news

Mike Hopkins Gives Lecture at 2022 Abel Prize Celebration

Dr. Michael J. Hopkins, George Putnam Professor of Pure and Applied Mathematics and Harvard Department of Mathematics Chair, gave a lecture titled "The great wild...

Congratulations to this year’s prize and award recipients! Thomas Temple Hoopes Prize From the estate of Thomas T. Hoopes, Class of 1919, Harvard received a...

Mark Kisin Elected to American Academy of Arts and Sciences

We are thrilled to announce that Perkins Professor of Mathematics and Director of Graduate Studies Mark Kisin is among sixteen Harvard faculty elected to the...

Demystifying Math 55

By Anastasia Yefremova Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies...

See Older News

announcements

Advances in Mathematical Physics: A Conference in Honor of Elliott H. Lieb on his 90th Birthday.
July 30, 2022 - August 1, 2022
Advances in Mathematical Physics A Conference in Honor of Elliott H. Lieb on his 90th Birthday Dates: July 30-August 1, 2022 Harvard University July 30...
See Older Announcements

upcoming events

< 2021 >
April
«
»
Sun
Mon
Tue
Wed
Thu
Fri
Sat
March
March
March
March
1
• CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Applied physics of high-Tc theories

CMSA EVENTCMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Applied physics of high-Tc theories

9:00 AM-10:30 AM
April 1, 2021

Since the discovery of high-temperature superconductors in cuprates in 1986, many theoretical ideas have been proposed which have enriched condensed matter theory. Especially, the resonating valence bond (RVB) state for (doped) spin liquids is one of the most fruitful idea. In this talk, I would like to describe the development of RVB idea to broader class of materials, especially more conventional magnets. It is related to the noncollinear spin structures with spin chirality and associated quantal Berry phase applied to many phenomena and spintronics applications. It includes the (quantum) anomalous Hall effect, spin Hall effect, topological insulator, multiferroics, various topological spin textures, e.g., skyrmions, and nonlinear optics. I will show that even though the phenomena are extensive, the basic idea is rather simple and common in all of these topics.

• HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUM

HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUMLoop-erased random walk—a random fractal

4:30 PM-5:30 PM
April 1, 2021

Many models in equilibrium statistical physics produce random fractal curves “at criticality.”  I will discuss one particular model, the loop-erased random walk, which is closely related to uniform spanning trees and Laplacian motion, and survey what is known today including some more recent results.  I will also discuss some of the important open problems and explain why the problem is hardest in exactly three dimensions. This talk is intended for a general mathematics audience and does not assume the audience knows the terms in the previous sentence.

2
3
4
5
• CMSA EVENT: CMSA Mathematical Physics Seminar: Topological recursion in 4d N = 2 supersymmetric gauge theories

CMSA EVENTCMSA Mathematical Physics Seminar: Topological recursion in 4d N = 2 supersymmetric gauge theories

10:00 AM-11:00 AM
April 5, 2021

According to the Alday-Gaiotto-Tachikawa conjecture (proved in this case by Schiffman and Vasserot), the instanton partition function in 4d N = 2 SU(r) supersymmetric gauge theory on P^2 with equivariant parameters ε₁, ε₂ is the norm of a Whittaker vector for W(gl_r) algebra. I will explain how these Whittaker vectors can be computed (at least perturbatively in the energy scale) by topological recursion for ε₁ + ε₂ = 0, and by a non-commutation version of the topological recursion in the Nekrasov-Shatashvili regime where ε₁/ε₂ is fixed. This is a joint work to appear with Bouchard, Chidambaram and Creutzig.

6
• DIFFERENTIAL GEOMETRY SEMINAR

DIFFERENTIAL GEOMETRY SEMINARQuasimodular forms from Betti numbers

8:00 AM-9:00 AM
April 6, 2021

This talk will be about refined curve counting on local P^2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P^2. This gives a proof of some stringy predictions about the refined topological string theory of local P^2 in the Nekrasov-Shatashvili limit. Partly based on work with Honglu Fan, Shuai Guo, and Longting Wu.

• CMSA EVENT: CMSA Math Science Literature Lecture Series

CMSA EVENTCMSA Math Science Literature Lecture Series

9:00 AM-10:30 AM
April 6, 2021

TITLE: Isadore Singer’s Work on Analytic Torsion

ABSTRACT: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.

Talk chair: Cumrun Vafa

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

Register here to attend.
• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINARConjugation of words, self-intersections of planar curves, and non-commutative divergence

10:00 AM-11:00 AM
April 6, 2021

The space spanned by homotopy classes of free oriented loops on a 2-manifold carries an interesting algebraic structure (a Lie bialgebra structure) due to Goldman and Turaev. This structure is defined in terms of intersections and self-intersections of planar curves. In the talk, we will explain a surprising link between the Gaoldman-Turaev theory and the Kashiwara-Vergne problem on properties of the Baker-Campbell-Hausdorff series. Important tools in establishing this link are the non-commutative divergence cocycle and a novel characterization of conjugacy classes in free Lie algebras in terms of cyclic words. The talk is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.

• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINARConjugation of words, self-intersections of planar curves, and non-commutative divergence

10:00 AM-11:00 AM
April 6, 2021

The space spanned by homotopy classes of free oriented loops on a 2-manifold carries an interesting algebraic structure (a Lie bialgebra structure) due to Goldman and Turaev. This structure is defined in terms of intersections and self-intersections of planar curves. In the talk, we will explain a surprising link between the Gaoldman-Turaev theory and the Kashiwara-Vergne problem on properties of the Baker-Campbell-Hausdorff series. Important tools in establishing this link are the non-commutative divergence cocycle and a novel characterization of conjugacy classes in free Lie algebras in terms of cyclic words. The talk is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.

• CMSA EVENT: CMSA Computer Science for Mathematicians: Confidence-Budget Matching for Sequential Budgeted Learning

CMSA EVENTCMSA Computer Science for Mathematicians: Confidence-Budget Matching for Sequential Budgeted Learning

11:30 AM-12:30 PM
April 6, 2021

A core element in decision-making under uncertainty is the feedback on the quality of the performed actions. However, in many applications, such feedback is restricted. For example, in recommendation systems, repeatedly asking the user to provide feedback on the quality of recommendations will annoy them. In this work, we formalize decision-making problems with querying budget, where there is a (possibly time-dependent) hard limit on the number of reward queries allowed. Specifically, we consider multi-armed bandits, linear bandits, and reinforcement learning problems. We start by analyzing the performance of greedy’ algorithms that query a reward whenever they can. We show that in fully stochastic settings, doing so performs surprisingly well, but in the presence of any adversity, this might lead to linear regret. To overcome this issue, we propose the Confidence-Budget Matching (CBM) principle that queries rewards when the confidence intervals are wider than the inverse square root of the available budget. We analyze the performance of CBM based algorithms in different settings and show that they perform well in the presence of adversity in the contexts, initial states, and budgets.

Joint work with Yonathan Efroni, Aadirupa Saha and Shie Mannor.

7
• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Higher Form Symmetries in string/M-theory

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Higher Form Symmetries in string/M-theory

10:30 AM-12:00 PM
April 7, 2021

In this talk, I will give an overview of recent developments in geometric constructions of field theory in string/M-theory and identifying higher form symmetries. The main focus will be on d>= 4 constructed from string/M-theory. I will also discuss realization in terms of holographic models in string theory. In the talk next week Lakshya Bhardwaj will speak about 1-form symmetries in class S, N=1 deformations thereof and the relation to confinement.

• RANDOM MATRIX SEMINAR

RANDOM MATRIX SEMINARJoint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Random Matrices

2:00 PM-3:00 PM
April 7, 2021

In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.

A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.

Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARThe motivic Satake equivalence

3:00 PM-4:00 PM
April 7, 2021

The geometric Satake equivalence due to Lusztig, Drinfeld, Ginzburg, Mirković and Vilonen is an indispensable tool in the Langlands program. Versions of this equivalence are known for different cohomology theories such as Betti cohomology or algebraic D-modules over characteristic zero fields and $\ell$-adic cohomology over arbitrary fields. In this talk, I explain how to apply the theory of motivic complexes as developed by Voevodsky, Ayoub, Cisinski-Déglise and many others to the construction of a motivic Satake equivalence. Under suitable cycle class maps, it recovers the classical equivalence. As dual group, one obtains a certain extension of the Langlands dual group by a one dimensional torus. A key step in the proof is the construction of intersection motives on affine Grassmannians. A direct consequence of their existence is an unconditional construction of IC-Chow groups of moduli stacks of shtukas. My hope is to obtain on the long run independence-of-$\ell$ results in the work of V. Lafforgue on the Langlands correspondence for function fields. This is ongoing joint work with Jakob Scholbach from Münster.

Password: The order of the permutation group on 9 elements.

• CMSA EVENT: CMSA New Technologies in Mathematics: Type Theory from the Perspective of Artificial Intelligence

CMSA EVENTCMSA New Technologies in Mathematics: Type Theory from the Perspective of Artificial Intelligence

3:00 PM-4:00 PM
April 7, 2021

This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science.  From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics.  Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements, isomorphism between structures, and Birkoff and Rota’s notion of cryptomorphism between structure classes.  This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians.  It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT.  The talk will also discuss BTT as a model of MLTT.  The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details.  The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.

• OPEN NEIGHBORHOOD SEMINAR

OPEN NEIGHBORHOOD SEMINAROutward-facing mathematics

4:30 PM-5:30 PM
April 7, 2021

I will talk, pretty casually, about random walks, which I first learned about as part of my Harvard undergrad thesis in finite group theory, and which turn out to be at the heart of the mathematical analysis of gerrymandering; along the way I will talk about the project of doing mathematics in a way that engages with the world outside the math department walls.

Please go to the College Calendar to register.

Website: https://math.harvard.edu/ons

8
• CMSA EVENT: CMSA Math Science Literature Lecture Series

CMSA EVENTCMSA Math Science Literature Lecture Series

9:00 AM-10:30 AM
April 8, 2021

TITLE: Quantum error correcting codes and fault tolerance

ABSTRACT: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field.

Talk chair: Zhengwei Liu

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

Register here to attend.
• CMSA EVENT: CMSA Interdisciplinary Science Seminar: Supergeometry and Super Riemann Surfaces of Genus Zero

CMSA EVENTCMSA Interdisciplinary Science Seminar: Supergeometry and Super Riemann Surfaces of Genus Zero

12:00 PM-1:00 PM
April 8, 2021

Supergeometry is a mathematical theory of geometric spaces with anti-commuting coordinates and functions which is motivated by the concept of supersymmetry from theoretical physics. I will explain the functorial approach to supermanifolds by Molotkov and Sachse. Super Riemann surfaces are an interesting supergeometric generalization of Riemann surfaces. I will present a differential geometric approach to their classification in the case of genus zero and with Neveu-Schwarz punctures.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Chiral edge modes, thermoelectric transport, and the Third Law of Thermodynamics

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Chiral edge modes, thermoelectric transport, and the Third Law of Thermodynamics

1:00 PM-2:30 PM
April 8, 2021

In this talk, I will discuss several issues related to thermoelectric transport, with application to topological invariants of chiral topological phases in two dimensions. In the first part of the talk, I will argue in several different ways that the only topological invariants associated with anomalous edge transport are the Hall conductance and the thermal Hall conductance. Thermoelectric coefficients are shown to vanish at zero temperature and do not give rise to topological invariants. In the second part of the talk, I will describe microscopic formulas for transport coefficients (Kubo formulas) which are valid for arbitrary interacting lattice systems. I will show that in general “textbook” Kubo formulas require corrections. This is true even for some dissipative transport coefficients, such as Seebeck and Peltier coefficients. I will also make a few remarks about “matching” (in the sense of Effective Field Theory) between microscopic descriptions of transport and hydrodynamics.

9
• ALGEBRAIC DYNAMICS SEMINAR

ALGEBRAIC DYNAMICS SEMINARCanonical heights and vector heights in families

10:00 AM-12:00 PM
April 9, 2021

Canonical heights are a standard tool of arithmetic dynamics over global fields. When studying families of dynamical systems, or systems over larger fields, however, there are significant geometric obstacles to constructing canonical heights. Either Northcott fails to hold, or the construction requires a model for the family with such strict properties (good reduction, minimality, extensions of rational maps,…) that such a model is unlikely to exist outside of very special settings. Instead, I’ll show how to resolve this problem using vector-valued heights, first introduced in characteristic zero by Yuan and Zhang. These generalize R-valued heights, produce canonical heights for any polarized dynamical system, and exhibit Northcott finiteness conditional on a strong non-isotriviality condition, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. This is achieved by working simultaneously over a system of models with much weaker requirements. As part of ongoing work, I’ll show how these arithmetic methods can produce results that hold over any field, and discuss how this can extend to quasi-projective varieties as well as projective varieties.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.

10
11
12
• CMSA EVENT: CMSA Mathematical Physics Seminar: Networks and quantization

CMSA EVENTCMSA Mathematical Physics Seminar: Networks and quantization

10:00 AM-11:00 AM
April 12, 2021

I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a link “invariant” (with possible wall-crossing behavior) for links L in a 3-manifold M, where M is a Riemann surface C times a real line. This construction unifies the computation of familiar link invariant with the refined counting of framed BPS states for line defects in 4d N=2 theories of class S. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of Seiberg-Witten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedinger-like equations. At the end of my talk I will also try to sketch a possibility to bridge these two scenarios.

13
• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINARAspects of M Theory

10:00 AM-11:00 AM
April 13, 2021

After giving a brief introduction to Membrane Theory and its matrix regularization, commenting on an inherent dynamical symmetry for all M-branes (the related “reconstruction-algebra” for M=1, strings, being the Virasoro algebra), I will explain some very recent work, including the observation that super-symmetrizable systems canonically (i.e. more or less automatically) have a Lax-pair formulation, with calculable r-matrix, – the appearance of infinite-dimensional CKL-algebras naturally entering the double bracket equations of Quantum Minimal Surfaces (IKKT model) and the (“BFFS”) membrane matrix model.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARApplication of a Bogomolov-Gieseker type inequality to counting invariants

3:00 PM-4:00 PM
April 13, 2021

I will work on a  Calabi-Yau 3-fold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\i-Toda for weak stability conditions, such as the quintic threefold. I will explain how wall-crossing with respect to weak stability conditions gives an expression of Joyce’s generalised Donaldson-Thomas invariants counting Gieseker semistable sheaves of any rank greater than or equal to one on X in terms of those counting sheaves of rank 0 and pure dimension 2.  This is joint work with Richard Thomas.

14
• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Confinement and 1-form Symmetries in 4d from 6d (2,0)

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Confinement and 1-form Symmetries in 4d from 6d (2,0)

10:30 AM-12:00 PM
April 14, 2021

In this talk, I will discuss several issues related to thermoelectric transport, with application to topological invariants of chiral topological phases in two dimensions. In the first part of the talk, I will argue in several different ways that the only topological invariants associated with anomalous edge transport are the Hall conductance and the thermal Hall conductance. Thermoelectric coefficients are shown to vanish at zero temperature and do not give rise to topological invariants. In the second part of the talk, I will describe microscopic formulas for transport coefficients (Kubo formulas) which are valid for arbitrary interacting lattice systems. I will show that in general “textbook” Kubo formulas require corrections. This is true even for some dissipative transport coefficients, such as Seebeck and Peltier coefficients. I will also make a few remarks about “matching” (in the sense of Effective Field Theory) between microscopic descriptions of transport and hydrodynamics.

• CMSA EVENT: CMSA New Technologies in Mathematics: A Bayesian neural network predicts the dissolution of compact planetary systems

CMSA EVENTCMSA New Technologies in Mathematics: A Bayesian neural network predicts the dissolution of compact planetary systems

3:00 PM-4:00 PM
April 14, 2021

Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. I will discuss our deep learning architecture (arxiv:2101.04117) which pushes forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. The Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on three-planet configurations, the model demonstrates robust generalization to five-planet systems, even outperforming models designed for that specific set of integrations. I will also discuss some work on recovering symbolic representations of such models using arxiv:2006.11287.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARArithmetic curves lying in compact subsets of affine schemes

3:00 PM-4:00 PM
April 14, 2021

We will describe the notion of affine schemes and their modifications in the context of Arakelov geometry. Using geometry of numbers in infinite rank, we will study their cohomological properties. Concretely, given an affine scheme X over Z and a compact subset K of the set of complex points of X, we will investigate the geometry of those proper arithmetic curves in X whose complex points lie in K. This is joint work with Jean-Benoît Bost.

Password: The order of the permutation group on 9 elements.

15
• CMSA EVENT: CMSA Interdisciplinary Science Seminar: Weak solutions to the isentropic system of gas dynamics

CMSA EVENTCMSA Interdisciplinary Science Seminar: Weak solutions to the isentropic system of gas dynamics

9:00 AM-10:00 AM
April 15, 2021

In this talk, I will discuss the global weak solutions to the isentropic system of gas dynamics: existence and non-uniqueness. In the first part, we generalized the renormalized techniques introduced by DiPerna-Lions to build up the global weak solutions to the compressible Navier-Stokes equations with degenerate viscosities. This existence result holds for any $\gamma>1$ in any dimensional spaces for the large initial data. In the second part, we proved that for any initial data belonging to a dense subset of the energy space, there exists infinitely many global weak solutions to the isentropic Euler equations for any $1<\gamma\leq 1+2/n$. Our result is based on a generalization of convex integration techniques by De Lellis-Szekelyhidi and weak vanishing viscosity limit of the Navier-Stokes equations. The first part is based on the joint works with D. Bresch and A. Vasseur, and the second one is based on our recent joint work with R. M Chen and A. Vasseur.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: QCD without diagrams

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: QCD without diagrams

10:30 AM-12:00 PM
April 15, 2021

QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking. This talk will be an elementary overview of the present framework for understanding how these effects come about.

• HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUM

HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUMOn the cohomology of moduli of abelian varieties

4:30 PM-5:30 PM
April 15, 2021

I’ll discuss recent work using tropical techniques to find new rational cohomology classes in moduli spaces A_g of abelian varieties, building on previous joint work with Galatius and Payne on M_g. I will try to take a broad view. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

16
• CMSA EVENT: CMSA Math Science Literature Lecture Series

CMSA EVENTCMSA Math Science Literature Lecture Series

1:00 PM-2:30 PM
April 16, 2021

TITLE: Deep Networks from First Principles

ABSTRACT: In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective. This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.

Talk chair: Harry Shum

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

Register here to attend.
• RANDOM MATRIX SEMINAR

RANDOM MATRIX SEMINARJoint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

2:00 PM-3:00 PM
April 16, 2021

In a disordered quantum system, delocalization can be understood in many ways. One of these is quantum unique ergodicity, which was proven in the random matrix context by Bourgade and Yau. It states that for a given eigenvector and set of coordinates J, the mass placed on J by the eigenvector tends to N^{-1}|J|, the mass placed on those coordinates by the uniform distribution. Notably, this convergence holds for any size of J, showing that the eigenvectors distribute evenly on all scales.

I will present a result which establishes that the fluctuations of these averages are Gaussian on scales where |J| is asymptotically less than N, for generalized Wigner matrices with smooth entries. The proof uses new eigenvector observables, which are analyzed dynamically using the eigenvector moment flow and the maximum principle.

This is joint work with Lucas Benigni.

17
18
19
• CMSA EVENT: CMSA Mathematical Physics Seminar: Branching Rules and Young Tableaux Methods: 10D & 11D Supergravity

CMSA EVENTCMSA Mathematical Physics Seminar: Branching Rules and Young Tableaux Methods: 10D & 11D Supergravity

10:00 AM-11:00 AM
April 19, 2021

In this talk, I will review 4D, N = 1 off-shell supergravity. Then I present explorations to construct 10D and 11D supergravity theories in two steps. The first step is to decompose scalar superfield into Lorentz group representations which involves branching rules and related methods. Interpretations of component fields by Young tableaux methods will be presented. The second step is to implement an analogue of Breitenlohner’s approach for 4D supergravity to 10D and 11D theories.

20
• CMSA EVENT: CMSA Math Science Literature Lecture Series

CMSA EVENTCMSA Math Science Literature Lecture Series

9:00 AM-10:30 AM
April 20, 2021

TITLE: The Atiyah-Singer Index Theorem

ABSTRACT: The story of the index theorem ties together the Gang of Four— Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.

Talk chair: Cumrun Vafa

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

Register here to attend.
• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINAREmerging frontiers in nuclear magnetic resonance

10:00 AM-11:00 AM
April 20, 2021

Nuclear Magnetic Resonance (NMR) is a powerful spectroscopic technique that provides information about matter at an atomic resolution. One of the applications of NMR is to decipher the molecular architecture of biomolecules including nucleic acids and proteins. In addition to providing information on the structure of biomolecules, NMR also provides information on the dynamics of these molecule machines. The seminar will introduce some basics of NMR, discuss some of the current limitations, and present new methods to push the frontiers of NMR.

• CMSA EVENT: CMSA Computer Science for Mathematicians: EigenGame: SVD as a Nash Equilibrium

CMSA EVENTCMSA Computer Science for Mathematicians: EigenGame: SVD as a Nash Equilibrium

11:30 AM-12:30 PM
April 20, 2021

We present a novel view on singular value decomposition (SVD) as a competitive game in which each approximate singular vector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this EigenGame and the behavior of its gradient based updates. The resulting algorithm — which combines elements from Oja’s rule with a generalized Gram-Schmidt orthogonalization — is naturally decentralized and hence parallelizable through message passing. EigenGame’s updates are biased if computed using minibatches of data, which hinders convergence and more sophisticated parallelism in the stochastic setting. Therefore, in follow-up work, we propose an unbiased stochastic update that is asymptotically equivalent to EigenGame, enjoys greater parallelism allowing computation on datasets of larger sample sizes, and outperforms the original EigenGame in experiments. We demonstrate the a) scalability of the algorithm by conducting principal component analyses of large image datasets, language datasets, and neural network activations and b) generality by reusing the same algorithm to perform spectral clustering of a social network. We discuss how this new view of SVD as a differentiable game can lead to further algorithmic developments and insights.

This talk is based on two recent works, both joint work with Brian McWilliams, Claire Vernade, and Thore Graepel —

https://arxiv.org/abs/2010.00554 (EigenGame – ICLR ‘21)

https://arxiv.org/abs/2102.04152 (EigenGame Unloaded – ICML ‘21 submission)

— and will focus in detail on some of the more interesting mathematical properties of the game.

• DIFFERENTIAL GEOMETRY SEMINAR

DIFFERENTIAL GEOMETRY SEMINARGroup actions and stability on elliptic surfaces

9:00 PM-10:00 PM
April 20, 2021

There are two natural group actions on the Bridgeland stability manifold of a triangulated category: a left action by the group of autoequivalences, and a right action by the universal covering space of $\mathrm{GL}^+(2,\mathbb{R})$.  The left action is much harder to compute than the right action in general.  In this talk, we will discuss a method for recognising when a left action is equivalent to that of a right action, and apply it to a non-standard autoequivalence on elliptic surfaces.

This work is partly motivated by an attempt to understand equivalences of triangulated categories in representation theory and algebraic geometry at the same time.

21
• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Exotic new animals in the CFT zoo: quasiparticles and anisotropic scaling

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Exotic new animals in the CFT zoo: quasiparticles and anisotropic scaling

10:30 AM-11:30 AM
April 21, 2021
• RANDOM MATRIX SEMINAR

RANDOM MATRIX SEMINARJoint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Mean-field spin glasses: beyond Parisi’s formula?

2:00 PM-3:00 PM
April 21, 2021

Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARThe absolute prismatic site

3:00 PM-4:00 PM
April 21, 2021

The absolute prismatic site of a p-adic formal scheme carries interesting arithmetic and geometric information attached to the formal scheme. In this talk, after recalling the definition of this site, I will discuss an algebro-geometric (stacky) approach to absolute prismatic cohomology and its concomitant structures (joint with Lurie, and partially due independently to Drinfeld). As a geometric application, I’ll explain Drinfeld’s refinement of the Deligne-Illusie theorem on Hodge-to-de Rham degeneration. On the arithmetic side, I’ll describe a new classification of crystalline representations of the Galois group of a local field in terms of F-crystals on the site (joint with Scholze).

Password: The order of the permutation group on 9 elements.

• CMSA EVENT: CMSA New Technologies in Mathematics: Homotopy type theory and the quest for extensionality

CMSA EVENTCMSA New Technologies in Mathematics: Homotopy type theory and the quest for extensionality

3:00 PM-4:00 PM
April 21, 2021

Over the past decades, dependent type theory has proven to be a powerful framework for verified software and formalized mathematics.  However, its treatment of equality has always been somewhat uncomfortable.  Recently, homotopy type theory has made progress towards a more useful notion of equality, which natively implements both isomorphism-invariance in mathematics and representation-independence in programming. This progress is based on ideas from abstract homotopy theory and higher category theory, and with the development of cubical type theories it can be implemented as a true programming language.  In this talk, I will survey these developments and their potential applications, and suggest some directions for further improvement.

• OPEN NEIGHBORHOOD SEMINAR

OPEN NEIGHBORHOOD SEMINARAlgebraic topology and sums of squares formulas

4:30 PM-5:30 PM
April 21, 2021

It is a classical fact that the product of a sum of two squares with a sum of two squares is naturally a sum of two squares. (One can also replace “two” by “four” or “eight.”) But in general, it is not known exactly when a product of the sum of m squares with a sum of n squares can be represented as a sum of p squares. I will discuss how methods of algebraic topology have been used to study this question. In particular, the tools of algebraic topology produce tools to obstruct the existence of such formulas in general. Moreover, these tools can be adapted to study the analogous question in positive characteristic.

Please go to the College Calendar to register.

Website: https://math.harvard.edu/ons

22
• CMSA EVENT: CMSA Interdisciplinary Science Seminar: Convex Integration and Fluid Turbulence

CMSA EVENTCMSA Interdisciplinary Science Seminar: Convex Integration and Fluid Turbulence

9:00 AM-10:00 AM
April 22, 2021

The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy even in the vanishing viscosity limit, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. These methods originated in the works of Nash and Gromov and were adapted to the context of fluid equations by De Lellis and Szekelyhidi Jr. In this talk, we will survey the history of both phenomenological theories of turbulence and convex integration. Finally, we discuss recent joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from Kolmogorov’s predictions.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Non-abelian bosonization in two and three spatial dimensions and some applications

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Non-abelian bosonization in two and three spatial dimensions and some applications

10:30 AM-12:00 PM
April 22, 2021

In this talk, we generalize Witten’s non-abelian bosonization in $(1+1)$-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. As applications, we apply the bosonization results to the $SU(2)$ gauge theory of the $\pi$ flux mean-field theory of half-filled Hubbard model, critical spin liquids of “bipartite-Mott insulators” in 1,2,3 spatial dimensions, and twisted bilayer graphene.

23
24
25
26
• CMSA EVENT: CMSA Mathematical Physics Seminar: Topological-holomorphic field theories and their BV quantizations

CMSA EVENTCMSA Mathematical Physics Seminar: Topological-holomorphic field theories and their BV quantizations

10:00 AM-11:00 AM
April 26, 2021

Topological field theories and holomorphic field theories have each had a substantial impact in both physics and mathematics, so it is natural to consider theories that are hybrids of the two, which we call topological-holomorphic and denote as THFTs. Examples include Kapustin’s twist of N=2, D=4 supersymmetric Yang-Mills theory and Costello’s 4-dimensional Chern-Simons theory. In this talk about joint work with Rabinovich and Williams, I will define THFTs, describe several examples, and then explain how to quantize them rigorously and explicitly, by building on techniques of Si Li. Time permitting, I will indicate how these results offer a novel perspective on the Gaudin model via 3-dimensional field theories.

27
• DIFFERENTIAL GEOMETRY SEMINAR

DIFFERENTIAL GEOMETRY SEMINARBall Quotients from Deligne-Mostow Theory and Periods of K3 Surfaces

8:00 AM-9:00 AM
April 27, 2021

In this talk, I will first briefly review the Deligne-Mostow theory of moduli spaces of weighted points on the projective line, and a construction of ball quotients from periods of (possibly singular) K3 surfaces with non-symplectic group action. Then I will discuss how these two constructions can be unified for some examples. I will focus on a new case about a 6-dimensional family of K3 surfaces with D4-singularity. This is a joint work with Yiming Zhong.

• CMSA EVENT: CMSA Math Science Literature Lecture Series

CMSA EVENTCMSA Math Science Literature Lecture Series

9:00 AM-10:30 AM
April 27, 2021

TITLE: Moment maps and the Yang-Mills functional

ABSTRACT: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

Register here to attend.
• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINARDimerization in quantum spin chains with O(n) symmetry

10:00 AM-11:00 AM
April 27, 2021

We consider spin-S quantum spin chains with a family of O(2S+1)-invariant nearest-neighbor interactions and discuss the ground state phase diagram of this family of models. Using a graphical representation for the partition function, we give a proof of dimerization for an open region in the phase diagram, for all sufficiently large values of S. (Joint work with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARProperness of the K-moduli space

3:00 PM-4:00 PM
April 27, 2021

K-stability is an algebraic condition that characterizes the existence of K\”ahler-Einstein metrics on Fano varieties. Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties.  Motivated by results in differential geometry, it is conjectured that this K-moduli space is proper and projective. In this talk, I’ll discuss some recent progress in birational geometry that leads to a full solution of this conjecture. Based on joint work with Yuchen Liu and Chenyang Xu.

28
• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: 1-form symmetry-protected topological phases and measurement-based quantum computation

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: 1-form symmetry-protected topological phases and measurement-based quantum computation

1:00 PM-2:30 PM
April 28, 2021

I will use Walker-Wang models to demonstrate the connection between 1-form symmetry-protected topological phases and topological measurement-based quantum computation. I will describe the classification of these phases in terms of symmetry domain walls and how these lead to “anomalous” 1-form symmetry actions on the boundary. I will also demonstrate that when the symmetries are strictly enforced these phases persist to finite temperatures and use this to explain symmetry-protected self-correction properties of the boundary topological phases.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARFrom Ramanujan to K-theory

3:00 PM-4:00 PM
April 28, 2021

The Rogers-Ramanujan identity is an equality between a certain “q-series” (given as an infinite sum) and a certain modular form (given as an infinite product). Motivated by ideas from physics, Nahm formulated a necessary condition for when such q-hypergeometric series were modular. Perhaps surprisingly, this turns out to be related to algebraic K-theory. We discuss a proof of this conjecture. This is joint work with Stavros Garoufalidis and Don Zagier.

Password: The order of the permutation group on 9 elements

29
• CMSA EVENT: CMSA Interdisciplinary Science Seminar: An isoperimetric problem with a competing nonlocal singular term

CMSA EVENTCMSA Interdisciplinary Science Seminar: An isoperimetric problem with a competing nonlocal singular term

9:00 AM-10:00 AM
April 29, 2021

We are interested in the minimization problem of a functional in which the perimeter is competing with a nonlocal singular term comparable to a fractional perimeter, with volume constraint. We prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. We connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Subsystem-Symmetry protected phases of matter

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Subsystem-Symmetry protected phases of matter

10:30 AM-12:00 PM
April 29, 2021

We know that different systems with the same unbroken global symmetry can nevertheless be in distinct phases of matter.  These different “symmetry-protected topological” phases are characterized by protected (gapless) surface states.  After reviewing this physics in interacting systems with global symmetries, I will describe how a different class of symmetries known as subsystem symmetries, which are neither local nor global, can also lead to protected gapless boundaries.  I will discuss how some of these subsystem-symmetry protected phases are related (though not equivalent) to interacting higher-order topological insulators, with protected gapless modes along corners or hinges in higher dimensional systems.

30
• OPEN NEIGHBORHOOD SEMINAR

OPEN NEIGHBORHOOD SEMINARSpecial Joint Math Table/Open Neighborhood Seminar

1:00 PM-2:00 PM
April 30, 2021

Speaker: Kenz Kallal (Harvard)
Title: The Arthur–Selberg trace formula and some applications to arithmetic statistics

Speaker: Lux Zhao (Harvard)
Title: The history and mathematics of the axiom of choice

Go here to register and obtain Zoom information.

May