Seminars Sorted by Series

Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion —...

Abstract: I will discuss a result with Bonatti and Crovisier from 2009 showing that the $C^1$ generic diffeomorphism f of a closed manifold has trivial centralizer; i.e. fg = gf implies that g is a power of f. I’ll discuss features of the $C^1$...

The problem of control of large multi-agent systems, such as vehicular traffic, poses many challenges both for the development of mathematical models and their analysis and the application to real systems. First, we discuss how conservation laws can...

Any non-negative univariate polynomial over the reals can be written as a sum of squares.  This gives a simple-to-verify certificate of non-negativity of the polynomial. Rooted in Hilbert's 17th problem, there's now more than a century's work that...

The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of...

A deep result of Furstenberg from 1967 states that if $\Gamma$ is a lattice in a semisimple Lie group $G$, then there exists a measure on $\Gamma$ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of...

Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide...

The so-called Problem of Compact Quotients asks for which homogeneous spaces G/H there exists a discrete subgroup Gamma of G such that Gamma\G/H is a compact manifold. We will discuss this problem, and describe recent progress on it arising from...

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information...

Triangulated surfaces are Riemann surfaces formed by gluing together equilateral triangles. They are also the Riemann surfaces defined over the algebraic numbers. Brooks, Makover, Mirzakhani and many others proved results about the geometric...

Lorentzian polynomials serve as a bridge between continuous and discrete convex analysis, with tropical geometry providing the critical link. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions, leading...

Canceled

I will discuss some well-known and less-known papers of Turing, exemplify the scope of deep, prescient ideas he put forth, and mention follow-up work on these by the Theoretical CS community.

No special background will be assumed.

Joint work with Amir Mohammadi, Zhiren Wang, and Lei Yang

Let Q be an indefinite ternary quadratic form. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values...

A promenade in Serre's paper, plus some related results of mine.

Expansion in graphs is a well studied topic, with a wealth of applications in many areas of mathematics and the theory of computation.

High dimensional expansion is a generalization of expansion from graphs to higher dimensional objects, such as...

This colloquium will explore some fundamental issues in the mathematics of plasmas, focusing on the stability and instability of solutions to Vlasov-type equations, which are crucial for describing the behavior of charged particles in a plasma. A...

We travel the years in order to understand the relationship between Nilpotency and Riemannian geometry: including Gromov's almost flat theorem for manifolds with bounded curvature and Fukaya-Yamaguchi's almost nilpotency of spaces with lower...

The Poisson-Furstenberg boundary  is a measure space that describes asymptotics of infinite trajectories of random walks. The boundary is non-trivial if and only if the defining  measure admits non-constant bounded harmonic functions. 

The origin of...

In the late 1800s, in the course of his study of classical problems of number theory, the young Hermann Minkowski discovered the importance of a new kind of geometric object that we now call a convex set. He soon developed a rich theory for...

The Prékopa-Leindler inequality (PL) and its strengthening, the Borell-Brascamp-Lieb inequality, are functional extensions of the Brunn-Minkowski inequality from convex geometry, which itself refines the classical isoperimetric inequality. These...

In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its...

Erd\H{o}s unit distance problem asks the following: Let $P$ be a set of $n$ distinct points in the plane, and let $U(P)$ denote the number of pairs of points in $P$ that are at distance 1. How large can $U(P)$ be? In 1946, Erd\H{o}s showed that for...

Originally introduced as analogs of symmetric spaces for  groups over non-archimedian fields buildings have proven useful in various areas by now. In this talk I will introduce (Bruhat-Tits) buildings,  combinatorial toolkits to study them and their...

Hel Braun (IAS member 1947-1948) was a mathematician who introduced approaches that continue to impact research today.  Braun's research contributions lie in three areas: classical number theory problems about integers, modular and automorphic forms...

Selmer groups are a cohomological tool used to reduce the task of finding solutions of certain diophantine equations to easier field and modular arithmetic. I'll explain how this works in down to earth terms and give some concrete applications...

Persistence modules and their associated barcodes were intensively studied since the early 2000s with a view towards applied mathematics. Recently they have also found numerous applications in pure mathematics. We will discuss a few examples from...

The field of unlikely Intersections presents a robust paradigm for problems in which several subjects intermingle: arithmetic, o-minimality, and hodge theory. The goal of this talk will be to introduce some of those connections. My aim is to...

Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. What can we say about this? When all the angles of the polygon are...

In 1947 John Von Neumann and Herman Goldstine, while developing the IAS computing machines, wrote a seminal paper on numerical errors in matrix computations. They suggested modeling the "computing noise" (coming from rounding errors, transcendental...

I'll give a brief introduction to o-minimality and how it can be used to prove asymptotic estimates for the number of rational points in definable sets. I'll then show how problems from various areas of mathematics can be reformulated as questions...

The SAT (Boolean Satisfiability) problem asks whether a given logical formula on $n$ Boolean variables has an assignment of true/false values to its variables that makes the formula true.  The P vs NP question is equivalent to asking whether SAT has...

Langlands, inspired by his work on classification of representations of real groups, initiated a program of classifying representations of certain topological groups associated to reductive groups in terms of Langlands parameters. In the recent...

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the “best” for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its...

Riemannian metrics are the simplest generalizations of Euclidean geometry to smooth manifolds. The Ricci curvature of a metric measures, in an averaged sense, how the geometry deviates from being flat. The tensor $-2\,\mathrm{Ric}$ can be viewed as...

Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover...

Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However...

The fundamental group $\pi_1(X)$ is an important invariant of a complex algebraic variety X.  Despite its topological nature, it is closely connected to the geometry of many algebraic structures on X.  In this talk I want to discuss two elementary...

The P vs. NP problem was formulated about 50 years ago, and was chosen to be one of the seven Clay millenium problems 25 years ago. In this period our understanding of the depth, breadth and impact of the problem has changed dramatically. I plan to...

I will describe an application of point-counting on tame real sets and functional transcendence to a problem in harmonic analysis. I will touch on the traditional diophantine applications of these ideas to highlight the new features of this not-so...

Homological stability has emerged over the past decades as an organizing principle in topology and beyond. Broadly speaking, many sequences of moduli spaces exhibit the striking phenomenon that their homology stabilizes as the underlying complexity...

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This talk will recall the Birch--Swinnerton-Dyer Conjecture and describe highlights of the progress towards it, both pre- and post-millennium.  This will largely be a repeat of the speaker's plenary talk from the 2025 Clay Research conference on the...

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