Seminars Sorted by Series

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...

We consider compactifications of the Betti, de Rham and Dolbeault realizations of the character variety. Starting from an example, we look at what can be said, mostly conjecturally, about the relationship between these spaces.

Decoupling inequalities in harmonic analysis permit to bound the Fourier transform of measures carried by hyper surfaces by certain square functions defined using the geometry of the hyper surface. The original motivation has to do with issues in...

Chern-Schwartz-MacPherson class is a functorial Chern class defined for any algebraic variety. I will give a geometric proof of a positivity conjecture of Aluffi and Mihalcea that Chern classes of Schubert cells and varieties in Grassmannians are...

One of the most effective methods for solving boundary value problems for basic elliptic equations of mathematical physics in a given domain is the method of layer potentials. Its essence is to reduce the entire problem to an integral equation on...

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable $t$. We prove that for infinitely many values of $t$ the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is...

I'll review recent progress on properties of 3-manifold groups, especially following from geometric properties. Then I'll discuss some open questions regarding 3-manifold groups, including their profinite completions, torsion in covers, orderability...

What can you learn about a group from a presentation? Sometimes very little; almost every interesting problem about groups given by (finite) presentations is unsolvable in full generality. But if one asks about *typical* groups - so-called "random"...

A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the classical is ergodic, almost all of the Laplace eigenfunctions become uniformly distributed in...

There are various notions of an ``optimal'' position for a knot in the 3-sphere. Beginning with Schubert's bridge position, I'll talk about some 1- and 2-parameter definitions of complexity for a knot, and explain some of the reasons they have been...

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log...

There is a natural action of the group $\mathrm{SL}(2,\mathbb R)$ of $2 \times 2$ matrices on the unit tangent bundle of the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models, which allows one to make...

Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In...