Seminars Sorted by Series

Andrei Rapinchuk and I have introduced a new notion of ``weak-commensurability’’ of subgroups of two semi-simple groups. We have shown that existence of weakly-commensurable Zariski-dense subgroups in semi-simple groups G_1 and G_2 lead to strong...

A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and...

My goal in this talk is to survey some of the emerging applications of polynomial methods in both learning and in statistics. I will give two examples from my own work in which the solution to well-studied problems in learning and statistics can be...

Part of geometric representation theory involves constructing representations of algebras on the cohomology of algebraic varieties. A great example of such a construction is the work of Nakajima and Grojnowski, who independently constructed an...

Heegaard Floer homology groups were recently introduced by Ozsvath and Szabo to study properties of 3-manifolds and knots in them. The definition of the invariants rests on delicate holomorphic geometry, making the actual computations cumbersome. In...

I will give an introduction to the problem of parallel repetition of two-prover games and its applications and related results in theoretical computer science (the PCP theorem, hardness of approximation), mathematics (the geometry of foams, tiling...

I will give an introduction to symplectic geometry and Hamiltonian systems and then introduce an invariant called symplectic cohomology. This has many applications in symplectic geometry and has been used a lot especially in the last 5-10 years. I...

Quillen's higher K-groups, defined in 1971, paved the way for motivic cohomology of algebraic varieties. Their definition as homotopy groups of combinatorially constructed topological spaces initially seems abstract and inaccessible. In this talk...

Patching methods are usually used to construct global objects from more local ones. On the other hand, algebraic objects can sometimes be understood from their local behavior, i.e., they satisfy a local-global principle. In this talk, we explain a...

35 years ago Wehrl defined a classical entropy of a quantum density matrix using Gaussian (Schr\"odinger, Bargmann, ...) coherent states. This entropy, unlike other classical approximations, has the virtue of being positive. He conjectured that the...

This talk is intended for a general audience. The recent discovery of an interpretation of constructive type theory into abstract homotopy theory has led to a new approach to foundations with both intrinsic geometric content and a computational...

This talk is designed for a general mathematical audience; no prior knowledge of type theory is presumed. One of the main goals for the special year on univalent foundations is the development of a logical formalism, called homotopy type theory...

A general algebraic formalism for the mathematical modeling of physical systems is sketched. This formalism is sufficiently general to encompass classical and quantum-mechanical models. It is then explained in which way quantum theory differs in an...

A `toy model' for studying the probabilistic distribution of nodal curves of eigenfunctions of linear operators arises from the Laplacian on the standard real 2-torus. Here the eigenvalues are associate to integers m that are sum of two squares...

This is the first of two talks in which the speaker will discuss the development of the theory of Toeplitz matrices and determinants in response to questions arising in the analysis of the Ising model of statistical mechanics. The first talk will be...

``Quantum ergodicity'' usually deals with the study of eigenfunctions of the Laplacian on Riemannian manifolds, in the high-frequency asymptotics. The rough idea is that, under certain geometric assumptions (like negative curvature), the...

Mirror symmetry is a deep conjectural relationship between complex and symplectic geometry. It was first noticed by string theorists. Mathematicians became interested in it when string theorists used it to predict counts of curves on the quintic...

Fundamental questions in basic and applied ecology alike involve complex adaptive systems, in which localized interactions among individual agents give rise to emergent patterns that feed back to affect individual behavior. In such systems, a...

In its simplest form, Hirzebruch's 1953 Riemann-Roch theorem is an identity between certain combinations of Hodge numbers on the one hand and certain combinations of Chern numbers on the other. I will show that there are no other such identities...

A fundamental theorem in linear algebra is that any real n x d matrix has a singular value decomposition (SVD). Several important numerical linear algebra problems can be solved efficiently once the SVD of an input matrix is computed: e.g. least...

Discussions about constructive mathematics are usually derailed by philosophical opinions and meta-mathematics. But how does it actually feel to do constructive mathematics? A famous mathematician wrote that "taking the principle of excluded middle...

We illustrate how bounded cohomology with coefficients can be used to prove rigidity theorems for groups acting on non-positively curved spaces, among which CAT(0) cube complexes.

In the middle of the sixties, A. Lichnerowicz raised the following simple question: “Is the round sphere the only compact Riemannian manifold admitting a noncompact group of conformal transformations?” The talk will present the developments which...

The Weil height measures the “complexity” of an algebraic number. It vanishes precisely at 0 and at the roots of unity. Moreover, a finite field extension of the rationals contains no elements of arbitrarily small, positive heights. Amoroso...

In this seminar, we will discuss the recent work on the eigenvalue and eigenvector distributions of random matrices. We will discuss a dynamical approach to these problems and related open questions. We will discuss both Wigner type matrix ensembles...

We discuss the classical and non-commutative geometry of wire systems which are the complement of triply periodic surfaces. We consider a \(C^*\)-geometry that models their electronic properties. In the presence of an ambient magnetic field, the...

We will review some interactions between random matrix theory and distributions of zeroes of \(L\)-functions in families (the Katz-Sarnak philosophy) before presenting some recent results (joint with Dorian Goldfeld) in the higher rank setting. We...

This is intended to be a survey talk, accessible to a general mathematical audience. The cdh topology was created by Voevodsky to extend motivic cohomology from smooth varieties to singular varieties, assuming resolution of singularities (for...

A widely studied model from statistical physics consists of many (one-dimensional) Brownian motions interacting through a pair potential. The large scale behavior of this model has has been investigated by Varadhan, Yau, and others in the 90's. As a...

The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the...

I will present some recent applications of symplectic geometry to the restricted three body problem. More specifically, I will discuss how Gromov's original study of pseudoholomorphic curves in the complex projective plane has led to the...

I describe recent results on spiked covariance matrices, which model multivariate data containing nontrivial correlations. In principal components analysis, one extracts the leading contribution to the covariance by analysing the top eigenvalues and...

Consider a rational homogeneous variety \(X\). (For example, take \(X\) to be the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space.) The Schubert classes of \(X\) form a free additive basis of the integral homology of \(X\)...

This will be a gentle intro, aimed at a general mathematical audience, to supergeometry: supermanifolds, super Riemann surfaces, super moduli, etc. As time permits, we will discuss various aspects of supergeometry, including deformation theory and...

After a brief review of various classical connections between problems of polynomial zero localization, continued fractions, and matrix theory, I will show a few ways to generalize these classical techniques to get new results about some interesting...

We discuss various Gaussian ensembles for real homogeneous polynomials in several variables and the question of the distribution of the topologies of the connected components of the zero sets of a typical such random real hypersurface. For the "real...

The Grothendieck ring of varieties over \(k\) is defined to be the free abelian group generated by varieties over \(k\), modulo the relation \([X] = [Y] + [X \backslash Y]\) for all \(X\) and closed subvarieties \(Y\). Multiplication is induced by...

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the...

We revisit the question, originally posed by Yao (1982), of whether encryption security may be characterized using computational information. Yao provided an affirmative answer, using a compression-based notion of computational information to give a...