Calendar

This is a report about work in progress with: Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi. We will describe a virtual Atiyah-Bott formula for Artin stacks.  In the Deligne-Mumford case our methods allow us to remove the global resolution hypothesis for the virtual normal bundle.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

We will present some rigorous results about entropy and  relative entropy in QFT, motivated in part by recent physicists’ work which however depends on heuristic arguments such as introducing cut off and using path integrals.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

I will explain a notion of rank for the relations amongst the equations of a projective variety, generalizing the classical notion of rank of a quadric. I will then explain a result telling us that, for a general canonical curve, all syzygies are linear combinations of syzygies of minimal rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank four. As a special case, this perspective gives us a new, and simpler, proof of Green’s conjecture for general curves.

https://harvard.zoom.us/j/91082896381?pwd=TmJCanRPVlM4VGR2L1RzZGVGbHRVQT09

The SYZ proposal suggests that mirror symmetry is T-duality. It is a folklore that locally free sheaves are mirror to a Lagrangian multi-section of the SYZ fibration. In this talk, I will introduce the notion of tropical Lagrangian multi-sections and discuss how to obtain from such object to a class of locally free sheaves on the log Calabi-Yau spaces that Gross-Siebert have considered. I will also discuss a joint work with Kwokwai Chan and Ziming Ma, where we proved the smoothability of a class of locally free sheaves on some log Calabi-Yau surfaces by using combinatorial data obtained from tropical Lagrangian multi-sections.

Zoom Link: https://cuhk.zoom.us/j/95804207557

Meeting ID: 958 0420 7557; Passcode: 20220302

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase “w2w3” in (4+1)D that has an anomalous boundary topological order with fermionic particle and fermionic loop excitations that have mutual statistics. Finally, we will demonstrate a few examples of fermionic loop excitations.

https://harvard.zoom.us/j/977347126
Password: cmsa

Scaling laws and associated downstream trends can be used as an organizing principle when thinking about current and future ML progress.  I will briefly review scaling laws for generative models in a number of domains, emphasizing language modeling.  Then I will discuss scaling results for transfer from natural language to code, and results on python programming performance from “codex” and other models.  If there’s time I’ll discuss prospects for the future — limitations from dataset sizes, and prospects for RL and other techniques.

Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split’ maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related’. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.

In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of split maps and curves. To this end, we establish instances of a ‘relative dynamical Bogomolov conjecture’. This is joint work with Harry Schmidt (University of Basel).

The Newell-Littlewood numbers are defined in terms of the  Littlewood-Richardson coefficients. Both arise as tensor product  multiplicities for a classical Lie group. A. Klyachko connected  eigenvalues of sums of Hermitian matrices to the saturated LR-cone and  established defining linear inequalities. We prove analogues for the  saturated NL-cone. This is based on work with Gidon Orelowitz, Nicolas  Ressayre and Alexander Yong.

We will explore the fascinating history of counting shapes, starting with counting dots (you can have no dots or one dot or two dots…) and then moving on to counting shapes of higher dimension. Hopefully you will see some connections to interesting questions in geometry, physics, homotopy theory, and number theory.

While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason — they either work in the Neural Tangent Kernel regime where neurons don’t move much, or require an enormous number of neurons. In this talk I will describe our recent works towards understanding training dynamics that go beyond kernel regimes with only polynomially many neurons (mildly overparametrized). In particular, we first give a local convergence result for mildly overparametrized two-layer networks. We then analyze the global training dynamics for a related overparametrized tensor model. For both works, we rely on a key intuition that neurons in overparametrized models work in groups and it’s important to understand the behavior of an average neuron in the group. Based on two works: https://arxiv.org/abs/2102.02410 and https://arxiv.org/abs/2106.06573.

Professor Rong Ge is Associate Professor of Computer Science at Duke University. He received his Ph.D. from the Computer Science Department of Princeton University, supervised by Sanjeev Arora. He was a post-doc at Microsoft Research, New England. In 2019, he received both a Faculty Early Career Development Award from the National Science Foundation and the prestigious Sloan Research Fellowship. His research interest focus on theoretical computer science and machine learning. Modern machine learning algorithms such as deep learning try to automatically learn useful hidden representations of the data. He is interested in formalizing hidden structures in the data and designing efficient algorithms to find them. His research aims to answer these questions by studying problems that arise in analyzing text, images, and other forms of data, using techniques such as non-convex optimization and tensor decompositions.

Zoom ID: 950 2372 5230 (Password: cmsa)

In this talk we will discuss the interaction between magnetic monopoles and massless fermions. In the 1980’s Callan and Rubakov showed that in the simplest example and that fermion-monopole interactions catalyze proton decay in GUT completions of the standard model. Here we will explain how fermions in general representations interact with general spherically symmetric monopoles and classify the types of symmetries that are broken: global symmetries with ABJ-type anomalies.

https://harvard.zoom.us/j/977347126
Password: cmsa

The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science, and chemistry. The algorithm builds a maximal independent set by inspecting the graph’s vertices one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.

In this talk, I will present a simple yet general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a valuable notion of local convergence. I will demonstrate the applicability of this framework by giving short and straightforward proofs for results on previously studied families of graphs, such as paths and various random graphs, and by providing new results for other models such as random trees.

If time allows, I will discuss a more delicate (and combinatorial) result, according to which, in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.

The Nishimori line (NL) describes a special one parameter family of disordered spin models which are invariant under a gauge transformation acting jointly on the spins and the disorder. For group valued spins such, as SO(2) and SU(2), we prove that there is long range order in 3 or more dimensions along the NL. The proof relies on work of Abbe, Massoulie, Montanari, Sly and Srivastava on group synchronization. It does not use reflection positivity. This is joint work with Christophe Garban, arXiv:2109.01617v2.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

In 1998, using arguments from M-theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi-Yau 3-folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov-Witten invariants by a transformation of the generating series. The Gopakumar-Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

Zoom Link: https://cuhk.zoom.us/j/95974708402

Meeting ID: 959 7470 8402

Passcode: 20220309

The  low  dimensional  manifold  hypothesis  posits  that  the  data  found  in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input.  However, one often needs to  consider  evaluating  the  optimized  network  at  points  outside  the  training distribution.  We analyze the cases where the training data are distributed in a linear subspace of Rd.  We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace.  We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Motivated by a puzzle arising from recent work on staggered lattice fermions we introduce Kaehler-Dirac fermions and describe their connection both to Dirac fermions and staggered fermions. We show that they suffer from a gravitational anomaly that breaks a chiral U(1) symmetry specific to Kaehler-Dirac fermions down to Z_4 in any even dimension. In odd dimensions we show that the effective theory that results from integrating out massive Kaehler-Dirac fermions is a topological gravity theory. Such theories generalize Witten’s construction of (2+1) gravity as a Chern Simons theory. In the presence of a domain wall massless modes appear on the wall which can be consistently coupled to gravity due to anomaly inflow from the bulk gravitational theory. Much of this story parallels the usual discussion of topological insulators. The key difference is that the twisted chiral symmetry and anomaly structure of Kaehler-Dirac theories survives intact under discretization and governs the behavior of the lattice models. $Z_4$ invariant four fermion interactions can be used to gap out states in such theories without breaking symmetries and in flat space yields the known constraints on the number of Majorana fermions needed symmetric mass generation namely eight and sixteen Majorana spinors in two and four dimensions.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

For a smooth projective curve $C/\mathbb{Q}$, how many field extensions of $\mathbb{Q}$ — of given degree and bounded discriminant — arise from adjoining a point of $C(verline{\mathbb{Q}})$? Can we further count the number of such extensions with a specified Galois group?
Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves $C$.

The gig economy provides workers with the benefits of autonomy and flexibility, but it does so at the expense of work identity and co-worker bonds. Among the many reasons why gig workers leave their platforms, an unexplored aspect is the organization identity. In a series of studies, we develop a team formation and inter-team contest at a ride-sharing platform. We employ an end-to-end data science approach, combining methodologies from randomized field experiments, recommender systems, and counterfactual machine learning. Together, our results show that platform designers can leverage team identity and team contests to increase revenue and worker engagement in a gig economy.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

We consider the Cauchy problem for the Einstein equations for cosmological spacetimes, i.e. spacetimes with compact spatial hypersurfaces. Various classes of those dynamical spacetimes have been constructed and analyzed using CMC foliations or equivalently the CMC-Einstein flow. We will briefly review the Andersson-Moncrief stability result of negative Einstein metrics under the vacuum Einstein flow and then present various recent generalizations to the nonvacuum case. We will emphasize what difficulties arise in those generalizations, how they can be handled depending on the matter model at hand, and what implications we can draw from these results for cosmology. We then turn to a scenario where the CMC Einstein flow leads to a large data result in 2+1-dimensions.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

We present computations of the thermal Hall coefficient of phonons scattering off defects with multiple energy levels. Using a microscopic formulation based on the Kubo formula, we find that the leading contribution perturbative in the phonon-defect coupling is of the `side-jump’ type, which is proportional to the phonon lifetime. This contribution is at resonance when the phonon energy equals a defect level spacing. Our results are obtained for different defect models, and include models of an impurity quantum spin in the presence of quasi-static magnetic order with an isotropic Zeeman coupling to the applied field.
This work is based on arxiv: 2201.11681

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

It is by now well-known that mirror symmetry may be expressed as an equivalence between categories associated to dual Kahler manifolds. Following a proposal of Teleman, we inaugurate a program to understand 3d mirror symmetry as an equivalence between 2-categories associated to dual holomorphic symplectic stacks. We consider here the abelian case, where our theorem expresses the 2-category of spherical functors as a 2-category of coherent sheaves of categories. Applications include categorifications of hypertoric category O and of many related constructions in representation theory. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Billiards on an elliptical billiard table are completely integrable: phase space is foliated by invariant submanifolds for the billiard flow. Birkhoff conjectured that ellipses are the only plane domains with integrable

billiards. Avila-deSimoi- Kaloshin proved the conjecture for ellipses of sufficiently small eccentricity.   Kaloshin-Sorrentino proved local results for all eccentricities.  On the quantum level, the analogous conjecture is that ellipses are uniquely determined by their Dirichlet (or, Neumann) eigenvalues. Using the results on the Birkhoff conjecture, Hamid Hezari and I proved that for ellipses of small eccentricity are indeed uniquely determined by their eigenvalues. Except for disks, which Kac proved to be uniquely determined, these are the only domains for which it is known that one can hear their shape.

Zoom Link: https://cuhk.zoom.us/j/97001510671

(Meeting ID: 970 0151 0671; Passcode: 20220316)

Some recent work in the quantum gravity literature has considered what happens when the amplitudes of a TQFT are summed over the bordisms between fixed in-going and out-going boundaries. We will comment on these constructions. The total amplitude, that takes into account all in-going and out-going boundaries can be presented in a curious factorized form. This talk reports on work done with Anindya Banerjee and is based on the paper on the e-print arXiv   2201.00903.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision and natural language processing. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization and generalization, during the optimization process. In this talk, I will discuss some mathematical properties of optimization and generalization for deep neural networks.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision and natural language processing. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization and generalization, during the optimization process. In this talk, I will discuss some mathematical properties of optimization and generalization for deep neural networks.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The Swampland program aims at uncovering the universal implications of quantum gravity at low-energy physics. I will review the basic ideas of the Swampland program, formal and phenomenological implications, and provide a survey of the techniques commonly used in Swampland research including tools from quantum information, holography, supersymmetry, and string theory.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Current Developments in Mathematics 2021-22

March 18-19, 2022

Harvard University Science Center

Lecture Hall B

 
Speakers:
Jessica Fintzen (University of Cambridge and Duke University)
Ryan O’Donnell (Carnegie Mellon)
Jack Thorne (University of Cambridge)
Cynthia Vinzant (University of Washington)
Mu-Tao Wang (Columbia)

This conference will also be available in hybrid format as an online Zoom Webinar. To attend virtually, please register using the Zoom Registration links for each day of the conference. Separate Zoom registrations are required for Friday’s and Saturday’s sessions.

Because of COVID protocols, advance registration is required and capacity may be restricted. If you’d like to participate, please fill out the registration form below by March 16.
Register Here

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to cdm@math.harvard.edu. Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.

Schedule
• Friday, March 18, 2022
• Friday Zoom Webinar Registration
• Jessica Fintzen
1:30-2:20pm   I: Representations of p-adic groups
2:30-3:20pm   II: Representations of p-adic groups

• Mu-Tao Wang
3:35-4:25pm   I: Angular momentum and supertranslation in general relativity
4:35-5:25pm   II: Angular momentum and supertranslation in general relativity

• Saturday, March 19, 2022
• Saturday Zoom Webinar Registration
• Jack Thorne
9:05-9:55am   I: L-functions and symmetric power functoriality
10:05-10:55am   II: L-functions and symmetric power functoriality

• Ryan O’Donnell
11:10am-12:00pm   I: Learning and Testing Quantum States
12:00-1:30pm   Lunch
1:30-2:20pm   II: Learning and Testing Quantum States

• Cynthia Vinzant
2:35-3:25pm   I: Log-concavity in matroids and expanders
3:40-4:35pm   II: Log-concavity in matroids and expanders

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard)

Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.

Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

The AdS/CFT conjecture in physics posits the existence of a  correspondence between gravitational theories in asymptotically Anti-de  Sitter (aAdS) spacetimes and field theories on their conformal boundary.  In this presentation, we prove rigorous mathematical statements toward  this conjecture.

In particular, we show there is a one-to-one correspondence between aAdS  solutions of the Einstein-vacuum equations and a suitable space of data  on the conformal boundary (consisting of the boundary metric and the  boundary stress-energy tensor). We also discuss consequences of this  result, as well as the main ingredient behind its proof: a unique  continuation property for wave equations on aAdS spacetimes.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

1. The conjecture and an overview of the results
2. Static regular: a sufficient condition for existence and local uniqueness
3. Convex boundary, isometric embedding, and static regular
4. Perturbations of any hypersurface are static regular

March 22 – 25, 2022
22nd & 23rd, 10:00 am – 11:30am ET
24th & 25th, 11:00 am – 12:30pm ET

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Consider a smooth family of hypersurfaces of degree d in P^{n+1}.
When is every smooth projective limit of this family also a hypersurface?
While it is easy to construct example of limits that are not
hypersurfaces when the degree d is composite, Mori conjectured that, if d
is prime and n>2, every smooth projective limit is indeed a hypersurface.
However, there are counterexamples when n=1 or 2; for example, one can
take a family of degree 5 plane curves and degenerate to a smooth
hyperelliptic (non-planar) curve.  In this talk, we will propose a
re-formulation of Mori’s conjecture that explains the failure in low
dimensions, provide results in dimension one, and discuss a general
approach to the problem using moduli spaces of pairs.  This is joint work
with David Stapleton.

The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

1. The conjecture and an overview of the results
2. Static regular: a sufficient condition for existence and local uniqueness
3. Convex boundary, isometric embedding, and static regular
4. Perturbations of any hypersurface are static regular

March 22 – 25, 2022
22nd & 23rd, 10:00 am – 11:30am ET
24th & 25th, 11:00 am – 12:30pm ET

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam.

for information on how to join, please go to:

https://cmsa.fas.harvard.edu/cmsa-colloquium

I will show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero lattice momentum, i.e. if it is an eigenstate of the translation symmetry with eigenvalue not equal to 1. Equivalently, any state that can be connected with a non-zero momentum state through a finite-depth local unitary transformation must also be long-range entangled. The statement can also be generalized to fermion systems. I will then present two applications of this result: (1) several different types of Lieb-Schultz-Mattis (LSM) theorems, including a previously unknown version involving only a discrete Z_n symmetry, can be derived in a simple manner; (2) a gapped topological order (in space dimension d>1) must weakly break translation symmetry if one of its ground states on torus has nontrivial momentum – this generalizes the familiar physics of Tao-Thouless in fractional quantum Hall systems.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Cohen and Lenstra gave a conjectural distribution for the  class group of a random quadratic number field. Since the class group  is the Galois group of the maximum abelian unramified extension, a  natural generalization would be to give a conjecture for the  distribution of the Galois group of the maximal unramified extension.  Previous work has produced a plausible conjecture in special cases,  with the most general being recent work of Liu, Wood, and Zurieck-Brown.

There is a deep analogy between number fields and 3-manifolds. Thus,  an analogous question would be to describe the distribution of the  profinite completion of the fundamental group of a random 3-manifold.  In this talk, I will explain how Melanie Wood and I answered this  question for a model of random 3-manifolds defined by Dunfield and  Thurston, and how the techniques we used should allow us, in future  work, to give a more general conjecture in the number field case.

We will see how groups and symmetries are at the heart of various problems in statistics. I will also describe approaches to estimate parameters in statistical models using invariant theory, based on joint work with Carlos Amendola, Kathlen Kohn, and Philipp Reichenbach.

The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

1. The conjecture and an overview of the results
2. Static regular: a sufficient condition for existence and local uniqueness
3. Convex boundary, isometric embedding, and static regular
4. Perturbations of any hypersurface are static regular

March 22 – 25, 2022
22nd & 23rd, 10:00 am – 11:30am ET
24th & 25th, 11:00 am – 12:30pm ET

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

The operadic structure on the moduli spaces of algebraic curves  encodes in a combinatorial way how nodal curves in the boundary can be obtained by glueing smooth curves along marked points. In this talk, I will present a generalization of the operadic structure to moduli spaces of SUSY curves (or super Riemann surfaces). This requires colored graphs and generalized operads in the sense of Borisov-Manin. Based joint work with Yu. I. Manin and Y. Wu. https://arxiv.org/abs/2202.10321

For information on how to join, please go to:

https://cmsa.fas.harvard.edu/interdisciplinary-science-seminar

I will talk about the edge physics of the deconfined quantum phase transition (DQCP) between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet superconductor (SC). Although the bulk of this transition is in the same universality class as the paradigmatic deconfined Neel to valence-bond-solid transition, the boundary physics has a richer structure due to proximity to a quantum spin Hall state. We use the parton trick to write down an effective field theory for the QSH-SC transition in the presence of a boundary and calculate various edge properties in a large-N limit. We show that the boundary Luttinger liquid in the QSH state survives at the phase transition, but only as fractional degrees of freedom that carry charge but not spin. The physical fermion remains gapless on the edge at the critical point, with a universal jump in the fermion scaling dimension as the system approaches the transition from the QSH side. The critical point could be viewed as a gapless analogue of the QSH state but with the full SU(2) spin rotation symmetry, which cannot be realized if the bulk is gapped. This talk reports on the work done with Liujun Zou and Chong Wang (arxiv:2110.08280).

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $2<s'<s$.  The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $mega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density,  the result is known to hold for $mega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be  $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of a sound wave from the transported part and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The developmental cycle of Myxococcus xanthus involves the coordination of many hundreds of thousands of cells aggregating to form mounds known as fruiting bodies. This aggregation process begins with the sequential formation of more and more cell layers. Using three-dimensional confocal imaging we study this layer formation process by observing the formation of holes and second layers within a base monolayer of M xanthus cells. We find that cells align with each other over the majority of the monolayer forming an active nematic liquid crystal with defect point where cell alignment is undefined. We find that new layers and holes form at positive and negative topological defects respectively. We model the cell layer using hydrodynamic modeling and find that this layer and hole formation process is driven by active nematic forces through cell motility and anisotropic substrate friction.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

1. The conjecture and an overview of the results
2. Static regular: a sufficient condition for existence and local uniqueness
3. Convex boundary, isometric embedding, and static regular
4. Perturbations of any hypersurface are static regular

March 22 – 25, 2022
22nd & 23rd, 10:00 am – 11:30am ET
24th & 25th, 11:00 am – 12:30pm ET

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

I will discuss the asymptotics of multimatrix models. In a small parameters (or high temperature) region, their free energy is well known to converge and to be related with the enumeration of maps. The latter can be proved by relating the so-called Dyson-Schwinger equations with Tutte surgery by nice pictures. In general, the convergence of the free energy is unknown, a problem which is closely related with the lack of a full entropy theory in free probability. We will discuss few models that can be analyzed, in particular in the large parameters (or low temperature) region.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

How often does a general cubic surface appear in a 4-dimensional linear system?  As a slice of a general cubic 3-fold?  I will describe how we can solve problems of this kind using equivariant geometry.  This is joint work with Anand Patel and Dennis Tseng.

We present an effective quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].

Zoom link: https://cuhk.zoom.us/j/99612299554

(Meeting ID: 996 1229 9554; Passcode: 20220330)

We present an effective quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].

For more information, please see: http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

In this talk I first review some of the many appearances of localized degrees of freedom — edge modes —  in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

We show that Polchinski’s equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy.  This gives a surprising information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport.  We will provide reviews of both the exact renormalization group, as well as the theory of optimal transportation.  Our results allow us to establish a new, non-perturbative RG monotone, and also reformulate RG flow as a variational problem.  The latter enables new numerical techniques and allows us to establish a systematic connection between neural network methods and RG flows of conventional field theories.  Our techniques generalize to other RG flow equations beyond Polchinski’s.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Language models typically need to be trained or fine-tuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this talk, I will discuss how we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate NN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. We also find that the model is capable of making use of newly defined functions and theorems during test time.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This is joint work with Frank Calegari and Vesselin Dimitrov.

I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

All in-person attendees must register online.

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

Matroids are combinatorial abstractions of vector spaces embedded in a coordinate space.  Many fundamental questions have been open for these classical objects.  We highlight some recent progress that arise from the interaction between matroid theory and algebraic geometry.  Key objects involve compactifications of embedded vector spaces, and an exceptional Hirzebruch-Riemann-Roch isomorphism between the K-ring of vector bundles and the cohomology ring of stellahedral varieties.

For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

I will first explain the significance of Thurston’s pushforward operator for studying postcritically finite rational maps. I will then present some results due to Buff-Epstein-Koch which relate the eigenvalues of this operator to the multipliers of a rational map and also describe the set of eigenvalues of all unicritical polynomials of a given degree.