Seminars Sorted by Series

We show that every complete Riemannian manifold of finite volume contains a complete embedded minimal hypersurface of finite volume. This is a joint work with Gregory Chambers.

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the...

Quantum modular forms were defined in 2010 by Zagier; they are somewhat analogous to ordinary modular forms, but they are defined on the rational numbers Q as opposed to the upper half complex plane H, and they transform in Q under the action of the...

Positive geometries are real semialgebraic sets inside complex varieties characterized by the existence of a meromorphic top-form called the canonical form. The defining property of positive geometries and their canonical forms is that the residue...

I will discuss some geometric inequalities that hold on Riemannian 2-disks and 2-spheres.

For example, I will prove that on any Riemannian 2-sphere there M exist at least three simple periodic geodesics of length at most 20d, where d is the...

The goal is to describe how techniques from symplectic dynamics can be used to study orbit travel in three dimensions, for systems like the restricted 3-body problem from celestial mechanics. The pseudo-holomorphic curve theory initiated by Hofer...

In the early 1930's, the Ergodic theorems of von Neumann and Birkhoff put Boltzmann's Ergodic Hypothesis in mathematical terms, and the natural question was born: is ergodicity the "general case" among conservative dynamical systems? Oxtoby and Ulam...

Invariant theory is a fundamental subject in mathematics, and is potentially applicable whenever there is symmetry at hand (group actions). In recent years, new problems and conjectures inspired by complexity have come to light. In this talk, I will...

Deligne's "Weil II" paper includes a far-reaching conjecture to the effect that for a smooth variety on a finite field of characteristic p, for any prime l distinct from p, l-adic representations of the etale fundamental group do not occur in...

Schur polynomials are the characters of finite-dimensional irreducible representations of the general linear group. We will discuss both continuous and discrete concavity property of Schur polynomials. There will be one theorem and eight conjectures...

A fundamental task in (unsupervised) analysis of data is to detect and estimate interesting "structure" hidden in it. In low dimensions, this task has been explored for over 100 years with dozens of developed methods. In this talk I'll focus on...

Symplectic capacities are measurements of symplectic size. They are often defined as the lengths of certain periodic trajectories of dynamical systems, and so they connect symplectic embedding problems with dynamics. I will explain joint work...

In this talk, I will give two vignettes on the theme of sparse matrices in sparse analysis. The first vignette covers work from compressive sensing in which we want to design sparse matrices (i.e., matrices with few non-zero entries) that we use to...

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with various constraints on boundary and interior marked points. The presence of boundary poses an obstacle to invariance. In...

In this talk, I'll discuss the role that Lie algebras play in algebraic topology and motivate the development of a "homotopy coherent" version of the theory. I'll also explain an "equation-free" formulation of the classical theory of Lie algebras...

The "nearest neighbor (NN) classifier" labels a new data instance by taking a majority vote over the k most similar instances seen in the past. With an appropriate setting of k, it is capable of modeling arbitrary decision rules. Traditional...

Human memory is a multi-staged phenomenon of extreme complexity, which results in highly unpredictable behavior in real-life situations. Psychologists developed classical paradigms for studying memory in the lab, which produce easily quantifiable...

Symplectic geometry, and its close relative contact geometry, are geometries closely tied to complex geometry, smooth topology, and mathematical physics. The h-principle is a general method used for construction of smooth geometric objects...

We give the first examples of codimension-1 knotting in the 4-sphere, i.e. there is a 3-ball B1 with boundary the standard linear 2-sphere, which is not isotopic rel boundary to the standard linear 3-ball B0. Actually, there is an infinite family of...

The purpose of this talk is to introduce the classification problem of partially hyperbolic diffeomorphisms in dimension 3 (including introducing the concept of partially hyperbolic diffeomorphisms and its relevance). The main goal will be to...

The geometric optics trace formula gives the singular support of wave trace on a compact Riemannian manifold. In the case of of a one dimensional singular manifold, that is a metric (or quantum) graph, this formula is exact and yields a crystalline...

The Information Bottleneck (IB) is an information theoretic framework for optimal representation learning. It stems from the problem of finding minimal sufficient statistics in supervised learning, but has insightful implications for Deep Learning...

In this talk I will discuss the Sunflower Lemma and similar lemmas that prove (in various contexts) that a set/distribution can be partitioned into a structured part and a "random-looking" part. I will introduce communication complexity as a key...

Striving to make contact with mathematics and to be consistent with neuroanatomy at the same time, I propose an idealized picture of the cerebral cortex consisting of a hierarchical network of brain regions each further subdivided into...

Hilbert’s 23rd Problem is the last in his famous list of problems and is of a different character than the others. The description is several pages, and basically says that the calculus of variations is a subject which needs development. We will...

Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will start by motivating the uses of diagonalization in representation theory. Then comes a brief introduction to categorical representation...

Several well-known open questions, such as: "are all groups sofic or hyperlinear?", have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $Sym(n)$ (in the sofic case) or the unitary groups $U(n)$...

Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids...

You can make a paper Moebius band by starting with a $1$ by $L$ rectangle, giving it a twist, and then gluing the ends together. The question is: How short can you make $L$ and still succeed in making the thing? This question goes back to B. Halpern...

(joint work with Assaf Naor) A key problem in metric geometry asks: given metric spaces $X$ and $Y$, how well does $X$ embed in $Y$? In this talk, we will consider this problem for the case of the Heisenberg group and explain its connections to...

There are striking analogies between topology and arithmetic algebraic geometry, which studies the behavior of solutions to polynomial equations in arithmetic rings. One expression of these analogies is through the theory of etale cohomology, which...

In a joint work with Tianyi Zheng we show that the growth function of the first Grigorchuk group satisfies \[ \ln \ln v_n/\ln v_n = a, \] where $a = \log 2/\log x$, $x$ being a positive root of the polynomial $x^3-x^2-2x-4$. This is done by...

We present an overview of elementary methods to study extensions of modular representations of various types of "groups". We shall begin by discussing actions of an elementary abelian $p$-group, $E = (Z/p)^r$, on finite dimensional vector spaces...

Low-dimensional topology and geometry have many problems with an easy formulation, but a hard solution. Despite our intuitive feeling that these problems are "hard", lower or upper bounds on algorithmic complexity are known only for some of them...

Physics inspired mathematics helps us understand the random evolution of Markov processes. For example, the Kolmogorov forward and backward differential equations that govern the dynamics of Markov transition probabilities are analogous to the...