Calendar

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.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

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.”

For more information, please visit the event page.

Register here to attend.

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.

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

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.

Zoom: https://harvard.zoom.us/j/98231541450

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.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Zoom: https://harvard.zoom.us/j/977347126

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.

Zoom: https://harvard.zoom.us/j/99333938108

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).

Zoom: https://harvard.zoom.us/j/99334398740

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

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.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

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.

Zoom: https://calendar.college.harvard.edu/event/open_neighborhood_seminar

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

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.

Zoom: https://harvard.zoom.us/j/98248914765?pwd=Q01tRTVWTVBGT0lXek40VzdxdVVPQT09

(Password: 419419)

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.

Zoom: https://harvard.zoom.us/j/977347126