Seminars Sorted by Series

In this talk I will discuss some classical and new applications of the abc conjecture, its relation to conjectures about elliptic curves, and some (admittedly weak) unconditional partial results.

One way to construct new 3-manifolds is by surgery on a knot in the 3-sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3-manifolds can be obtained in this manner? We provide obstructions using the Heegaard...

We will discuss some natural problems in arithmetic that can be reformulated in terms of orbits of certain "thin" (semi)groups of integer matrix groups.

Classically, the Gauss-Manin connection relates the de Rham cohomology of different smooth fibres of a map. Its point of origin is the fact that vector fields act trivially on de Rham cohomology (the Cartan homotopy formula). That fact has an...

I will give a brief overview of symplectic mapping class groups, then explain how one can use homological mirror symmetry to get information about them. This is joint work with Ivan Smith.

The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over $\mathbb Q$ are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is...

One of the main ideas that comes up in the proof of Fermat's Last Theorem is a way of "counting" 2-dimensional Galois representations over $\mathbb Q$ with certain prescribed properties. We discuss the problem of counting other types of Galois...

For certain applications of linear algebra, it is useful to understand the distribution of the largest eigenvalue of a finite sum of discrete random matrices. One of the useful tools in this area is the "Matrix Chernoff" bound which gives tight...

Computing the class number is a hard question. In 1956, Iwasawa announced a surprising formula for an infinite family of class numbers, starting an entire theory that lies behind this phenomenon. We will not focus too much on this theory (Iwasawa...

Representation theory over $\mathbb Z$ is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover...

Representations of open compact subgroups play a fundamental role in studying representations of $p$-adic groups and their covering groups. We give an overview of this subject, called the theory of types, in connection with harmonic analysis. We...

The Fukaya category of a symplectic manifold is a robust intersection theory of its Lagrangian submanifolds. Over the past decade, ideas emerging from Wehrheim--Woodward's theory of quilts have suggested a method for producing maps between the...

The tree amplituhedron $A(n,k,m)$ is the image in the Grassmannian $Gr(k,k+m)$ of the totally nonnegative part of $Gr(k,n)$, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in...

I'll describe joint work with Sheel Ganatra and Nick Sheridan which rigorously establishes the relationship between different aspects of the mirror symmetry phenomenon for Calabi-Yau manifolds. Homological mirror symmetry---an abstract, categorical...

The Hofer-Zehnder capacity of a symplectic manifold is one of the early symplectic invariants: it is a non-negative real number, possibly infinite. Finiteness of this capacity has strong consequences for Hamiltonian dynamics, and it is an old...

Berkovich geometry is an enhancement of classical rigid analytic geometry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds. I will explain (1) what is mirror symmetry, (2) what are Berkovich spaces, (3) how Berkovich spaces...

Over the past two decades, information theory has reemerged within computational complexity theory as a mathematical tool for obtaining unconditional lower bounds in a number of models, including streaming algorithms, data structures, and...

The sum-of-squares (SOS) hierarchy (due to Shor'85, Parrilo'00, and Lasserre'00) is a widely-studied meta-algorithm for (non-convex) polynomial optimization that has its roots in Hilbert's 17th problem about non-negative polynomials.SOS plays an...

Combinatorial geometry is a field that studies combinatorial problems that involve some simple geometric objects/notions, such as: lines, points, distances, collinearity. While such problems are often easy to state, some of them are very difficult...

Four years ago, Cachazo, He and Yuan found a system of algebraic equations, now named the "scattering equations", that effectively encoded the kinematics of massless particles in such a way that the scattering amplitudes, the quantities of physical...

Locally symmetric spaces are a class of Riemannian manifolds which play a special role in number theory. In this talk, I will introduce these spaces through example, and show some of their unusual properties from the point of view of both analysis...

We introduce the notion of a "crystallographic sphere packing," which generalizes the classical Apollonian circle packing. Tools from arithmetic groups, hyperbolic geometry, and dynamics are used to show that, on one hand, there is an infinite zoo...

In this talk, I will discuss the behavior of hard-core lattice particle systems at high fugacities. I will first present a collection of models in which the high fugacity phase can be understood by expanding in powers of the inverse of the fugacity...

Decomposition theorem for perverse sheaves on algebraic varieties, proved by Beilinson-Bernstein-Deligne-Gabber, is one of the most important and useful theorems in the contemporary mathematics. By the Riemann-Hilbert correspondence, we may regard...

The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and a representation of the skein algebra features in Witten's topological quantum...

This talk is going to be an extended and more technical version of my brief public lecture https://www.ias.edu/ideas/2017/arora-zemel-machine-learningI will present some of the basic ideas of machine learning, focusing on the mathematical...

We will give a brief overview of the classical topics, problems and results in Algebraic Combinatorics. Emerging from the representation theory of $S_n$ and $GL_n$, they took a life on their own via the theory of symmetric functions and Young...

Symplectic Geometry and its dynamics originated from classical mechanics as the geometry of physical phase space, in particular from celestial mechanics, and one of the most driving questions is up to today that of stability for such systems. One of...

This talk will survey ideas surrounding a conjecture in number theory about the structure of class groups of number fields. Each number field has associated to it a finite abelian group, the class group, and as long ago as Gauss, deep questions...

Classical and quantum Hamiltonian actions of reductive groups, respectively, give rise to ubiquitous families of commuting flows and of commutative rings of operators. I will explain how a construction of Ngô (from the proof of the Fundamental Lemma...

Last semester I gave a talk that was a 1-hour condensation of undergrad machine learning. https://www.ias.edu/video/membsem/2017/1127-SanjeevArora . This semester's talk will be a 1-hour treatment of some research directions in theoretical machine...

This talk will be an introduction to the methods used in the study of spectral properties of Schroedinger operators with a potential defined via the action of an ergodic transformation. Open problems relating to Lyapunov exponents over a skew shift...

I will survey what is known about the construction of (the building blocks of) representations of p-adic groups, mention recent developments, and explain some of the concepts underlying all constructions. In particular, I will introduce filtrations...