetric but gapped, suggesting symmetric mass generation (SMG). This could be the consequence of the lattice fermions used in this study. Staggered fermions in the massless limit are known to be anomaly free, allowing an SMG phase in the continuum limit.

References:
Phys.Rev.D 106 (2022) 1, 014513 • e-Print: 2204.04801
Phys.Rev.D 104 (2021) 7, 074509 • e-Print: 2109.02790
For anomalies and staggered fermion, see
Phys.Rev.D 104 (2021) 9, 094504 • e-Print: 2101.01026
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• Member Seminar

##### Categorical Symmetry of the Standard Model from Gravitational Anomaly

12:00 pm-1:00 pm
03/07/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Juven Wang

Title: Categorical Symmetry of the Standard Model from Gravitational Anomaly

Abstract: In the Standard Model, the total “sterile right-handed” neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in our previous work on Cobordism and Deformation Class of the Standard Model [2112.147652204.08393] and Ultra Unification [2012.15860] in order to predict new highly entangled sectors beyond the Standard Model. Moreover, the invertible baryon minus lepton number B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow concept and the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, or the 3d boundary topological order of 4d ℤTF4-time-reversal symmetric topological superconductor [2302.14862].

8
• Colloquia

##### Conformal symmetry, Optimization algorithms and the Critical Phenomena

12:30 pm-1:30 pm
03/08/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Ning Su, University of Pisa

Title: Conformal symmetry, Optimization algorithms and the Critical Phenomena

Abstract: In the phase diagram of many substances, the critical points have emergent conformal symmetry and are described by conformal field theories. Traditionally, physical quantities near the critical point can be computed by perturbative field theory method, where conformal symmetry is not fully utilized. In this talk, I will explain how conformal symmetry can be used to determine certain physical quantities, without even knowing the fine details of the microscopic structure. To compute the observables precisely, one needs to develop powerful numerical techniques. In the last few years, we have invented many computational tools and algorithms, and predicted critical exponents of Helium-4 superfluid phase transition and Heisenberg magnet to very high precision.

• New Technologies in Mathematics Seminar

##### How to steer foundation models?

2:00 pm-3:00 pm
03/08/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

New Technologies in Mathematics Seminar

Speaker: Jimmy Ba, University of Toronto

Title: How to steer foundation models?

Abstract: By conditioning on natural language instructions, foundation models and large language models (LLMs) have displayed impressive capabilities as general-purpose computers. However, task performance depends significantly on the quality of the prompt used to steer the model. Due to the lack of knowledge of how foundation models work, most effective prompts have been handcrafted by humans through a demanding trial-and-error process. To reduce the human effort in this alignment process, I will discuss a few approaches to steer these powerful models to excel in various downstream language and image tasks.

9
• General Relativity Seminar

##### Pseudospectrum and black hole quasinormal mode instability: An ultraviolet universality conjecture

9:30 am-10:30 am
03/09/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

General Relativity Seminar

Speaker: Jose Luis Jaramillo (Bourgogne U.)

Title: Pseudospectrum and black hole quasinormal mode instability: an ultraviolet universality conjecture

Abstract: Can we measure the ‘effective regularity’ of spacetime from the perturbation of quasi-normal mode (QNM) overtones? Black hole (BH) QNMs encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed branches of different qualitative behaviour and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudospectrum approach. Specifically, we cast the QNM problem as an eigenvalue problem for a non-selfadjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We find evidence that perturbed Nollert & Price BH QNMs track the pseudospectrum contour lines, therefore probing the analytic structure of the resolvent, showing the following (in)stability behaviour: i) the slowest decaying (fundamental) mode is stable, whereas ii) (all) QNM overtones are ultraviolet unstable (for sufficiently high frequency). Building on recent work characterizing Burnett’s conjecture as a low-regularity problem in general relativity, we conjecture that (in the infinite-frequency limit) generic ultraviolet spacetime perturbations make BH QNMs migrate to ‘Regge QNM branches’ with a precise universal logarithmic pattern. This is a classical general relativity (effective) low-regularity phenomenon, agnostic to possible detailed (quantum) descriptions of gravity at higher-energies and potentially observationally accessible.

• Probability Seminar

##### On the free energy of spin glasses with multiple types

11:00 am-12:00 pm
03/09/2023

Probability Seminar

Speaker: Jean-Christophe Mourrat (ENS Lyon)

Title: On the free energy of spin glasses with multiple types

Abstract: In the simplest spin-glass model, due to Sherrington and Kirkpatrick, the energy function involves interaction terms between all pairs of spins. A bipartite version of this model can be obtained by splitting the spins into two groups, which we can visualize as forming two layers, and by keeping only interaction terms that go from one to the other layer. For this and other models that incorporate a finite number of types of spins, the asymptotic behavior of the free energy remains mysterious (at least from the mathematical point of view). I will present the difficulties arising there, and some partial progress.

• Workshop

##### Special Lectures on Machine Learning and Protein Folding

3:30 pm-5:00 pm
03/09/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

The CMSA will host a series of three 90-minute lectures on the subject of machine learning for protein folding.

Thursday Feb. 9, Thursday Feb. 16, & Thursday March 9, 2023, 3:30-5:00 pm ET

Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138

Directions and Recommended Lodging

These special lectures will be hybrid:  they will be both in-person and online.

Registration is required.

In-person registration

Zoom webinar registration form: Zoom Webinar.

Speaker: Nazim Bouatta, Harvard Medical School

Abstract: AlphaFold2, a neural network-based model which predicts protein structures from amino acid sequences, is revolutionizing the field of structural biology. This lecture series, given by a leader of the OpenFold project which created an open-source version of AlphaFold2, will explain the protein structure problem and the detailed workings of these models, along with many new results and directions for future research.

Thursday, Feb. 9, 2023

 Thursday, Feb. 9, 20233:30–5:00 pm ET Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm spaceA brief intro to protein biology. AlphaFold2 impacts on experimental structural biology. Co-evolutionary approaches. Space of ‘algorithms’ for protein structure prediction. Proteins as images (CNNs for protein structure prediction). End-to-end differentiable approaches. Attention and long-range dependencies. AlphaFold2 in a nutshell.

 Thursday, Feb. 16, 20233:30–5:00 pm ET Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architectureTurning the co-evolutionary principle into an algorithm: EvoFormer. Structure module and symmetry principles (equivariance and invariance). OpenFold: retraining AlphaFold2 and insights into its learning mechanisms and capacity for generalization. Applications of variants of AlphaFold2 beyond protein structure prediction: AlphaFold Multimer for protein complexes, RNA structure prediction.

 Thursday, March 9, 20233:30–5:00 pm ET Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFoldLimitations of AlphaFold2 and evolutionary ML pipelines. OpenFold: retraining AlphaFold2 yields new insights into its capacity for generalization.

Biography: Nazim Bouatta received his doctoral training in high-energy theoretical physics, and transitioned to systems biology at Harvard Medical School, where he received training in cellular and molecular biology in the group of Prof. Judy Lieberman. He is currently a Senior Research Fellow in the Laboratory of Systems Pharmacology led by Prof. Peter Sorger at Harvard Medical School, and an affiliate of the Department of Systems Biology at Columbia, in the group of Prof. Mohammed AlQuraishi. He is interested in applying machine learning, physics, and mathematics to biology at multiple scales. He recently co-supervised the OpenFold project, an optimized, trainable, and completely open-source version of AlphaFold2. OpenFold has paved the way for many breakthroughs in biology, including the release of the ESM Metagenomic Atlas containing over 600 million predicted protein structures.

Chair: Michael Douglas (Harvard CMSA)

Moderators: Farzan Vafa & Sergiy Verstyuk (Harvard CMSA)

### Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space

https://youtu.be/yqeUH4RsJp8

### Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold

https://youtu.be/kIkn5DGEJJw

10
• Quantum Matter

##### Quantum entropy thermalization

10:00 am-11:00 am
03/10/2023

Quantum Matter Seminar

Speaker: Yichen Huang (Harvard)

Title: Quantum entropy thermalization

Abstract: In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. We prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. In this model, there is a regime of “thermalization without eigenstate thermalization.” Thus, the eigenstate thermalization hypothesis is not a necessary condition for thermalization.

References: arXiv:2302.10165, 2209.09826; Joint work with Aram W. Harrow

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• Swampland Seminar

##### String theory scalar potentials and their critical points

11:00 am-12:00 pm
03/13/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Swampland Seminar

Speaker: David Andriot (Annecy, LAPTH)

Title: String theory scalar potentials and their critical points

Abstract: Positive scalar potentials in string effective theories could provide an origin to Dark Energy, responsible for the accelerated expansion of our universe today or during inflation. It is thus crucial to characterize these scalar potentials, namely their slope, their critical points (de Sitter solutions) and the associated stability, as also advocated by the Swampland Program. We will present such characterizations. Going further, we will also discuss negative scalar potentials, and make related observations on anti-de Sitter solutions, in particular on a new mass bound, as well as comments on scale separation.

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• Member Seminar

##### Randomized algorithms in combinatorics

12:00 pm-1:00 pm
03/14/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Michael Simkin

Title: Randomized algorithms in combinatorics

Abstract: Randomized algorithms have been a computational workhorse for almost as long as there have been computers. Surprisingly, such algorithms can also be used to attack problems that are neither algorithmic nor probabilistic. Time permitting I will discuss the following combintorial examples:

1. Enumerative combinatorics and the n-queens problem: In how many ways can one place n queens on an n x n chessboard so that no queen threatens any other?
2. Constructions of combinatorial designs and the Erdos high-girth Steiner triple system problem: An order-n Steiner triple system (STS) is a collection of triples on n vertices such that every pair of vertices is contained in exactly one triple. In 1973 Erdos conjectured that there exist STSs with arbitrary large girth (informally, no small set of vertices spans many triples). I will discuss the use of randomized algorithms to prove this conjecture. Joint work with Kwan, Sah, and Sawhney.
3. Thresholds in random graphs and hypergraphs: I will discuss how randomized algorithms can be combined with the recent resolution of the Kahn–Kalai conjecture to determine thresholds in random (hyper) graph theory. Joint work with Pham, Sah, and Sawhney.
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• Active Matter Seminar

##### Active chemical reactions in phase-separating systems

1:00 pm-2:00 pm
03/16/2023

Active Matter Seminar

Speaker:  Jonathan Bauermann, Max Planck Institute for the Physics of Complex Systems

Title: Active chemical reactions in phase-separating systems

Abstract: Motivated by the existence of membrane-less compartments in the chemically active environment of living cells, I will discuss the dynamics of droplets in the presence of active chemical reactions. Therefore, I will first introduce the underlying interplay between phase separation and active reactions, which can alter the droplet dynamics compared to equilibrium systems. A key feature of such systems is the emergence of concentration gradients even at steady states. In the second part of this talk, I will discuss how these gradients can trigger instabilities in the core of chemically active droplets, giving rise to a new non-equilibrium steady state of liquid spherical shells. Finally, I will present experimental and theoretical results discussing the existence and energetic cost of this non-equilibrium steady state in a coacervate system.

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• Quantum Matter

##### Tensorial TQFT and disentangling modular Walker-Wang models

10:00 am-11:30 am
03/17/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Quantum Matter Seminar

Speaker: Andreas Bauer  (Freie Universität Berlin)

Title: Tensorial TQFT and disentangling modular Walker-Wang models

Abstract: I will introduce simple “tensorial” definitions for many algebraic and categorical structures appearing in the classification of topological phases of matter. Such “tensorial TQFTs” will be defined as maps that associate tensors to geometric/topological objects of some type, subject to gluing axioms. Tensorial TQFTs are very directly related to microscopic physical models in terms of discrete path integrals. I will use those tensorial definitions to construct invertible boundaries which disentangle modular Walker-Wang models.

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• Member Seminar

##### Quantum Gravity constraints beyond asymptotic regimes

12:00 pm-1:00 pm
03/21/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Max Wiesner

Title: Quantum Gravity constraints beyond asymptotic regimes

Abstract: Not every effective field theory that is consistent in the absence of gravity can be completed to a consistent theory of quantum gravity. The goal of the Swampland program is to find general criteria that distinguish effective field theories, that can be obtained as a low-energy approximation of quantum gravity, from those that are inconsistent in the presence of gravity. These criteria are oftentimes motivated by patterns observed in explicit compactifications of perturbative string theory and have passed many non-trivial tests in asymptotic regions of the field space such as, e.g., weak coupling limits. Still, the Swampland criteria should equally apply to effective theories that do not arise in asymptotic regions of the field space of string theory compactifications. In this talk I will summarize some of my recent works that studies the interior of regions of the field space of string theory in the context of the Swampland program.

• CMSA EVENT: 2023 Ding Shum Lecture

##### Event2023 Ding Shum Lecture

5:00 pm-6:00 pm
03/21/2023
1 Oxford Street, Cambridge MA 02138

On March 21, 2023, the CMSA will host the fourth annual Ding Shum Lecture, given by Cynthia Dwork (Harvard SEAS and Microsoft Research).

Time: 5:00-6:00 pm ET

Location: Harvard University Science Center Hall D

This event will be held in person and via Zoom webinar.

Registration is required.

Zoom Webinar registration (link)

Title: Measuring Our Chances: Risk Prediction in This World and its Betters

Abstract: Prediction algorithms score individuals, assigning a number between zero and one that is often interpreted as an individual probability: a 0.7 “chance” that this child is in danger in the home; an 80% “probability” that this woman will succeed if hired; a 1/3 “likelihood” that they will graduate within 4 years of admission. But what do words like “chance,” “probability,” and “likelihood” actually mean for a non-repeatable activity like going to college? This is a deep and unresolved problem in the philosophy of probability. Without a compelling mathematical definition we cannot specify what an (imagined) perfect risk prediction algorithm should produce, nor even how an existing algorithm should be evaluated. Undaunted, AI and machine learned algorithms churn these numbers out in droves, sometimes with life-altering consequences.

An explosion of recent research deploys insights from the theory of pseudo-random numbers – sequences of 0’s and 1’s that “look random” but in fact have structure – to yield a tantalizing answer to the evaluation problem, together with a supporting algorithmic framework with roots in the theory of algorithmic fairness.

We can aim even higher. Both (1) our qualifications, health, and skills, which form the inputs to a prediction algorithm, and (2) our chances of future success, which are the desired outputs from the ideal risk prediction algorithm, are products of our interactions with the real world. But the real world is systematically inequitable. How, and when, can we hope to approximate probabilities not in this world, but in a better world, one for which, unfortunately, we have no data at all? Surprisingly, this novel question is inextricably bound with the very existence of nondeterminism.

Professor Cynthia Dwork is Gordon McKay Professor of Computer Science at the Harvard University John A. Paulson School of Engineering and Applied Sciences, Affiliated Faculty at Harvard Law School, and Distinguished Scientist at Microsoft. She uses theoretical computer science to place societal problems on a firm mathematical foundation.

Her recent awards and honors include the 2020 ACM SIGACT and IEEE TCMF Knuth Prize, the 2020 IEEE Hamming Medal, and the 2017 Gödel Prize.

Talk Chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)
Moderator: Faidra Monachou (Harvard CMSA)

The 2020-2022 Ding Shum lectures were postponed due to Covid-19.

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• Colloquia

##### Synchronization in a Kuramoto Mean Field Game

12:30 pm-1:30 pm
03/22/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Mete Soner (Princeton University)

Title: Synchronization in a Kuramoto Mean Field Game

Abstract:  Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a  key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system. While the system is not synchronized when this term is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter.  We explore this system in the mean field formalism.  We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles determine endogenously their behaviors by minimizing a cost functional and eventually, settling in a Nash equilibrium.  The mean field game also exhibits a bifurcation from incoherence to self-organization.  This approach has found interesting applications including circadian rhythms and jet-lag recovery.  This is joint work with Rene Carmona of Princeton and Quentin Cormier of INRIA, Paris.

• Probability Seminar

##### Some rigorous results on the Lévy spin glass model

3:30 pm-4:30 pm
03/22/2023

Probability Seminar

Speaker: Wei-Kuo Chen (Minnesota)

Title: Some rigorous results on the Lévy spin glass model

Abstract: The Lévy spin glass model, proposed by Cizeau-Bouchaud, is a mean-field model defined on a fully connected graph, where the spin interactions are formulated through a power-law distribution. This model is well-motivated from the study of the experimental metallic spin glasses. It is also expected to bridge between some mean-field and diluted models. In this talk, we will discuss some recent progress on the Lévy model including its high temperature behavior and the existence and variational expression for the limiting free energy. Based on a joint work with Heejune Kim and Arnab Sen.

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• General Relativity Seminar

##### New Phases of N=4 SYM

1:30 pm-2:30 pm
03/23/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

General Relativity Seminar

Speaker: Prahar Mitra (University of Cambridge)

Title: New Phases of N=4 SYM

Abstract: We construct new static solutions to gauged supergravity that, via the AdS/CFT correspondence, are dual to thermal phases in N=4 SYM at finite chemical potential. These solutions dominate the micro-canonical ensemble and are required to ultimately reproduce the microscopic entropy of AdS black holes. These are constructed in two distinct truncations of gauged supergravity and can be uplifted to solutions of type IIB supergravity. Together with the known phases of the truncation with three equal charges, our findings permit a good understanding of the full phase space of SYM thermal states with three arbitrary chemical potentials. We will also discuss the status of hairy supersymmetric black hole solutions in this theory.

Based on: https://arxiv.org/pdf/2207.07134.pdf [hep-th]

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• Quantum Matter

##### Traversable wormhole dynamics on a quantum processor

10:00 am-11:30 am
03/24/2023

Quantum Matter Seminar

Speaker: Alexander Zlokapa, MIT

Title: Traversable wormhole dynamics on a quantum processor

Abstract: The holographic principle, theorized to be a property of quantum gravity, postulates that the description of a volume of space can be encoded on a lower-dimensional boundary. The anti-de Sitter (AdS)/conformal field theory correspondence or duality is the principal example of holography. The Sachdev–Ye–Kitaev (SYK) model of N >> 1 Majorana fermions has features suggesting the existence of a gravitational dual in AdS2, and is a new realization of holography. We invoke the holographic correspondence of the SYK many-body system and gravity to probe the conjectured ER=EPR relation between entanglement and spacetime geometry through the traversable wormhole mechanism as implemented in the SYK model. A qubit can be used to probe the SYK traversable wormhole dynamics through the corresponding teleportation protocol. This can be realized as a quantum circuit, equivalent to the gravitational picture in the semiclassical limit of an infinite number of qubits. Here we use learning techniques to construct a sparsified SYK model that we experimentally realize with 164 two-qubit gates on a nine-qubit circuit and observe the corresponding traversable wormhole dynamics. Despite its approximate nature, the sparsified SYK model preserves key properties of the traversable wormhole physics: perfect size winding, coupling on either side of the wormhole that is consistent with a negative energy shockwave, a Shapiro time delay, causal time-order of signals emerging from the wormhole, and scrambling and thermalization dynamics. Our experiment was run on the Google Sycamore processor. By interrogating a two-dimensional gravity dual system, our work represents a step towards a program for studying quantum gravity in the laboratory. Future developments will require improved hardware scalability and performance as well as theoretical developments including higher-dimensional quantum gravity duals and other SYK-like models.

• CMSA EVENT: CMSA/MATH Bi-Annual Gathering

##### EventCMSA/MATH Bi-Annual Gathering

4:30 pm-6:00 pm
03/24/2023
20 Garden Street, Cambridge, MA 02138 USA

On Friday, March 24th, 4:30PM – 6PM, the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the Common Room at 20 Garden Street, Cambridge MA 02138.

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• Swampland Seminar

##### Recent developments on the tidal Love numbers of black holes

11:00 am-12:00 pm
03/27/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Swampland Seminar

Speaker: Valerio De Luca (UPenn)

Title: Recent developments on the tidal Love numbers of black holes

Abstract: Tidal Love numbers describe the deformability of compact objects under the presence of external tidal perturbations, and are found to be exactly zero for black holes in pure General Relativity. This property is however fragile, since they receive corrections from higher-order derivative terms in the theory. We show that the tidal deformability of neutral black holes is constrained by the Weak Gravity Conjecture.

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• Member Seminar

##### Member Seminar Title TBA

12:00 pm-1:00 pm
03/28/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Jue Liu

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• Colloquia

##### Title TBA

12:30 pm-1:30 pm
03/29/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Ruth Britto, Princeton

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• Quantum Matter

##### A Plane Defect in the 3d O(N) Model

10:00 am-11:30 am
03/31/2023
Virtual and in 20 Garden Street, Room G10

Quantum Matter Seminar

Speaker: Abijith Krishnan (MIT)

Title: A Plane Defect in the 3d O(N) Model

Abstract: It was recently found that the classical 3d O(N) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as (log x)^(-q). This universality class exists for a range 2≤N<Nc and Monte-Carlo simulations and conformal bootstrap indicate Nc>3. In this talk, I’ll extend this result to the 3d O(N) model in an infinite geometry with a plane defect. I’ll explain using the renormalization group (RG) that the extraordinary-log universality class is present for any finite N≥2, and that a line of defect fixed points is present at N=∞. This line of defect fixed points is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N corrections. I’ll show that the line of defect fixed points and the 1/N corrections agree with an a-theorem by Jensen and O’Bannon for 3d CFTs with a boundary. Finally, I’ll conclude by noting some physical systems where the extraordinary-log universality class can be observed.

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1
• Conference

##### Conference on Geometry and Statistics

9:00 am-5:30 pm
03/01/2023-03/01/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.

Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138

This conference will be held in person. Directions and Recommended Lodging

Registration is required.

Register here to attend in-person.

Organizing Committee:
Stephan Huckemann (Georg-August-Universität Göttingen)
Ezra Miller (Duke University)
Zhigang Yao (Harvard CMSA and Committee Chair)

Horng-Tzer Yau (Harvard CMSA)
Shing-Tung Yau (Harvard CMSA)

Speakers:

• Tamara Broderick (MIT)
• David Donoho (Stanford)
• Ian Dryden (Florida International University in Miami)
• David Dunson (Duke)
• Charles Fefferman (Princeton)
• Stefanie Jegelka (MIT)
• Sebastian Kurtek (OSU)
• Lizhen Lin (Notre Dame)
• Steve Marron (U North Carolina)
• Ezra Miller (Duke)
• Hans-Georg Mueller (UC Davis)
• Nicolai Reshetikhin (UC Berkeley)
• Wolfgang Polonik (UC Davis)
• Amit Singer (Princeton)
• Zhigang Yao (Harvard CMSA)
• Bin Yu (Berkeley)

Moderator: Michael Simkin (Harvard CMSA)

#### SCHEDULE

Monday, Feb. 27, 2023 (Eastern Time)

 8:30 am Breakfast 8:45–8:55 am Zhigang Yao Welcome Remarks 8:55–9:00 am Shing-Tung Yau* Remarks Morning Session Chair: Zhigang Yao 9:00–10:00 am David Donoho Title: ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated NoiseAbstract: Truncation of the singular value decomposition is a true scientific workhorse. But where to Truncate?For 55 years the answer, for many scientists, has been to eyeball the scree plot, an approach which still generates hundreds of papers per year.I will describe ScreeNOT, a mathematically solid alternative deriving from the many advances in Random Matrix Theory over those 55 years. Assuming a model of low-rank signal plus possibly correlated noise, and adopting an asymptotic viewpoint with number of rows proportional to the number of columns, we show that ScreeNOT has a surprising oracle property.It typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance – i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure.The talk is based on joint work with Matan Gavish and Elad Romanov, Hebrew University. 10:00–10:10 am Break 10:10–11:10 am Steve Marron Title: Modes of Variation in Non-Euclidean SpacesAbstract: Modes of Variation provide an intuitive means of understanding variation in populations, especially in the case of data objects that naturally lie in non-Euclidean spaces. A variety of useful approaches to finding useful modes of variation are considered in several non-Euclidean contexts, including shapes as data objects, vectors of directional data, amplitude and phase variation and compositional data. 11:10–11:20 am Break 11:20 am–12:20 pm Zhigang Yao Title: Manifold fitting: an invitation to statisticsAbstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y \subset \mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $\widehat{M} \subset \mathbb{R}^D$ to approximate $Y$, and how well can $\widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference:https://arxiv.org/abs/1909.10228 12:20–1:50 pm 12:20 pm Group Photofollowed by Lunch Afternoon Session Chair: Stephan Huckemann 1:50–2:50 pm Bin Yu* Title: Interpreting Deep Neural Networks towards TrustworthinessAbstract: Recent deep learning models have achieved impressive predictive performance by learning complex functions of many variables, often at the cost of interpretability. This lecture first defines interpretable machine learning in general and introduces the agglomerative contextual decomposition (ACD) method to interpret neural networks. Extending ACD to the scientifically meaningful frequency domain, an adaptive wavelet distillation (AWD) interpretation method is developed. AWD is shown to be both outperforming deep neural networks and interpretable in two prediction problems from cosmology and cell biology. Finally, a quality-controlled data science life cycle is advocated for building any model for trustworthy interpretation and introduce a Predictability Computability Stability (PCS) framework for such a data science life cycle. 2:50–3:00 pm Break 3:00-4:00 pm Hans-Georg Mueller Title: Exploration of Random Objects with Depth Profiles and Fréchet RegressionAbstract: Random objects, i.e., random variables that take values in a separable metric space, pose many challenges for statistical analysis, as vector operations are not available in general metric spaces. Examples include random variables that take values in the space of distributions, covariance matrices or surfaces, graph Laplacians to represent networks, trees and in other spaces. The increasing prevalence of samples of random objects has stimulated the development of metric statistics, an emerging collection of statistical tools to characterize, infer and relate samples of random objects. Recent developments include depth profiles, which are useful for the exploration of random objects. The depth profile for any given object is the distribution of distances to all other objects (with P. Dubey, Y. Chen 2022).These distributions can then be subjected to statistical analysis. Their mutual transports lead to notions of transport ranks, quantiles and centrality. Another useful tool is global or local Fréchet regression (with A. Petersen 2019) where random objects are responses and scalars or vectors are predictors and one aims at modeling conditional Fréchet means. Recent theoretical advances for local Fréchet regression provide a basis for object time warping (with Y. Chen 2022). These approaches are illustrated with distributional and other data. 4:00-4:10 pm Break 4:10-5:10 pm Stefanie Jegelka Title: Some benefits of machine learning with invariancesAbstract: In many applications, especially in the sciences, data and tasks have known invariances. Encoding such invariances directly into a machine learning model can improve learning outcomes, while it also poses challenges on efficient model design. In the first part of the talk, we will focus on the invariances relevant to eigenvectors and eigenspaces being inputs to a neural network. Such inputs are important, for instance, for graph representation learning. We will discuss targeted architectures that can universally express functions with the relevant invariances – sign flips and changes of basis – and their theoretical and empirical benefits.Second, we will take a broader, theoretical perspective. Empirically, it is known that encoding invariances into the machine learning model can reduce sample complexity. For the simplified setting of kernel ridge regression or random features, we will discuss new bounds that illustrate two ways in which invariances can reduce sample complexity. Our results hold for learning on manifolds and for invariances to (almost) any group action, and use tools from differential geometry.This is joint work with Derek Lim, Joshua Robinson, Behrooz Tahmasebi, Lingxiao Zhao, Tess Smidt, Suvrit Sra, and Haggai Maron.

Tuesday, Feb. 28, 2023 (Eastern Time)

 8:30-9:00 am Breakfast Morning Session Chair: Zhigang Yao 9:00-10:00 am Charles Fefferman* Title: Lipschitz Selection on Metric SpacesAbstract: The talk concerns the problem of finding a Lipschitz map F from a given metric space X into R^D, subject to the constraint that F(x) must lie in a given compact convex “target” K(x) for each point x in X. Joint work with Pavel Shvartsman and with Bernat Guillen Pegueroles. 10:00-10:10 am Break 10:10-11:10 am David Dunson Title: Inferring manifolds from noisy data using Gaussian processesAbstract: In analyzing complex datasets, it is often of interest to infer lower dimensional structure underlying the higher dimensional observations. As a flexible class of nonlinear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. This article proposes a new methodology for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing application of Gaussian processes for probabilistic manifold reconstruction. In addition to theory justifying the algorithm, we provide simulated and real data examples to illustrate the performance. Joint work with Nan Wu – see https://arxiv.org/abs/2110.07478 11:10-11:20 am Break 11:20 am-12:20 pm Wolfgang Polonik Title: Inference in topological data analysisAbstract: Topological data analysis has seen a huge increase in popularity finding applications in numerous scientific fields. This motivates the importance of developing a deeper understanding of benefits and limitations of such methods. Using this angle, we will present and discuss some recent results on large sample inference in topological data analysis, including bootstrap for Betti numbers and the Euler characteristics process. 12:20–1:50 pm Lunch Afternoon Session Chair: Stephan Huckemann 1:50-2:50 pm Ezra Miller Title: Geometric central limit theorems on non-smooth spacesAbstract: The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think “curvature”) is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural outgrowth, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran. 2:50-3:00 pm Break 3:00-4:00 pm Lizhen Lin Title: Statistical foundations of deep generative modelsAbstract: Deep generative models are probabilistic generative models where the generator is parameterized by a deep neural network. They are popular models for modeling high-dimensional data such as texts, images and speeches, and have achieved impressive empirical success. Despite demonstrated success in empirical performance, theoretical understanding of such models is largely lacking. We investigate statistical properties of deep generative models from a nonparametric distribution estimation viewpoint. In the considered model, data are assumed to be observed in some high-dimensional ambient space but concentrate around some low-dimensional structure such as a lower-dimensional manifold structure. Estimating the distribution supported on this low-dimensional structure is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. We obtain convergence rates with respect to the Wasserstein metric of distribution estimators based on two methods: a sieve MLE based on the perturbed data and a GAN type estimator. Such an analysis provides insights into i) how deep generative models can avoid the curse of dimensionality and outperform classical nonparametric estimates, and ii) how likelihood approaches work for singular distribution estimation, especially in adapting to the intrinsic geometry of the data. 4:00-4:10 pm Break 4:10-5:10 pm Conversation session

Wednesday, March 1, 2023 (Eastern Time)

 8:30-9:00 am Breakfast Morning Session Chair: Ezra Miller 9:00-10:00 am Amit Singer* Title: Heterogeneity analysis in cryo-EM by covariance estimation and manifold learningAbstract: In cryo-EM, the 3-D molecular structure needs to be determined from many noisy 2-D tomographic projection images of randomly oriented and positioned molecules. A key assumption in classical reconstruction procedures for cryo-EM is that the sample consists of identical molecules. However, many molecules of interest exist in more than one conformational state. These structural variations are of great interest to biologists, as they provide insight into the functioning of the molecule. Determining the structural variability from a set of cryo-EM images is known as the heterogeneity problem, widely recognized as one of the most challenging and important computational problem in the field. Due to high level of noise in cryo-EM images, heterogeneity studies typically involve hundreds of thousands of images, sometimes even a few millions. Covariance estimation is one of the earliest methods proposed for heterogeneity analysis in cryo-EM. It relies on computing the covariance of the conformations directly from projection images and extracting the optimal linear subspace of conformations through an eigendecomposition. Unfortunately, the standard formulation is plagued by the exorbitant cost of computing the N^3 x N^3 covariance matrix. In the first part of the talk, we present a new low-rank estimation method that requires computing only a small subset of the columns of the covariance while still providing an approximation for the entire matrix. This scheme allows us to estimate tens of principal components of real datasets in a few minutes at medium resolutions and under 30 minutes at high resolutions. In the second part of the talk, we discuss a manifold learning approach based on the graph Laplacian and the diffusion maps framework for learning the manifold of conformations. If time permits, we will also discuss the potential application of optimal transportation to heterogeneity analysis. Based on joint works with Joakim Andén, Marc Gilles, Amit Halevi, Eugene Katsevich, Joe Kileel, Amit Moscovich, and Nathan Zelesko. 10:00-10:10 am Break 10:10-11:10 am Ian Dryden Title: Statistical shape analysis of molecule dataAbstract: Molecular shape data arise in many applications, for example high dimension low sample size cryo-electron microscopy (cryo-EM) data and large temporal sequences of peptides from molecular dynamics simulations. In both applications it is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principal components analysis (PCA) can be problematic as the data may lie far from in a flat manifold. Principal nested spheres gives a fundamentally different decomposition of data from the usual Euclidean subspace based PCA. Subspaces of successively lower dimension are fitted to the data in a backwards manner with the aim of retaining signal and dispensing with noise at each stage. We adapt the methodology to 3D sub-shape spaces and provide some practical fitting algorithms. The methodology is applied to cryo-EM data of a large sliding clamp multi-protein complex and to cluster analysis of peptides, where different states of the molecules can be identified. Further molecular modeling tasks include resolution matching, where coarse resolution models are back-mapped into high resolution (atomistic) structures. This is joint work with Kwang-Rae Kim, Charles Laughton and Huiling Le. 11:10-11:20 am Break 11:20 am-12:20 pm Tamara Broderick Title: An Automatic Finite-Sample Robustness Metric: Can Dropping a Little Data Change Conclusions?Abstract: One hopes that data analyses will be used to make beneficial decisions regarding people’s health, finances, and well-being. But the data fed to an analysis may systematically differ from the data where these decisions are ultimately applied. For instance, suppose we analyze data in one country and conclude that microcredit is effective at alleviating poverty; based on this analysis, we decide to distribute microcredit in other locations and in future years. We might then ask: can we trust our conclusion to apply under new conditions? If we found that a very small percentage of the original data was instrumental in determining the original conclusion, we might not be confident in the stability of the conclusion under new conditions. So we propose a method to assess the sensitivity of data analyses to the removal of a very small fraction of the data set. Analyzing all possible data subsets of a certain size is computationally prohibitive, so we provide an approximation. We call our resulting method the Approximate Maximum Influence Perturbation. Our approximation is automatically computable, theoretically supported, and works for common estimators. We show that any non-robustness our method finds is conclusive. Empirics demonstrate that while some applications are robust, in others the sign of a treatment effect can be changed by dropping less than 0.1% of the data — even in simple models and even when standard errors are small. 12:20-1:50 pm Lunch Afternoon Session Chair: Ezra Miller 1:50-2:50 pm Nicolai Reshetikhin* Title: Random surfaces in exactly solvable models in statistical mechanics.Abstract: In the first part of the talk I will be an overview of a few models in statistical mechanics where a random variable is a geometric object such as a random surface or a random curve. The second part will be focused on the behavior of such random surfaces in the thermodynamic limit and on the formation of the so-called “limit shapes”. 2:50-3:00 pm Break 3:00-4:00 pm Sebastian Kurtek Title: Robust Persistent Homology Using Elastic Functional Data AnalysisAbstract: Persistence landscapes are functional summaries of persistence diagrams designed to enable analysis of the diagrams using tools from functional data analysis. They comprise a collection of scalar functions such that birth and death times of topological features in persistence diagrams map to extrema of functions and intervals where they are non-zero. As a consequence, variation in persistence diagrams is encoded in both amplitude and phase components of persistence landscapes. Through functional data analysis of persistence landscapes, under an elastic Riemannian metric, we show how meaningful statistical summaries of persistence landscapes (e.g., mean, dominant directions of variation) can be obtained by decoupling their amplitude and phase variations. This decoupling is achieved via optimal alignment, with respect to the elastic metric, of the persistence landscapes. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under geometric, scale and sampling variabilities, the phase function prescribes an optimal rate of increase of the resolution parameter for enhancing the topological signal in a persistence diagram. The proposed approach adds to the statistical analysis of data objects with rich structure compared to past studies. In particular, we focus on two sets of data that have been analyzed in the past, brain artery trees and images of prostate cancer cells, and show that separation of amplitude and phase of persistence landscapes is beneficial in both settings. This is joint work with Dr. James Matuk (Duke University) and Dr. Karthik Bharath (University of Nottingham). 4:00-4:10 pm Break 4:10-5:10 pm Conversation session 5:10-5:20 pm Stephan Huckemann, Ezra Miller, Zhigang Yao Closing Remarks

* Virtual Presentation

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• Member Seminar

##### New bounds on lattice covering volumes, and nearly uniform covers

12:00 pm-1:00 pm
03/02/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Barak Weiss

Title: New bounds on lattice covering volumes, and nearly uniform covers

Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

• Colloquia

##### The string/black hole transition in anti de Sitter space

4:00 pm-5:00 pm
03/02/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Erez Urbach, Weizmann Institute of Science

Title: The string/black hole transition in anti de Sitter space

Abstract: String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

3
• Quantum Matter

##### Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions

10:00 am
03/03/2023

Quantum Matter Seminar

Speaker: Anna Hasenfratz (University of Colorado)

Title: Strongly coupled ultraviolet fixed point and symmetric mass generation in four dimensions with 8 Kähler-Dirac fermions
Abstract: 4-dimensional gauge-fermion systems exhibit a quantum phase transition from a confining, chirally broken phase to a conformal phase as the number of fermions is increased. While the existence of the conformal phase is well established, very little is known about the nature of the phase transition or the strong coupling phase.
Lattice QCD methods can predict the RG $\beta$ function, but the calculations are often limited by non-physical bulk phase transition that prevent exploring the strong coupling region of the phase diagram. Even the critical flavor number is controversial, estimates vary between $N_f=8$ and 14 for fundamental fermions.
Using an improved lattice actions that include heavy Pauli-Villars (PV) type bosons to reduce ultraviolet fluctuations, I was able to simulate an SU(3) system with 8 fundamental flavors at much stronger renormalized coupling than previously possibly. The numerical results indicate a smooth phase transition from weak coupling to a strongly coupled phase.
I investigate the critical behavior of the transition using finite size scaling. The result of the scaling analysis is not consistent with a first order phase transition, but it is well described by   Berezinsky-Kosterlitz-Thouless or BKT scaling. BKT scaling could imply that the 8-flavor system is the opening of the conformal window, an exciting possibility that warrants further investigations.
The strongly coupled phase appear to be chirally symmetric but gapped, suggesting symmetric mass generation (SMG). This could be the consequence of the lattice fermions used in this study. Staggered fermions in the massless limit are known to be anomaly free, allowing an SMG phase in the continuum limit.

References:
Phys.Rev.D 106 (2022) 1, 014513 • e-Print: 2204.04801
Phys.Rev.D 104 (2021) 7, 074509 • e-Print: 2109.02790
For anomalies and staggered fermion, see
Phys.Rev.D 104 (2021) 9, 094504 • e-Print: 2101.01026
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• Member Seminar

##### Categorical Symmetry of the Standard Model from Gravitational Anomaly

12:00 pm-1:00 pm
03/07/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Juven Wang

Title: Categorical Symmetry of the Standard Model from Gravitational Anomaly

Abstract: In the Standard Model, the total “sterile right-handed” neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in our previous work on Cobordism and Deformation Class of the Standard Model [2112.147652204.08393] and Ultra Unification [2012.15860] in order to predict new highly entangled sectors beyond the Standard Model. Moreover, the invertible baryon minus lepton number B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow concept and the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, or the 3d boundary topological order of 4d ℤTF4-time-reversal symmetric topological superconductor [2302.14862].

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• Colloquia

##### Conformal symmetry, Optimization algorithms and the Critical Phenomena

12:30 pm-1:30 pm
03/08/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Ning Su, University of Pisa

Title: Conformal symmetry, Optimization algorithms and the Critical Phenomena

Abstract: In the phase diagram of many substances, the critical points have emergent conformal symmetry and are described by conformal field theories. Traditionally, physical quantities near the critical point can be computed by perturbative field theory method, where conformal symmetry is not fully utilized. In this talk, I will explain how conformal symmetry can be used to determine certain physical quantities, without even knowing the fine details of the microscopic structure. To compute the observables precisely, one needs to develop powerful numerical techniques. In the last few years, we have invented many computational tools and algorithms, and predicted critical exponents of Helium-4 superfluid phase transition and Heisenberg magnet to very high precision.

• New Technologies in Mathematics Seminar

##### How to steer foundation models?

2:00 pm-3:00 pm
03/08/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

New Technologies in Mathematics Seminar

Speaker: Jimmy Ba, University of Toronto

Title: How to steer foundation models?

Abstract: By conditioning on natural language instructions, foundation models and large language models (LLMs) have displayed impressive capabilities as general-purpose computers. However, task performance depends significantly on the quality of the prompt used to steer the model. Due to the lack of knowledge of how foundation models work, most effective prompts have been handcrafted by humans through a demanding trial-and-error process. To reduce the human effort in this alignment process, I will discuss a few approaches to steer these powerful models to excel in various downstream language and image tasks.

9
• General Relativity Seminar

##### Pseudospectrum and black hole quasinormal mode instability: An ultraviolet universality conjecture

9:30 am-10:30 am
03/09/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

General Relativity Seminar

Speaker: Jose Luis Jaramillo (Bourgogne U.)

Title: Pseudospectrum and black hole quasinormal mode instability: an ultraviolet universality conjecture

Abstract: Can we measure the ‘effective regularity’ of spacetime from the perturbation of quasi-normal mode (QNM) overtones? Black hole (BH) QNMs encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed branches of different qualitative behaviour and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudospectrum approach. Specifically, we cast the QNM problem as an eigenvalue problem for a non-selfadjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We find evidence that perturbed Nollert & Price BH QNMs track the pseudospectrum contour lines, therefore probing the analytic structure of the resolvent, showing the following (in)stability behaviour: i) the slowest decaying (fundamental) mode is stable, whereas ii) (all) QNM overtones are ultraviolet unstable (for sufficiently high frequency). Building on recent work characterizing Burnett’s conjecture as a low-regularity problem in general relativity, we conjecture that (in the infinite-frequency limit) generic ultraviolet spacetime perturbations make BH QNMs migrate to ‘Regge QNM branches’ with a precise universal logarithmic pattern. This is a classical general relativity (effective) low-regularity phenomenon, agnostic to possible detailed (quantum) descriptions of gravity at higher-energies and potentially observationally accessible.

• Probability Seminar

##### On the free energy of spin glasses with multiple types

11:00 am-12:00 pm
03/09/2023

Probability Seminar

Speaker: Jean-Christophe Mourrat (ENS Lyon)

Title: On the free energy of spin glasses with multiple types

Abstract: In the simplest spin-glass model, due to Sherrington and Kirkpatrick, the energy function involves interaction terms between all pairs of spins. A bipartite version of this model can be obtained by splitting the spins into two groups, which we can visualize as forming two layers, and by keeping only interaction terms that go from one to the other layer. For this and other models that incorporate a finite number of types of spins, the asymptotic behavior of the free energy remains mysterious (at least from the mathematical point of view). I will present the difficulties arising there, and some partial progress.

• Workshop

##### Special Lectures on Machine Learning and Protein Folding

3:30 pm-5:00 pm
03/09/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

The CMSA will host a series of three 90-minute lectures on the subject of machine learning for protein folding.

Thursday Feb. 9, Thursday Feb. 16, & Thursday March 9, 2023, 3:30-5:00 pm ET

Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138

Directions and Recommended Lodging

These special lectures will be hybrid:  they will be both in-person and online.

Registration is required.

In-person registration

Zoom webinar registration form: Zoom Webinar.

Speaker: Nazim Bouatta, Harvard Medical School

Abstract: AlphaFold2, a neural network-based model which predicts protein structures from amino acid sequences, is revolutionizing the field of structural biology. This lecture series, given by a leader of the OpenFold project which created an open-source version of AlphaFold2, will explain the protein structure problem and the detailed workings of these models, along with many new results and directions for future research.

Thursday, Feb. 9, 2023

 Thursday, Feb. 9, 20233:30–5:00 pm ET Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm spaceA brief intro to protein biology. AlphaFold2 impacts on experimental structural biology. Co-evolutionary approaches. Space of ‘algorithms’ for protein structure prediction. Proteins as images (CNNs for protein structure prediction). End-to-end differentiable approaches. Attention and long-range dependencies. AlphaFold2 in a nutshell.

 Thursday, Feb. 16, 20233:30–5:00 pm ET Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architectureTurning the co-evolutionary principle into an algorithm: EvoFormer. Structure module and symmetry principles (equivariance and invariance). OpenFold: retraining AlphaFold2 and insights into its learning mechanisms and capacity for generalization. Applications of variants of AlphaFold2 beyond protein structure prediction: AlphaFold Multimer for protein complexes, RNA structure prediction.

 Thursday, March 9, 20233:30–5:00 pm ET Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFoldLimitations of AlphaFold2 and evolutionary ML pipelines. OpenFold: retraining AlphaFold2 yields new insights into its capacity for generalization.

Biography: Nazim Bouatta received his doctoral training in high-energy theoretical physics, and transitioned to systems biology at Harvard Medical School, where he received training in cellular and molecular biology in the group of Prof. Judy Lieberman. He is currently a Senior Research Fellow in the Laboratory of Systems Pharmacology led by Prof. Peter Sorger at Harvard Medical School, and an affiliate of the Department of Systems Biology at Columbia, in the group of Prof. Mohammed AlQuraishi. He is interested in applying machine learning, physics, and mathematics to biology at multiple scales. He recently co-supervised the OpenFold project, an optimized, trainable, and completely open-source version of AlphaFold2. OpenFold has paved the way for many breakthroughs in biology, including the release of the ESM Metagenomic Atlas containing over 600 million predicted protein structures.

Chair: Michael Douglas (Harvard CMSA)

Moderators: Farzan Vafa & Sergiy Verstyuk (Harvard CMSA)

### Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space

https://youtu.be/yqeUH4RsJp8

### Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold

https://youtu.be/kIkn5DGEJJw

10
• Quantum Matter

##### Quantum entropy thermalization

10:00 am-11:00 am
03/10/2023

Quantum Matter Seminar

Speaker: Yichen Huang (Harvard)

Title: Quantum entropy thermalization

Abstract: In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. We prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. In this model, there is a regime of “thermalization without eigenstate thermalization.” Thus, the eigenstate thermalization hypothesis is not a necessary condition for thermalization.

References: arXiv:2302.10165, 2209.09826; Joint work with Aram W. Harrow

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• Swampland Seminar

##### String theory scalar potentials and their critical points

11:00 am-12:00 pm
03/13/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Swampland Seminar

Speaker: David Andriot (Annecy, LAPTH)

Title: String theory scalar potentials and their critical points

Abstract: Positive scalar potentials in string effective theories could provide an origin to Dark Energy, responsible for the accelerated expansion of our universe today or during inflation. It is thus crucial to characterize these scalar potentials, namely their slope, their critical points (de Sitter solutions) and the associated stability, as also advocated by the Swampland Program. We will present such characterizations. Going further, we will also discuss negative scalar potentials, and make related observations on anti-de Sitter solutions, in particular on a new mass bound, as well as comments on scale separation.

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• Member Seminar

##### Randomized algorithms in combinatorics

12:00 pm-1:00 pm
03/14/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Michael Simkin

Title: Randomized algorithms in combinatorics

Abstract: Randomized algorithms have been a computational workhorse for almost as long as there have been computers. Surprisingly, such algorithms can also be used to attack problems that are neither algorithmic nor probabilistic. Time permitting I will discuss the following combintorial examples:

1. Enumerative combinatorics and the n-queens problem: In how many ways can one place n queens on an n x n chessboard so that no queen threatens any other?
2. Constructions of combinatorial designs and the Erdos high-girth Steiner triple system problem: An order-n Steiner triple system (STS) is a collection of triples on n vertices such that every pair of vertices is contained in exactly one triple. In 1973 Erdos conjectured that there exist STSs with arbitrary large girth (informally, no small set of vertices spans many triples). I will discuss the use of randomized algorithms to prove this conjecture. Joint work with Kwan, Sah, and Sawhney.
3. Thresholds in random graphs and hypergraphs: I will discuss how randomized algorithms can be combined with the recent resolution of the Kahn–Kalai conjecture to determine thresholds in random (hyper) graph theory. Joint work with Pham, Sah, and Sawhney.
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16
• Active Matter Seminar

##### Active chemical reactions in phase-separating systems

1:00 pm-2:00 pm
03/16/2023

Active Matter Seminar

Speaker:  Jonathan Bauermann, Max Planck Institute for the Physics of Complex Systems

Title: Active chemical reactions in phase-separating systems

Abstract: Motivated by the existence of membrane-less compartments in the chemically active environment of living cells, I will discuss the dynamics of droplets in the presence of active chemical reactions. Therefore, I will first introduce the underlying interplay between phase separation and active reactions, which can alter the droplet dynamics compared to equilibrium systems. A key feature of such systems is the emergence of concentration gradients even at steady states. In the second part of this talk, I will discuss how these gradients can trigger instabilities in the core of chemically active droplets, giving rise to a new non-equilibrium steady state of liquid spherical shells. Finally, I will present experimental and theoretical results discussing the existence and energetic cost of this non-equilibrium steady state in a coacervate system.

17
• Quantum Matter

##### Tensorial TQFT and disentangling modular Walker-Wang models

10:00 am-11:30 am
03/17/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Quantum Matter Seminar

Speaker: Andreas Bauer  (Freie Universität Berlin)

Title: Tensorial TQFT and disentangling modular Walker-Wang models

Abstract: I will introduce simple “tensorial” definitions for many algebraic and categorical structures appearing in the classification of topological phases of matter. Such “tensorial TQFTs” will be defined as maps that associate tensors to geometric/topological objects of some type, subject to gluing axioms. Tensorial TQFTs are very directly related to microscopic physical models in terms of discrete path integrals. I will use those tensorial definitions to construct invertible boundaries which disentangle modular Walker-Wang models.

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• Member Seminar

##### Quantum Gravity constraints beyond asymptotic regimes

12:00 pm-1:00 pm
03/21/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Max Wiesner

Title: Quantum Gravity constraints beyond asymptotic regimes

Abstract: Not every effective field theory that is consistent in the absence of gravity can be completed to a consistent theory of quantum gravity. The goal of the Swampland program is to find general criteria that distinguish effective field theories, that can be obtained as a low-energy approximation of quantum gravity, from those that are inconsistent in the presence of gravity. These criteria are oftentimes motivated by patterns observed in explicit compactifications of perturbative string theory and have passed many non-trivial tests in asymptotic regions of the field space such as, e.g., weak coupling limits. Still, the Swampland criteria should equally apply to effective theories that do not arise in asymptotic regions of the field space of string theory compactifications. In this talk I will summarize some of my recent works that studies the interior of regions of the field space of string theory in the context of the Swampland program.

• CMSA EVENT: 2023 Ding Shum Lecture

##### Event2023 Ding Shum Lecture

5:00 pm-6:00 pm
03/21/2023
1 Oxford Street, Cambridge MA 02138

On March 21, 2023, the CMSA will host the fourth annual Ding Shum Lecture, given by Cynthia Dwork (Harvard SEAS and Microsoft Research).

Time: 5:00-6:00 pm ET

Location: Harvard University Science Center Hall D

This event will be held in person and via Zoom webinar.

Registration is required.

Zoom Webinar registration (link)

Title: Measuring Our Chances: Risk Prediction in This World and its Betters

Abstract: Prediction algorithms score individuals, assigning a number between zero and one that is often interpreted as an individual probability: a 0.7 “chance” that this child is in danger in the home; an 80% “probability” that this woman will succeed if hired; a 1/3 “likelihood” that they will graduate within 4 years of admission. But what do words like “chance,” “probability,” and “likelihood” actually mean for a non-repeatable activity like going to college? This is a deep and unresolved problem in the philosophy of probability. Without a compelling mathematical definition we cannot specify what an (imagined) perfect risk prediction algorithm should produce, nor even how an existing algorithm should be evaluated. Undaunted, AI and machine learned algorithms churn these numbers out in droves, sometimes with life-altering consequences.

An explosion of recent research deploys insights from the theory of pseudo-random numbers – sequences of 0’s and 1’s that “look random” but in fact have structure – to yield a tantalizing answer to the evaluation problem, together with a supporting algorithmic framework with roots in the theory of algorithmic fairness.

We can aim even higher. Both (1) our qualifications, health, and skills, which form the inputs to a prediction algorithm, and (2) our chances of future success, which are the desired outputs from the ideal risk prediction algorithm, are products of our interactions with the real world. But the real world is systematically inequitable. How, and when, can we hope to approximate probabilities not in this world, but in a better world, one for which, unfortunately, we have no data at all? Surprisingly, this novel question is inextricably bound with the very existence of nondeterminism.

Professor Cynthia Dwork is Gordon McKay Professor of Computer Science at the Harvard University John A. Paulson School of Engineering and Applied Sciences, Affiliated Faculty at Harvard Law School, and Distinguished Scientist at Microsoft. She uses theoretical computer science to place societal problems on a firm mathematical foundation.

Her recent awards and honors include the 2020 ACM SIGACT and IEEE TCMF Knuth Prize, the 2020 IEEE Hamming Medal, and the 2017 Gödel Prize.

Talk Chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)
Moderator: Faidra Monachou (Harvard CMSA)

The 2020-2022 Ding Shum lectures were postponed due to Covid-19.

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• Colloquia

##### Synchronization in a Kuramoto Mean Field Game

12:30 pm-1:30 pm
03/22/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Mete Soner (Princeton University)

Title: Synchronization in a Kuramoto Mean Field Game

Abstract:  Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a  key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system. While the system is not synchronized when this term is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter.  We explore this system in the mean field formalism.  We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles determine endogenously their behaviors by minimizing a cost functional and eventually, settling in a Nash equilibrium.  The mean field game also exhibits a bifurcation from incoherence to self-organization.  This approach has found interesting applications including circadian rhythms and jet-lag recovery.  This is joint work with Rene Carmona of Princeton and Quentin Cormier of INRIA, Paris.

• Probability Seminar

##### Some rigorous results on the Lévy spin glass model

3:30 pm-4:30 pm
03/22/2023

Probability Seminar

Speaker: Wei-Kuo Chen (Minnesota)

Title: Some rigorous results on the Lévy spin glass model

Abstract: The Lévy spin glass model, proposed by Cizeau-Bouchaud, is a mean-field model defined on a fully connected graph, where the spin interactions are formulated through a power-law distribution. This model is well-motivated from the study of the experimental metallic spin glasses. It is also expected to bridge between some mean-field and diluted models. In this talk, we will discuss some recent progress on the Lévy model including its high temperature behavior and the existence and variational expression for the limiting free energy. Based on a joint work with Heejune Kim and Arnab Sen.

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• General Relativity Seminar

##### New Phases of N=4 SYM

1:30 pm-2:30 pm
03/23/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

General Relativity Seminar

Speaker: Prahar Mitra (University of Cambridge)

Title: New Phases of N=4 SYM

Abstract: We construct new static solutions to gauged supergravity that, via the AdS/CFT correspondence, are dual to thermal phases in N=4 SYM at finite chemical potential. These solutions dominate the micro-canonical ensemble and are required to ultimately reproduce the microscopic entropy of AdS black holes. These are constructed in two distinct truncations of gauged supergravity and can be uplifted to solutions of type IIB supergravity. Together with the known phases of the truncation with three equal charges, our findings permit a good understanding of the full phase space of SYM thermal states with three arbitrary chemical potentials. We will also discuss the status of hairy supersymmetric black hole solutions in this theory.

Based on: https://arxiv.org/pdf/2207.07134.pdf [hep-th]

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• Quantum Matter

##### Traversable wormhole dynamics on a quantum processor

10:00 am-11:30 am
03/24/2023

Quantum Matter Seminar

Speaker: Alexander Zlokapa, MIT

Title: Traversable wormhole dynamics on a quantum processor

Abstract: The holographic principle, theorized to be a property of quantum gravity, postulates that the description of a volume of space can be encoded on a lower-dimensional boundary. The anti-de Sitter (AdS)/conformal field theory correspondence or duality is the principal example of holography. The Sachdev–Ye–Kitaev (SYK) model of N >> 1 Majorana fermions has features suggesting the existence of a gravitational dual in AdS2, and is a new realization of holography. We invoke the holographic correspondence of the SYK many-body system and gravity to probe the conjectured ER=EPR relation between entanglement and spacetime geometry through the traversable wormhole mechanism as implemented in the SYK model. A qubit can be used to probe the SYK traversable wormhole dynamics through the corresponding teleportation protocol. This can be realized as a quantum circuit, equivalent to the gravitational picture in the semiclassical limit of an infinite number of qubits. Here we use learning techniques to construct a sparsified SYK model that we experimentally realize with 164 two-qubit gates on a nine-qubit circuit and observe the corresponding traversable wormhole dynamics. Despite its approximate nature, the sparsified SYK model preserves key properties of the traversable wormhole physics: perfect size winding, coupling on either side of the wormhole that is consistent with a negative energy shockwave, a Shapiro time delay, causal time-order of signals emerging from the wormhole, and scrambling and thermalization dynamics. Our experiment was run on the Google Sycamore processor. By interrogating a two-dimensional gravity dual system, our work represents a step towards a program for studying quantum gravity in the laboratory. Future developments will require improved hardware scalability and performance as well as theoretical developments including higher-dimensional quantum gravity duals and other SYK-like models.

• CMSA EVENT: CMSA/MATH Bi-Annual Gathering

##### EventCMSA/MATH Bi-Annual Gathering

4:30 pm-6:00 pm
03/24/2023
20 Garden Street, Cambridge, MA 02138 USA

On Friday, March 24th, 4:30PM – 6PM, the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the Common Room at 20 Garden Street, Cambridge MA 02138.

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• Swampland Seminar

##### Recent developments on the tidal Love numbers of black holes

11:00 am-12:00 pm
03/27/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Swampland Seminar

Speaker: Valerio De Luca (UPenn)

Title: Recent developments on the tidal Love numbers of black holes

Abstract: Tidal Love numbers describe the deformability of compact objects under the presence of external tidal perturbations, and are found to be exactly zero for black holes in pure General Relativity. This property is however fragile, since they receive corrections from higher-order derivative terms in the theory. We show that the tidal deformability of neutral black holes is constrained by the Weak Gravity Conjecture.

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• Member Seminar

##### Member Seminar Title TBA

12:00 pm-1:00 pm
03/28/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Jue Liu

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• Colloquia

##### Title TBA

12:30 pm-1:30 pm
03/29/2023
CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Speaker: Ruth Britto, Princeton

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• Quantum Matter

##### A Plane Defect in the 3d O(N) Model

10:00 am-11:30 am
03/31/2023
Virtual and in 20 Garden Street, Room G10

Quantum Matter Seminar

Speaker: Abijith Krishnan (MIT)

Title: A Plane Defect in the 3d O(N) Model

Abstract: It was recently found that the classical 3d O(N) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as (log x)^(-q). This universality class exists for a range 2≤N<Nc and Monte-Carlo simulations and conformal bootstrap indicate Nc>3. In this talk, I’ll extend this result to the 3d O(N) model in an infinite geometry with a plane defect. I’ll explain using the renormalization group (RG) that the extraordinary-log universality class is present for any finite N≥2, and that a line of defect fixed points is present at N=∞. This line of defect fixed points is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N corrections. I’ll show that the line of defect fixed points and the 1/N corrections agree with an a-theorem by Jensen and O’Bannon for 3d CFTs with a boundary. Finally, I’ll conclude by noting some physical systems where the extraordinary-log universality class can be observed.

April