Calendar

Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. Kapustin and Orlov conjecture that coisotropic branes should be included into the Fukaya category from a physics view point. In this talk, I will construct for linear symplectic tori a version of the Fukaya category including coisotropic branes and show that the usual Fukaya category embeds fully faithfully into it. I will also explain the motivation of the construction through the perspective of Homological mirror symmetry.

We make the case that over the coming decade, computer assisted reasoning will become far more widely used in the mathematical sciences. This includes interactive and automatic theorem verification, symbolic algebra, and emerging technologies such as formal knowledge repositories, semantic search and intelligent textbooks.

After a short review of the state of the art, we survey directions where we expect progress, such as mathematical search and formal abstracts, developments in computational mathematics, integration of computation into textbooks, and organizing and verifying large calculations and proofs. For each we try to identify the barriers and potential solutions.

Fluid dynamical processes are key to understanding the formation and evolution of stars and planets.  While the astrophysical community has made exceptional progress in simulating highly compressible flows, models of low-Mach-number stellar and planetary flows typically use simplified equations based on numerical techniques for incompressible fluids.  In this talk, we will discuss improved numerical models of three low-Mach-number astrophysical phenomena: tidal instabilities in binary neutron stars, waves and convection in massive stars, and ice-ocean interactions in icy moons.  We will cover the basic physics of these systems and how ongoing additions to the open-source Dedalus Project are enabling their efficient simulation in spherical domains with spectral accuracy, implicit timestepping, phase-field methods, and complex equations of state.

In this work, we propose a systematical framework to construct Bell inequalities from stabilizers which are maximally violated by general stabilizer states. We show that the constructed Bell inequalities can self-test any stabilizer state which is essentially device-independent, if and only if these stabilizers can uniquely determine the state in a device-dependent manner. This bridges the gap between device-independent and device-dependent verification methods. Our framework can not only inspire more fruitful multipartite Bell inequalities from conventional verification methods, but also pave the way for their practical applications.

Joint work with Qi Zhao, arXiv:2002.01843

Let $M$ be a complete Ricci-flat manifold with Euclidean
volume growth. A theorem of Colding-Minicozzi states that if a tangent
cone at infinity of $M$ is smooth, then it is the unique tangent cone.
The key component in their proof is an infinite dimensional
Lojasiewicz-Simon inequality, which implies rapid decay of the
$L^2$-norm of the trace-free Hessian of the Green function. In this
talk we discuss how this inequality can be exploited to identify two
arbitrarily far apart scales in $M$ in a natural manner through a
diffeomorphism. We also prove a pointwise Hessian estimate for the
Green function when there is an additional condition on sectional
curvature, which is an analogue of various matrix Harnack inequalities
obtained by Hamilton and Li-Cao in different time-dependent settings.

— Organized by Prof. Shing-Tung Yau

I will describe how certain recursive distributional equations can be solved by importing rigorous results on the convergence of approximation schemes for degenerate PDEs, from numerical analysis. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell.

One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which
contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak’s conjecture, which deals with the disjointness of actions of (N,+) and (N,*). This talk is based on joint work with Vitaly Bergelson.

I plan to sketch Kontsevich’s proof of formality of the little n-disk operad, and time permitting, intrinsic formality.

The groups of homeomorphisms or diffeomorphisms of a manifold have many striking parallels with finite dimensional Lie groups. In this talk, I’ll describe some of these, and explain new work, joint with Lei Chen, that gives an orbit classification theorem and a structure theorem for actions of homeomorphism and diffeomorphism groups on other spaces, analogous to some classical results for actions of locally compact Lie groups. As applications, we answer many concrete questions towards classifying all actions of Diff(M) on other manifolds (many of which are nontrivial, for instance Diff(M) acts naturally on the unit tangent bundle of M…) and resolve several threads in a research program initiated by Ghys. I’ll aim to give both a broad overview and several toy applications in the talk.

Tea at 4:00 pm – Math Common Room

Talk at 4:30 pm – Hall A

In this talk, we will discuss a quasi-local Penrose inequality with charges for time-symmetric initial data of the Einstein-Maxwell equation. Namely, we derive a lower bound for Brown-York type quasi-local mass in terms of the horizon area and the electric charge. The inequality we obtained is sharp in the sense that equality holds for surfaces in the Reissner-Nordström manifold. This talk is based on joint work with Stephen McCormick.

We give the first examples of codimension-1 knotting in the 4-sphere, i.e. there is a 3-ball B_1 with boundary the standard linear 2-sphere, which is not isotopic rel boundary to the standard linear 3-ball B_0. Actually, there is an infinite family of distinct isotopy classes of such balls. This implies that there exist inequivalent fiberings of the unknot in 4-sphere, in contrast to the situation in dimension-3. Also, that there exists diffeomorphisms of S^1 x B^3 homotopic rel boundary to the identity, but not isotopic rel boundary to the identity. Joint work with Ryan Budney.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar