Insights from single cell lineage trees
2023-05-11 13:00 - 14:00 (America/New_York)
Active Matter Seminar

Speaker: Sahand Hormoz, Harvard Medical School, Dana-Farber Cancer Institute

Title: Insights from single cell lineage trees

Abstract: In this talk, I will discuss two recent projects from my lab that involve lineage trees of cells (the branching diagram that represents the ancestry and division history of individual cells). In the first project, we reconstructed the lineage trees of individual cancer cells from the patterns of randomly occurring mutations in these cells. We then inferred the age at which the cancer mutation first occurred and the rate of expansion of the population of cancer cells within each patient. To our surprise, we discovered that the cancer mutation occurs decades before diagnosis. For the second project, we developed microfluidic ‘mother machines’ that allow us to observe mammalian cells dividing across tens of generations. Using our observations, we calculated the correlation between the duration of cell cycle phases in pairs of cells, as a function of their lineage distance. These correlations revealed many surprises that we are trying to understand using hidden Markov models on trees. For both projects, I will discuss the mathematical challenges that we have faced and open problems related to inference from lineage trees.

How do the eigenvalues of a large non-Hermitian random matrix behave?
2023-05-11 13:30 - 14:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Probability Seminar
Speaker: Giorgio Cipolloni (Princeton)
Title: How do the eigenvalues of a large non-Hermitian random matrix behave?
Abstract: We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X.
Location: Science Center Room 232

Positivity of Static quasi-local Mass in general relativity
2023-05-11 13:30 - 14:30 | Virtual
General Relativity Seminar
Speaker: Aghil Alaee, Clark University
Title: Positivity of Static quasi-local Mass in general relativity
Abstract: In this talk, we review results on the PMT of quasi-local masses and prove the positivity of static quasi-local masses with respect to the AdS and AdS Schwarzschild spacetimes.

Anomalies of (1+1)D categorical symmetries
2023-05-12 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Carolyn Zhang (U Chicago)
Title: Anomalies of (1+1)D categorical symmetries
Abstract: We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category $A$ describes a $(2+1)D$ topological order whose gapped boundaries enumerate all $(1+1)D$ gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the emph{electric} Lagrangian algebra, that describes a phase with $A$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $A$. We observe that if there exists a different gapped boundary, given by a emph{magnetic} Lagrangian algebra, then there exists a gapped phase where $A$ is not spontaneously broken at all, which means that $A$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $A$ being anomaly-free. As an application, we consider the Drinfeld centers of $mathbb{Z}_Ntimesmathbb{Z}_N$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.

Quantum information and extended topological quantum field theory
2023-05-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Gabriel Wong
Title: Quantum information and extended topological quantum field theory
Abstract: Recently, ideas from quantum information theory have played an important role in condensed matter and quantum gravity research. Most of these applications focus on the entanglement structure of quantum states, and the computation of entanglement measures such as entanglement entropy has been an essential part of the story. In this talk, we will address some subtleties that arise when trying to define entanglement entropy in quantum field theory and quantum gravity. In particular, we will explain why extended topological field theory provides a useful framework to define and compute entanglement entropy in a continuous system. Time permitting, we will explain some recent applications of these ideas in low dimensional quantum gravity and to topological string theory.

GRAMSIA: Graphical Models, Statistical Inference, and Algorithms
2023-05-16 09:00 - 2023-05-19 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On May 16 – May 19, 2023 the CMSA hosted a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop was organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali (Carnegie Mellon).
The purpose of this workshop is to highlight various mathematical questions and issues associated with graphical models and message-passing algorithms, and to bring together a group of researchers for discussion of the latest progress and challenges ahead. In addition to the substantial impact of graphical models on applied areas, they are also connected to various branches of the mathematical sciences. Rather than focusing on the applications, the primary goal is to highlight and deepen these mathematical connections.
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Speakers:

Jake Abernethy (Georgia Tech)
Guy Bresler (MIT)
Florent Krzakala (Ecole Polytechnique Federale de Lausanne)
Lester Mackey (Microsoft Research New England)
Theo McKenzie (Harvard)
Andrea Montanari (Stanford)
Elchanan Mossel (MIT)
Yury Polyanskiy (MIT)
Patrick Rebeschini (Oxford)
Subhabrata Sen (Harvard)
Devavrat Shah (MIT)
Pragya Sur (Harvard)
Alex Wein (UC Davis)
Yihong Wu (Yale)
Sarath Yasodharan (Brown)
Horng-Tzer Yau (Harvard)
Christina Lee Yu (Cornell)
Ilias Zadik (MIT)

Schedule:
Tuesday, May 16, 2023

9:00 am
Breakfast

9:15 – 9:30 am
Introductory remarks by organizers

9:30 – 10:20 am
Subhabrata Sen (Harvard)
Title: Mean-field approximations for high-dimensional Bayesian regression
Abstract: Variational approximations provide an attractive computational alternative to MCMC-based strategies for approximating the posterior distribution in Bayesian inference. The Naive Mean-Field (NMF) approximation is the simplest version of this strategy—this approach approximates the posterior in KL divergence by a product distribution. There has been considerable progress recently in understanding the accuracy of NMF under structural constraints such as sparsity, but not much is known in the absence of such constraints. Moreover, in some high-dimensional settings, the NMF is expected to be grossly inaccurate, and advanced mean-field techniques (e.g. Bethe approximation) are expected to provide accurate approximations. We will present some recent work in understanding this duality in the context of high-dimensional regression. This is based on joint work with Sumit Mukherjee (Columbia) and Jiaze Qiu (Harvard University).

10:30 – 11:00 am
Coffee break

11:00 – 11:50 am
Elchanan Mossel (MIT)
Title: Some modern perspectives on the Kesten-Stigum bound for reconstruction on trees.
Abstract: The Kesten-Stigum bound is a fundamental spectral bound for reconstruction on trees. I will discuss some conjectures and recent progress on understanding when it is tight as well as some conjectures and recent progress on what it signifies even in cases where it is not tight.

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Christina Lee Yu (Cornell)
Title: Exploiting Neighborhood Interference with Low Order Interactions under Unit Randomized Design
Abstract: Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in many real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we utilize knowledge of the network structure to provide an unbiased estimator for the TTE when network interference effects are constrained to low order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. Central to our contribution is a new framework for balancing between model flexibility and statistical complexity as captured by this low order interactions structure.

3:00 – 3:30 pm
Coffee break

3:30 – 4:20 pm
Theo McKenzie (Harvard)
Title: Spectral statistics for sparse random graphs
Abstract: Understanding the eigenvectors and eigenvalues of the adjacency matrix of random graphs is fundamental to many algorithmic questions; moreover, it is related to longstanding questions in quantum physics. In this talk we focus on random models of sparse graphs, giving some properties that are unique to these sparse graphs, as well as some specific obstacles. Based on this, we show some new results on spectral statistics of sparse random graphs, as well as some conjectures.

4:40 – 6:30 pm
Lightning talk session + welcome reception

Wednesday, May 17, 2023

9:00 am
Breakfast

9:30 – 10:20
Ilias Zadik (MIT)
Title: Revisiting Jerrum’s Metropolis Process for the Planted Clique Problem
Abstract: Jerrum in 1992 (co-)introduced the planted clique model by proving the (worst-case initialization) failure of the Metropolis process to recover any o(sqrt(n))-sized clique planted in the Erdos-Renyi graph G(n,1/2). This result is classically cited in the literature of the problem, as the “first evidence” the o(sqrt(n))-sized planted clique recovery task is “algorithmically hard”.
In this work, we show that the Metropolis process actually fails to work (under worst-case initialization) for any o(n)-sized planted clique, that is the failure applies well beyond the sqrt(n) “conjectured algorithmic threshold”. Moreover we also prove, for a large number of temperature values, that the Metropolis process fails also under “natural initialization”, resolving an open question posed by Jerrum in 1992.

10:30 – 11:00
Coffee break

11:00 – 11:50
Florent Krzakala (Ecole Polytechnique Federale de Lausanne)
Title: Are Gaussian data all you need for machine learning theory?
Abstract: Clearly, no! Nevertheless, the Gaussian assumption remains prevalent among theoreticians, particularly in high-dimensional statistics and physics, less so in traditional statistical learning circles. To what extent are Gaussian features merely a convenient choice for certain theoreticians, or genuinely an effective model for learning? In this talk, I will review recent progress on these questions, achieved using rigorous probabilistic approaches in high-dimension and techniques from mathematical statistical physics. I will demonstrate that, despite its apparent limitations, the Gaussian approach is sometimes much closer to reality than one might expect. In particular, I will discuss key findings from a series of recent papers that showcase the Gaussian equivalence of generative models, the universality of Gaussian mixtures, and the conditions under which a single Gaussian can characterize the error in high-dimensional estimation. These results illuminate the strengths and weaknesses of the Gaussian assumption, shedding light on its applicability and limitations in the realm of theoretical statistical learning.

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Andrea Montanari (Stanford)
Title: Dimension free ridge regression
Abstract: Random matrix theory has become a widely useful tool in high-dimensional statistics and theoretical machine learning. However, random matrix theory is largely focused on the proportional asymptotics in which the number of columns grows proportionally to the number of rows of the data matrix. This is not always the most natural setting in statistics where columns correspond to covariates and rows to samples. With the objective to move beyond the proportional asymptotics, we revisit ridge regression. We allow the feature vector to be high-dimensional, or even infinite-dimensional, in which case it belongs to a separable Hilbert space and assume it to satisfy a certain convex concentration property. Within this setting, we establish non-asymptotic bounds that approximate the bias and variance of ridge regression in terms of the bias and variance of an ‘equivalent’ sequence model (a regression model with diagonal design matrix). Previously, such an approximation result was known only in the proportional regime and only up to additive errors: in particular, it did not allow to characterize the behavior of the excess risk when this converges to 0. Our general theory recovers earlier results in the proportional regime (with better error rates). As a new application, we obtain a completely explicit and sharp characterization of ridge regression for Hilbert covariates with regularly varying spectrum. Finally, we analyze the overparametrized near-interpolation setting and obtain sharp ‘benign overfitting’ guarantees.
[Based on joint work with Chen Cheng]

3:00 – 3:50 pm
Yury Polyanskiy (MIT)
Title: Recent results on broadcasting on trees and stochastic block model
Abstract: I will survey recent results and open questions regarding the q-ary stochastic block model and its local version (broadcasting on trees, or BOT). For example, establishing uniqueness of non-trivial solution to distribution recursions (BP fixed point) implies a characterization for the limiting mutual information between the graph and community labels. For q=2 uniqueness holds in all regimes. For q>2 uniqueness is currently only proved above a certain threshold that is asymptotically (for large q) is close to Kesten-Stigum (KS) threshold. At the same time between the BOT reconstruction and KS we show that uniqueness does not hold, at least in the presence of (arbitrary small) vertex-level side information. I will also discuss extension of the robust reconstruction result of Janson-Mossel’2004.
Based on joint works with Qian Yu (Princeton) and Yuzhou Gu (MIT).

4:00 – 4:30 pm
Coffee break

4:30 – 5:20 pm
Alex Wein (UC Davis)
Title: Is Planted Coloring Easier than Planted Clique?
Abstract: The task of finding a planted clique in the random graph G(n,1/2) is perhaps the canonical example of a statistical-computational gap: for some clique sizes, the task is statistically possible but believed to be computationally hard. Really, there are multiple well-studied tasks related to the planted clique model: detection, recovery, and refutation. While these are equally difficult in the case of planted clique, this need not be true in general. In the related planted coloring model, I will discuss the computational complexity of these three tasks and the interplay among them. Our computational hardness results are based on the low-degree polynomial model of computation.By taking the complement of the graph, the planted coloring model is analogous to the planted clique model but with many planted cliques. Here our conclusion is that adding more cliques makes the detection problem easier but not the recovery problem.

Thursday, May 18, 2023

9:00
Breakfast

9:30 – 10:20
Guy Bresler (MIT)
Title: Algorithmic Decorrelation and Planted Clique in Dependent Random Graphs
Abstract: There is a growing collection of average-case reductions starting from Planted Clique (or Planted Dense Subgraph) and mapping to a variety of statistics problems, sharply characterizing their computational phase transitions. These reductions transform an instance of Planted Clique, a highly structured problem with its simple clique signal and independent noise, to problems with richer structure. In this talk we aim to make progress in the other direction: to what extent can these problems, which often have complicated dependent noise, be transformed back to Planted Clique? Such a bidirectional reduction between Planted Clique and another problem shows a strong computational equivalence between the two problems. We develop a new general framework for reasoning about the validity of average-case reductions based on low sensitivity to perturbations. As a concrete instance of our general result, we consider the planted clique (or dense subgraph) problem in an ambient graph that has dependent edges induced by randomly adding triangles to the Erdos-Renyi graph G(n,p), and show how to successfully eliminate dependence by carefully removing the triangles while approximately preserving the clique (or dense subgraph). Joint work with Chenghao Guo and Yury Polyanskiy.

10:30 – 11:00
Coffee break

11:00 – 11:50
Sarath Yasodharan (Brown)
Title: A Sanov-type theorem for unimodular marked random graphs and its applications
Abstract: We prove a Sanov-type large deviation principle for the component empirical measures of certain sequences of unimodular random graphs (including Erdos-Renyi and random regular graphs) whose vertices are marked with i.i.d. random variables. Specifically, we show that the rate function can be expressed in a fairly tractable form involving suitable relative entropy functionals. As a corollary, we establish a variational formula for the annealed pressure (or limiting log partition function) for various statistical physics models on sparse random graphs. This is joint work with I-Hsun Chen and Kavita Ramanan.

12:00 – 12:15 pm
12:15 – 2:00 pm
Group Photo
Lunch

2:00 – 2:50 pm
Patrick Rebeschini (Oxford)
Title: Implicit regularization via uniform convergence
Abstract: Uniform convergence is one of the main tools to analyze the complexity of learning algorithms based on explicit regularization, but it has shown limited applicability in the context of implicit regularization. In this talk, we investigate the statistical guarantees on the excess risk achieved by early-stopped mirror descent run on the unregularized empirical risk with the squared loss for linear models and kernel methods. We establish a direct link between the potential-based analysis of mirror descent from optimization theory and uniform learning. This link allows characterizing the statistical performance of the path traced by mirror descent directly in terms of localized Rademacher complexities of function classes depending on the choice of the mirror map, initialization point, step size, and the number of iterations. We will discuss other results along the way.

3:00 – 3:50 pm
Pragya Sur (Harvard)
Title: A New Central Limit Theorem for the Augmented IPW estimator in high dimensions
Abstract: Estimating the average treatment effect (ATE) is a central problem in causal inference. Modern advances in the field studied estimation and inference for the ATE in high dimensions through a variety of approaches. Doubly robust estimators such as the augmented inverse probability weighting (AIPW) form a popular approach in this context. However, the high-dimensional literature surrounding these estimators relies on sparsity conditions, either on the outcome regression (OR) or the propensity score (PS) model. This talk will introduce a new central limit theorem for the classical AIPW estimator, that applies agnostic to such sparsity-type assumptions. Specifically, we will study properties of the cross-fit version of the estimator under well-specified OR and PS models, and the proportional asymptotics regime where the number of confounders and sample size diverge proportional to each other. Under assumptions on the covariate distribution, our CLT will uncover two crucial phenomena among others: (i) the cross-fit AIPW exhibits a substantial variance inflation that can be quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the estimators used while cross-fitting is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts, and open a vista of possibilities for studying similar other high-dimensional effects. On the technical front, our work utilizes a novel interplay between three distinct tools—approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach.

4:00 – 4:30 pm
Coffee break

4:30 – 5:20 pm
Yihong Wu (Yale)
Title: Random graph matching at Otter’s threshold via counting chandeliers

Abstract: We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erdős–Rényi graphs G(n,q) whose edges are correlated through a latent vertex correspondence, we show that this algorithm correctly matches all but a vanishing fraction of the vertices with high probability, provided that nqtoinfty and the edge correlation coefficient ρ satisfies ρ^2>α≈0.338, where α is Otter’s tree-counting constant. Moreover, this almost exact matching can be made exact under an extra condition that is information-theoretically necessary. This is the first polynomial-time graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs. In comparison, previous methods either require ρ=1−o(1) or are restricted to sparse graphs. The crux of the algorithm is a carefully curated family of rooted trees called chandeliers, which allows effective extraction of the graph correlation from the counts of the same tree while suppressing the undesirable correlation between those of different trees. This is joint work with Cheng Mao, Jiaming Xu, and Sophie Yu, available at https://arxiv.org/abs/2209.12313

Friday, May 19, 2023

9:00
Breakfast

9:30 – 10:20
Jake Abernethy (Georgia Tech)
Title: Optimization, Learning, and Margin-maximization via Playing Games
Abstract: A very popular trick for solving certain types of optimization problems is this: write your objective as the solution of a two-player zero-sum game, endow both players with an appropriate learning algorithm, watch how the opponents compete, and extract an (approximate) solution from the actions/decisions taken by the players throughout the process. This approach is very generic and provides a natural template to produce new and interesting algorithms. I will describe this framework and show how it applies in several scenarios, including optimization, learning, and margin-maximiation problems. Along the way we will encounter a number of novel tools and rediscover some classical ones as well.

10:30 – 11:00
Coffee break

11:00 – 11:50
Devavrat Shah (MIT)
Title: On counterfactual inference with unobserved confounding via exponential family
Abstract: We are interested in the problem of unit-level counterfactual inference with unobserved confounders owing to the increasing importance of personalized decision-making in many domains: consider a recommender system interacting with a user over time where each user is provided recommendations based on observed demographics, prior engagement levels as well as certain unobserved factors. The system adapts its recommendations sequentially and differently for each user. Ideally, at each point in time, the system wants to infer each user’s unknown engagement if it were exposed to a different sequence of recommendations while everything else remained unchanged. This task is challenging since: (a) the unobserved factors could give rise to spurious associations, (b) the users could be heterogeneous, and (c) only a single trajectory per user is available.
We model the underlying joint distribution through an exponential family. This reduces the task of unit-level counterfactual inference to simultaneously learning a collection of distributions of a given exponential family with different unknown parameters with single observation per distribution. We discuss a computationally efficient method for learning all of these parameters with estimation error scaling linearly with the metric entropy of the space of unknown parameters – if the parameters are an s-sparse linear combination of k known vectors in p dimension, the error scales as O(s log k/p). En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality.
Based on a joint work with Raaz Dwivedi, Abhin Shah and Greg Wornell (all at MIT) with manuscript available here: https://arxiv.org/abs/2211.08209

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Lester Mackey (Microsoft Research New England)
Title: Advances in Distribution Compression
Abstract: This talk will introduce two new tools for summarizing a probability distribution more effectively than independent sampling or standard Markov chain Monte Carlo thinning:
1. Given an initial n-point summary (for example, from independent sampling or a Markov chain), kernel thinning finds a subset of only square-root n-points with comparable worst-case integration error across a reproducing kernel Hilbert space.
2. If the initial summary suffers from biases due to off-target sampling, tempering, or burn-in, Stein thinning simultaneously compresses the summary and improves the accuracy by correcting for these biases.
These tools are especially well-suited for tasks that incur substantial downstream computation costs per summary point like organ and tissue modeling in which each simulation consumes 1000s of CPU hours.
Based on joint work with Raaz Dwivedi, Marina Riabiz, Wilson Ye Chen, Jon Cockayne, Pawel Swietach, Steven A. Niederer, Chris. J. Oates, Abhishek Shetty, and Carles Domingo-Enrich.

3:00 – 3:30 pm
Coffee break

3:30 – 4:20 pm
Horng-Tzer Yau (Harvard)
Title: On the spectral gap of mean-field spin glass models.
Abstract: We will discuss recent progress regarding spectral gaps for the Glauber dynamics of spin glasses at high temperature. In addition, we will also report on estimating the operator norm of the covariance matrix for the SK model.

Moderators: Benjamin McKenna, Harvard CMSA & Changji Xu, Harvard CMSA

CMSA COVID-19 Policies

Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24
2023-06-09 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speakers: Gerald Höhn (Kansas State University) & Sven Möller (University of Hamburg)
Title: Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24
Abstract: We discuss the classfication of self-dual vertex operator superalgebras (SVOAs) of central charge 24, or in physics parlance the purely chiral 2-dimensional fermionic conformal field theories with just one primary field.
There are exactly 969 such SVOAs under suitable regularity assumptions and the assumption that the shorter moonshine module VB^# is the unique self-dual SVOA of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish.
We construct and classify the self-dual SVOAs by determining the 2-neighbourhood graph of the self-dual (purely bosonic) VOAs of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice VOA. We show that all SVOAs besides VB^# x F and potential fake copies thereof stem from elements of the Conway group Co_0, the automorphism group of the Leech lattice.
By splitting off free fermions F, if possible, we obtain the classification for all central charges less than or equal to 24.
Reference: G. Höhn, S. Möller, arXiv:2303.17190.

Small Bosonic CFTs, Chiral Fermionization, and Symmetry/Subalgebra Duality
2023-06-13 10:00 - 12:00 | Virtual
Quantum Matter Seminar
Speaker: Brandon C. Rayhaun (C. N. Yang ITP, Stony Brook University)
Title: Small Bosonic CFTs, Chiral Fermionization, and Symmetry/Subalgebra Duality
Abstract: Conformal field theories in (1+1)D are key actors in many dramas of physics and mathematics. Their classification has therefore been an important and long-standing problem. In this talk, I will explain the main ideas behind the classification of (most) “small” bosonic CFTs. Here, I use the adjective “small” informally to refer to theories with low central charge (less than 24) and few primary operators (less than 5). Time and attention permitting, I will highlight two applications of this result. First, I will describe how it can be used in tandem with bosonization and fermionization techniques to establish the classification of chiral fermionic CFTs with central charge less than 23. Second, I will showcase how it can be used to bootstrap generalized global symmetries using the concept of “symmetry/subalgebra duality.”
Talk based on arXiv:2208.05486 [hep-th] (joint work with Sunil Mukhi) and arXiv:2303.16921 [hep-th].

Chiral fermionic CFTs of central charge ≤ 16
2023-06-26 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Title: Chiral fermionic CFTs of central charge ≤ 16
Abstract: We classified all chiral fermionic CFTs of central charge ≤ 16 using Kac’s theorem and bosonization/fermionization. This talk will discuss the derivation of this result, its application to the classification of non-supersymmetric heterotic string theories, and along the way we’ll address some oft-overlooked subtleties of bosonization from the point of view of anomalies and topological phases.

Monopoles, Scattering, and Generalized Symmetries
2023-06-30 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: Marieke Van Beest (SCGP)
Title: Monopoles, Scattering, and Generalized Symmetries
Abstract: In this talk, we will discuss the problem of electrically charged, massless fermions scattering off magnetic monopoles. The interpretation of the outgoing states has long been a puzzle, as they can carry fractional quantum numbers. We argue that such outgoing particles live in the twisted sector of a topological co-dimension 1 surface, which ends topologically on the monopole. This symmetry defect is often non-invertible, and as such the outgoing radiation not only carries unconventional flavor quantum numbers, but is often trailed by a topological field theory, which is a new prediction.

Grey Galaxy’ as the endpoint of the Kerr-AdS super radiant blackhole
2023-07-05 14:00 - 15:00 | Jefferson 453, 17 Oxford St, Cambridge, MA 02138, MA
General Relativity Seminar
Speaker: Suman Kundu (Weizmann Institute)
Title: ‘Grey Galaxy’ as the endpoint of the Kerr-AdS super radiant blackhole
Abstract: Kerr AdS$_{d+1}$ black holes for $dgeq 3$ suffer from classical superradiant instabilities over a range of masses near extremality. We conjecture that these instabilities settle down into Grey Galaxies (GG)s – a new class of solutions to Einstein’s equations which we construct for $d=3$. Grey Galaxies consist of an $omega=1$ black hole in the `centre’ of $AdS$, surrounded by a uniformly thick and very large disk of thermal bulk matter that revolves around the centre of AdS at the speed of light. The parametrically low energy density and parametrically large radius of the gas disk are inversely related; as a consequence, the gas carries a finite fraction of the total energy. Grey Galaxy saddles exist at masses that extend all the way down to the unitarity bound. Their thermodynamics is that of a weakly interacting mix of Kerr AdS black holes and the gas. In addition to a smooth piece, the boundary stress tensor of these solutions includes a contribution from a delta function localized at the `equator’ of the boundary sphere, a term which may be used as an order parameter that sharply distinguishes GG solutions from ordinary Kerr-Black hole saddles. We also construct `Revolving Black Hole (RBH) saddles’, macroscopically charged $SO(d,2)$ descendants of AdS-Kerr solutions, that describe black holes revolving around the centre of $AdS$, at the fixed radial location but in a quantum wave function in the angular directions. RBH saddles turn out to be (marginally) entropically subdominant to GG saddles. We argue that supersymmetric versions of RBH saddles exist and have interesting consequences for the spectrum of SUSY states in, e.g. ${cal N}=4$ Yang-Mills theory.

Two of my favorite numbers: 8 and 24
2023-08-24 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Speaker: John Baez (University of California, Riverside)
Title: Two of my favorite numbers: 8 and 24
Abstract: The numbers 8 and 24 play special roles in mathematics. The number 8 is special because of Bott periodicity, the octonions and the E8 lattice, while 24 is special for many reasons, including the binary tetrahedral group, the 3rd stable homotopy group of spheres, and the Leech lattice. The number 8 does for superstring theory what the number 24 does for bosonic string theory. In this talk, which is intended to be entertaining, I will overview these matters and also some connections between the numbers 8 and 24.

Big Data Conference 2023
2023-08-31 09:00 - 2023-09-01 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On August 31-Sep 1, 2023 the CMSA hosted the ninth annual Conference on Big Data. The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Speakers:

Jacob Andreas, MIT
Morgane Austern, Harvard
Albert-László Barabási, Northeastern
Rachel Cummings, Columbia
Melissa Dell, Harvard
Jianqing Fan, Princeton
Tommi Jaakkola, MIT
Ankur Moitra, MIT
Mark Sellke, Harvard
Marinka Zitnik, Harvard Medical School

Organizers:

Michael Douglas, CMSA, Harvard University
Yannai Gonczarowski, Economics and Computer Science, Harvard University
Lucas Janson, Statistics and Computer Science, Harvard University
Tracy Ke, Statistics, Harvard University
Horng-Tzer Yau, Mathematics and CMSA, Harvard University
Yue Lu, Electrical Engineering and Applied Mathematics, Harvard University

Schedule
(PDF download)
Thursday, August 31, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Albert-László Barabási (Northeastern, Harvard)
Title: From Network Medicine to the Foodome: The Dark Matter of Nutrition
Abstract: A disease is rarely a consequence of an abnormality in a single gene but reflects perturbations to the complex intracellular network. Network medicine offer a platform to explore systematically not only the molecular complexity of a particular disease, leading to the identification of disease modules and pathways, but also the molecular relationships between apparently distinct (patho) phenotypes. As an application, I will explore how we use network medicine to uncover the role individual food molecules in our health. Indeed, our current understanding of how diet affects our health is limited to the role of 150 key nutritional components systematically tracked by the USDA and other national databases in all foods. Yet, these nutritional components represent only a tiny fraction of the over 135,000 distinct, definable biochemicals present in our food. While many of these biochemicals have documented effects on health, they remain unquantified in any systematic fashion across different individual foods. Their invisibility to experimental, clinical, and epidemiological studies defines them as the ‘Dark Matter of Nutrition.’ I will speak about our efforts to develop a high-resolution library of this nutritional dark matter, and efforts to understand the role of these molecules on health, opening novel avenues by which to understand, avoid, and control disease.
https://youtu.be/UmgzUwi6K3E

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Rachel Cummings (Columbia)
Title: Differentially Private Algorithms for Statistical Estimation Problems
Abstract: Differential privacy (DP) is widely regarded as a gold standard for privacy-preserving computation over users’ data. It is a parameterized notion of database privacy that gives a rigorous worst-case bound on the information that can be learned about any one individual from the result of a data analysis task. Algorithmically it is achieved by injecting carefully calibrated randomness into the analysis to balance privacy protections with accuracy of the results.
In this talk, we will survey recent developments in the development of DP algorithms for three important statistical problems, namely online learning with bandit feedback, causal interference, and learning from imbalanced data. For the first problem, we will show that Thompson sampling — a standard bandit algorithm developed in the 1930s — already satisfies DP due to the inherent randomness of the algorithm. For the second problem of causal inference and counterfactual estimation, we develop the first DP algorithms for synthetic control, which has been used non-privately for this task for decades. Finally, for the problem of imbalanced learning, where one class is severely underrepresented in the training data, we show that combining existing techniques such as minority oversampling perform very poorly when applied as pre-processing before a DP learning algorithm; instead we propose novel approaches for privately generating synthetic minority points.
Based on joint works with Marco Avella Medina, Vishal Misra, Yuliia Lut, Tingting Ou, Saeyoung Rho, and Ethan Turok.
https://youtu.be/0cPE6rb1Roo

12:00–1:30 PM
Lunch

1:30–2:30 PM
Morgane Austern (Harvard)
Title: To split or not to split that is the question: From cross validation to debiased machine learning
Abstract: Data splitting is a ubiquitous method in statistics with examples ranging from cross-validation to cross-fitting. However, despite its prevalence, theoretical guidance regarding its use is still lacking. In this talk, we will explore two examples and establish an asymptotic theory for it. In the first part of this talk, we study the cross-validation method, a ubiquitous method for risk estimation, and establish its asymptotic properties for a large class of models and with an arbitrary number of folds. Under stability conditions, we establish a central limit theorem and Berry-Esseen bounds for the cross-validated risk, which enable us to compute asymptotically accurate confidence intervals. Using our results, we study the statistical speed-up offered by cross-validation compared to a train-test split procedure. We reveal some surprising behavior of the cross-validated risk and establish the statistically optimal choice for the number of folds. In the second part of this talk, we study the role of cross-fitting in the generalized method of moments with moments that also depend on some auxiliary functions. Recent lines of work show how one can use generic machine learning estimators for these auxiliary problems, while maintaining asymptotic normality and root-n consistency of the target parameter of interest. The literature typically requires that these auxiliary problems are fitted on a separate sample or in a cross-fitting manner. We show that when these auxiliary estimation algorithms satisfy natural leave-one-out stability properties, then sample splitting is not required. This allows for sample reuse, which can be beneficial in moderately sized sample regimes.
https://youtu.be/L_pHxgoQSgU

2:30–2:45 PM
Break

2:45–3:45 PM
Ankur Moitra (MIT)
Title: Learning from Dynamics
Abstract: Linear dynamical systems are the canonical model for time series data. They have wide-ranging applications and there is a vast literature on learning their parameters from input-output sequences. Moreover they have received renewed interest because of their connections to recurrent neural networks.
But there are wide gaps in our understanding. Existing works have only asymptotic guarantees or else make restrictive assumptions, e.g. that preclude having any long-range correlations. In this work, we give a new algorithm based on the method of moments that is computationally efficient and works under essentially minimal assumptions. Our work points to several missed connections, whereby tools from theoretical machine learning including tensor methods, can be used in non-stationary settings.
https://youtu.be/UmgzUwi6K3E

3:45–4:00 PM
Break

4:00–5:00 PM
Mark Sellke (Harvard)
Title: Algorithmic Thresholds for Spherical Spin Glasses
Abstract: High-dimensional optimization plays a crucial role in modern statistics and machine learning. I will present recent progress on non-convex optimization problems with random objectives, focusing on the spherical p-spin glass. This model is related to spiked tensor estimation and has been studied in probability and physics for decades. We will see that a natural class of “stable” optimization algorithms gets stuck at an algorithmic threshold related to geometric properties of the landscape. The algorithmic threshold value is efficiently attained via Langevin dynamics or by a second-order ascent method of Subag. Much of this picture extends to other models, such as random constraint satisfaction problems at high clause density.
https://youtu.be/JoghiwiIbT8

6:00 – 8:00 PM
Banquet for organizers and speakers

Friday, September 1, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Jacob Andreas (MIT)
Title: What Learning Algorithm is In-Context Learning?
Abstract: Neural sequence models, especially transformers, exhibit a remarkable capacity for “in-context” learning. They can construct new predictors from sequences of labeled examples (x,f(x)) presented in the input without further parameter updates. I’ll present recent findings suggesting that transformer-based in-context learners implement standard learning algorithms implicitly, by encoding smaller models in their activations, and updating these implicit models as new examples appear in the context, using in-context linear regression as a model problem. First, I’ll show by construction that transformers can implement learning algorithms for linear models based on gradient descent and closed-form ridge regression. Second, I’ll show that trained in-context learners closely match the predictors computed by gradient descent, ridge regression, and exact least-squares regression, transitioning between different predictors as transformer depth and dataset noise vary, and converging to Bayesian estimators for large widths and depths. Finally, we present preliminary evidence that in-context learners share algorithmic features with these predictors: learners’ late layers non-linearly encode weight vectors and moment matrices. These results suggest that in-context learning is understandable in algorithmic terms, and that (at least in the linear case) learners may rediscover standard estimation algorithms. This work is joint with Ekin Akyürek at MIT, and Dale Schuurmans, Tengyu Ma and Denny Zhou at Stanford.
https://youtu.be/UNVl64G3BzA

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Tommi Jaakkola (MIT)
Title: Generative modeling and physical processes
Abstract: Rapidly advancing deep distributional modeling techniques offer a number of opportunities for complex generative tasks, from natural sciences such as molecules and materials to engineering. I will discuss generative approaches inspired from physical processes including diffusion models and more recent electrostatic models (Poisson flow), and how they relate to each other in terms of embedding dimension. From the point of view of applications, I will highlight our recent work on SE(3) invariant distributional modeling over backbone 3D structures with ability to generate designable monomers without relying on pre-trained protein structure prediction methods as well as state of the art image generation capabilities (Poisson flow). Time permitting, I will also discuss recent analysis of efficiency of sample generation in such models.
https://youtu.be/GLEwQAWQ85E

12:00–1:30 PM
Lunch

1:30–2:30 PM
Marinka Zitnik (Harvard Medical School)
Title: Multimodal Learning on Graphs
Abstract: Understanding biological and natural systems requires modeling data with underlying geometric relationships across scales and modalities such as biological sequences, chemical constraints, and graphs of 3D spatial or biological interactions. I will discuss unique challenges for learning from multimodal datasets that are due to varying inductive biases across modalities and the potential absence of explicit graphs in the input. I will describe a framework for structure-inducing pretraining that allows for a comprehensive study of how relational structure can be induced in pretrained language models. We use the framework to explore new graph pretraining objectives that impose relational structure in the induced latent spaces—i.e., pretraining objectives that explicitly impose structural constraints on the distance or geometry of pretrained models. Applications in genomic medicine and therapeutic science will be discussed. These include TxGNN, an AI model enabling zero-shot prediction of therapeutic use across over 17,000 diseases, and PINNACLE, a contextual graph AI model dynamically adjusting its outputs to contexts in which it operates. PINNACLE enhances 3D protein structure representations and predicts the effects of drugs at single-cell resolution.
https://youtu.be/hjt4nsN_8iM

2:30–2:45 PM
Break

2:45–3:45 PM
Jianqing Fan (Princeton)
Title: UTOPIA: Universally Trainable Optimal Prediction Intervals Aggregation
Abstract: Uncertainty quantification for prediction is an intriguing problem with significant applications in various fields, such as biomedical science, economic studies, and weather forecasts. Numerous methods are available for constructing prediction intervals, such as quantile regression and conformal predictions, among others. Nevertheless, model misspecification (especially in high-dimension) or sub-optimal constructions can frequently result in biased or unnecessarily-wide prediction intervals. In this work, we propose a novel and widely applicable technique for aggregating multiple prediction intervals to minimize the average width of the prediction band along with coverage guarantee, called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). The method also allows us to directly construct predictive bands based on elementary basis functions. Our approach is based on linear or convex programming which is easy to implement. All of our proposed methodologies are supported by theoretical guarantees on the coverage probability and optimal average length, which are detailed in this paper. The effectiveness of our approach is convincingly demonstrated by applying it to synthetic data and two real datasets on finance and macroeconomics. (Joint work Jiawei Ge and Debarghya Mukherjee).
https://youtu.be/WY6dr1oEOrk

3:45–4:00 PM
Break

4:00–5:00 PM
Melissa Dell (Harvard)
Title: Efficient OCR for Building a Diverse Digital History
Abstract: Many users consult digital archives daily, but the information they can access is unrepresentative of the diversity of documentary history. The sequence-to-sequence architecture typically used for optical character recognition (OCR) – which jointly learns a vision and language model – is poorly extensible to low-resource document collections, as learning a language-vision model requires extensive labeled sequences and compute. This study models OCR as a character-level image retrieval problem, using a contrastively trained vision encoder. Because the model only learns characters’ visual features, it is more sample-efficient and extensible than existing architectures, enabling accurate OCR in settings where existing solutions fail. Crucially, it opens new avenues for community engagement in making digital history more representative of documentary history.
https://youtu.be/u0JY9vURUAs

Information about the 2022 Big Data Conference can be found here.

Light cones for open quantum systems
2023-09-06 16:00 - 17:00 | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Marius Lemm, University of Tuebingen
Title: Light cones for open quantum systems

Abstract: We consider non-relativistic Markovian open quantum dynamics in continuous space. We show that, up to small probability tails, the supports of quantum states propagate with finite speed in any finite-energy subspace. More precisely, if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound on the slope of this light cone (i.e., on the maximal speed). The general method can be used to derive propagation bounds for a variety of other quantum systems including Lieb-Robinson bounds for lattice bosons. Based on joint works with S. Breteaux, J. Faupin, D.H. Ou Yang, I.M. Sigal, and J. Zhang.

Correlation decay for finite lattice gauge theories
2023-09-07 13:30 - 14:30 | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Arka Adhikari (Stanford)
Title: Correlation decay for finite lattice gauge theories
Abstract: In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

A 6-year journey: from gravitational anomaly to a unified theory of generalized symmetry
2023-09-08 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Xiao-Gang Wen (MIT)
Title: A 6-year journey: from gravitational anomaly to a unified theory of generalized symmetry
Abstract: Emergent symmetry can be generalized symmetry beyond (higher) group description and/or can be anomalous. I will describe a unified theory for generalized symmetry based on symmetry/topological-order correspondence. I will also discuss some applications of emergent generalized symmetry.

Pole skipping, quasinormal modes, shockwaves and their connection to chaos
2023-09-12 11:00 - 12:00
General Relativity Seminar
Speaker: Diandian Wang(Harvard University)
Title: Pole skipping, quasinormal modes, shockwaves and their connection to chaos
Abstract: A chaotic quantum system can be studied using the out-of-time-order correlator (OTOC). I will tell you about pole skipping — a recently discovered feature of the retarded Green’s function — that seems to also know things: things like the Lyapunov exponent and the butterfly velocity, which are important quantifiers of the OTOC. Then I will talk about a systematic way of deriving pole-skipping conditions for general holographic CFTs dual to classical bulk theories and how to use this framework to derive a few interesting statements including: (1) theories with higher spins generally violate the chaos bound; (2) the butterfly velocity calculated using pole skipping agrees with that calculated using shockwaves for arbitrary higher-derivative gravity coupled to ordinary matter; (3) shockwaves are related to a special type of quasinormal modes. As we will see, the techniques are entirely classically gravitational, which I will go through with a certain level of details.

Homotopy classes of loops of Clifford unitaries
2023-09-12 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Roman Geiko, UCLA
Title: Homotopy classes of loops of Clifford unitaries
Abstract: We study Clifford locality-preserving unitaries and stabilizer Hamiltonians by means of Hermitian K-theory. We demonstrate how the notion of algebraic homotopy of modules over Laurent polynomial rings translates into the connectedness of two short-range entangled stabilizer Hamiltonians by a shallow Clifford circuit. We apply this observation to a classification of homotopy classes of loops of Clifford unitaries. The talk is based on a work in collaboration with Yichen Hu. https://arxiv.org/abs/2306.09903.

Phase transitions out of quantum Hall states in moire TMD bilayers
2023-09-13 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Xueyang Song (MIT)
Title: Phase transitions out of quantum Hall states in moire TMD bilayers
Abstract: Motivated by the recent experimental breakthroughs in observing Fractional Quantum Anomalous Hall (FQAH) states in moir’e Transition Metal Dichalcogenide (TMD) bilayers, we propose and study various unconventional phase transitions between quantum Hall phases and Fermi liquids or charge ordered phases upon tuning the bandwidth. At filling -2/3, we describe a direct transition between the FQAH state and a Charge Density Wave (CDW) insulator. The critical theory resembles that of the familiar deconfined quantum critical point (DQCP) but with an additional Chern-Simons term. At filling -1/2, we study the possibility of a continuous transition between the composite Fermi liquid (CFL) and the Fermi liquid (FL) building on and refining previous work by Barkeshli and McGreevy. Crucially we show that translation symmetry alone is enough to enable a second order CFL-FL transition. We argue that there must be critical CDW fluctuations though neither phase has long range CDW order. A striking signature is a universal jump of resistivities at the critical point. With disorder, we argue that the CDW order gets pinned and the CFL-FL evolution happens through an intermediate electrically insulating phase with mobile neutral fermions. A clean analog of this insulating phase with long range CDW order and a neutral fermi surface can potentially also exist. We also present a critical theory for the CFL to FL transition at filling -3/4. Our work opens up a new avenue to realize deconfined criticality and fractionalized phases beyond familiar Landau level physics in the moire Chern band system.

Anomalies of Non-Invertible Symmetries
2023-09-13 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Clay Córdova (U Chicago)
Title: Anomalies of Non-Invertible Symmetries

Frustration-free states of cell fate networks: the case of the epithelial-mesenchymal transition
2023-09-14 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Herbert Levine (Northeastern)

Title: Frustration-free states of cell fate networks: the case of the epithelial-mesenchymal transition

Abstract: Cell fate decisions are made by allowing external signals to govern the steady-state pattern adopted by networks of interacting regulatory factors governing transcription and translation. One of these decisions, of importance for both developmental processes and for cancer metastasis, is the epithelial-mesenchymal transition (EMT). In this talk, we will argue that these biological networks have highly non-generic interaction structures such that they allow for phenotypic states with very low frustration, i.e. where most interactions are satisfied. This property has important consequences for the allowed dynamics of these systems.

Quantum UV-IR map and curve counts in skeins
2023-09-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sunghyuk Park
Title: Quantum UV-IR map and curve counts in skeins
Abstract: Quantum UV-IR map (a.k.a. q-nonabelianization map), introduced by Neitzke and Yan, is a map from UV line defects in a 4d N=2 theory of class S to those of the IR. Mathematically, it can be described as a map between skein modules and is a close cousin of quantum trace map of Bonahon and Wong.
In this talk, I will discuss how quantum UV-IR map can be generalized to a map between HOMFLYPT skein modules, using skein-valued curve counts of Ekholm and Shende.

Quantization of causal diamonds in 2+1 dimensional gravity
2023-09-19 11:00 - 12:00
General Relativity Seminar
Speaker: Rodrigo Silva, University of Maryland
Title: Quantization of causal diamonds in 2+1 dimensional gravity
Abstract: We develop the reduced phase space quantization of causal diamonds in $2+1$ dimensional gravity with a nonpositive cosmological constant. The system is defined as the domain of dependence of a spacelike topological disk with a fixed boundary metric. By solving the constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of $Diff^+(S^1)/PSL(2, mathbb{R})$, i.e., the group of orientation-preserving diffeomorphisms of the circle modulo the projective special linear subgroup. Classically, the states correspond to causal diamonds embedded in $AdS_3$ (or $Mink_3$ if $Lambda = 0$), with a fixed corner length, that has the topological disk as a Cauchy surface. Because this phase space does not admit a global system of coordinates, a generalization of the standard canonical (coordinate) quantization is required — in particular, since the configuration space is a homogeneous space for a Lie group, we apply Isham’s group-theoretic quantization scheme. The Hilbert space of the associated quantum theory carries an irreducible unitary representation of the $BMS_3$ group and can be realized by wavefunctions on a coadjoint orbit of Virasoro with labels in irreducible unitary representations of the corresponding little group. A surprising result is that the twist of the diamond boundary loop is quantized in terms of the ratio of the Planck length to the corner length.

CMSA Q and A Seminar 9/19/2023
2023-09-19 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speakers: Dan Freed (Harvard Math and CMSA), Denis Auroux (Harvard Math)

Exact Results in Flat Band Hubbard Models
2023-09-20 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Jonah Herzog-Arbeitman, Princeton University
Title: Exact Results in Flat Band Hubbard Models
Abstract: Flat bands, like those in the kagome lattice or twisted bilayer graphene, are a natural setting for studying strongly coupled physics since the interaction strength is the only energy scale in the problem. They can exhibit unconventional behavior in the multi-orbital case: the mean-field theory of flat band attractive Hubbard models shows the possibility of superconductivity even though the Fermi velocity of the bands is strictly zero. However, it is not necessary to resort to this approximation. We demonstrate that the groundstates and low-energy excitations of a large class of attractive Hubbard models are exactly solvable, offering a rare, microscopic view of their physics. The solution reveals the importance of quantum geometry in escaping (some of) BCS phenomenology within a tractable and nontrivial strong coupling theory.

The TinyStories Dataset: How Small Can Language Models Be And Still Speak Coherent
2023-09-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Ronen Eldan, Microsoft Research
Title: The TinyStories Dataset: How Small Can Language Models Be And Still Speak Coherent
Abstract: While generative language models exhibit powerful capabilities at large scale, when either the model or the number of training steps is too small, they struggle to produce coherent and fluent text: Existing models whose size is below a few billion parameters often do not generate coherent text beyond a few sentences. Hypothesizing that one of the main reasons for the strong reliance on size is the vast breadth and abundance of patterns in the datasets used to train those models, this motivates the following question: Can we design a dataset that preserves the essential elements of natural language, such as grammar, vocabulary, facts, and reasoning, but that is much smaller and more refined in terms of its breadth and diversity?
In this talk, we introduce TinyStories, a synthetic dataset of short stories that only contain words that 3 to 4-year-olds typically understand, generated by GPT-3.5/4. We show that TinyStories can be used to train and analyze language models that are much smaller than the state-of-the-art models (below 10 million parameters), or have much simpler architectures (with only one transformer block), yet still produce fluent and consistent stories with several paragraphs that are diverse and have almost perfect grammar, and demonstrate certain reasoning capabilities. We also show that the trained models are substantially more interpretable than larger ones, as we can visualize and analyze the attention and activation patterns of the models, and show how they relate to the generation process and the story content. We hope that TinyStories can facilitate the development, analysis and research of language models, especially for low-resource or specialized domains, and shed light on the emergence of language capabilities in LMs.

Floquet codes, automorphisms, and quantum computation
2023-09-22 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Margarita Davydova (MIT)
Title: Floquet codes, automorphisms, and quantum computation
Abstract: In this talk, I will introduce a new kind of measurement-based quantum computation inspired by Floquet codes. In this model, the quantum logical gates are implemented by short sequences of low-weight measurements which simultaneously encode logical information and enable error correction. We introduce a new class of quantum error-correcting codes generalizing Floquet codes that achieve this, which we call dynamic automorphism (DA) codes.
As in Floquet codes, the instantaneous codespace of a DA code at any fixed point in time is that of a topological code. In this case, the quantum computation can be viewed as a sequence of time-like domain walls implementing automorphisms of the topological order, which can be understood in terms of reversible anyon condensation paths in a particular parent model. This talk will introduce all of these concepts as well as provide a new perspective for thinking about Floquet codes.
The explicit examples that we construct, which we call DA color codes, can implement the full Clifford group of logical gates in 2+1d by two- and, rarely three-body measurements. Using adaptive two-body measurements, we can achieve a non-Clifford gate in 3+1d, making the first step towards universal quantum computation in this model.
The talk is based on recent work with Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen [arxiv: 2307.10353].

Modularity of Landau-Ginzburg Models
2023-09-22 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Chuck Doran
Title: Modularity of Landau-Ginzburg Models
Abstract: Fano varieties are the basic building blocks of algebraic varieties. Smooth Fano varieties have been classified in dimensions one (the projective line), two (del Pezzo surfaces), and three (Mori-Mukai classification). What does Mirror Symmetry have to say about such classifications? By studying the Landau-Ginzburg models mirror to smooth Fano threefolds we can transform the Mori-Mukai classification into an effective uniruledness result for moduli spaces of certain K3 and abelian surfaces. This is joint work with Andrew Harder, Ludmil Katzarkov, Mikhail Ovcharenko, and Victor Przjalkowski (arXiv:2307.15607).

CMSA/Math Fall Gathering
2023-09-22 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Friday, Sep 22, 2023

4:00 pm

All CMSA and Math affiliates are invited.

Species Scale across String Moduli Spaces
2023-09-25 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Pre-talk Speaker: David Wu (Harvard Physics): 10:00-10:30 am
Speaker: Damian van de Heisteeg, CMSA

Title: Species Scale across String Moduli Spaces
Abstract: String theories feature a wide array of moduli spaces. We propose that the energy cutoff scale of these theories – the so-called species scale – can be determined through higher-curvature corrections. This species scale varies with the moduli; we use it both asymptotically to bound the diameter of the field space, as well as in the interior to determine a “desert point” where it is maximized.
