Seminars Sorted by Series

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...

It is well known that a finitely generated torsion abelian group A is finite, and thus carries no `rational' information. However, if the torsion group A is not finitely generated, then there exist meaningful ways to extract rational information out...

Constant mean curvature (CMC) tori in \(S^3\), \(R^3\) or \(H^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the...

If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By...

My talk will be a broad introduction to what is the (mostly conjectural) higher dimensional generalization of Abel's theorem on divisors on Riemann surfaces, namely, the relationship between the structure of the group of algebraic cycles on a...

Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know...

After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of \(\zeta(3)\). Then I will give an overview of subsequent results in this field, and finally propose a...

Shot-noise random fields can model a lot of different phenomena that can be described as the additive contributions of randomly distributed points. In the first part of the talk, I will give some properties of these random fields. And in a second...

Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re...

Start with a compact Riemann surface \(X\) and a complex reductive group \(G\), like \(\mathrm{GL}(n,\mathbb C)\). According to Hitchin-Simpson's ``non abelian Hodge theory", the pair \((X,G)\) comes with two new complex manifolds: the character...

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

Ball quotients are complex manifolds appearing in many different settings: algebraic geometry, hyperbolic geometry, group theory and number theory. I will describe various results and conjectures on them.

Algebraic cycles are linear combinations of algebraic subvarieties of an algebraic variety. We want to know whether all algebraic subvarieties can be built from finitely many, in a suitable sense. We present some recent results and counterexamples.

The data of a compact Lie group $G$ and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of the infinite dimensional group of loops in $G$. Previous work with...

A classical problem in the theory of automorphic forms is to count the number of Laplace eigenfunctions on the quotient of the upper half plane by a lattice $L$. For $L$ a congruence subgroup in $\mathrm{SL}(2,\mathbb Z)$ the Weyl law was proven by...