The seminar meets on Tuesdays, 14:30-15:30, in Math -101

2019–20–A meetings

Oct 29 Faculty meeting Faculty meeting
Nov 5 Simultaneous normalization of families of isolated singularities GERT-MARTIN GREUEL (Technische Universitat Kaiserslautern)

A singularity refers always to a special situation, something that is not true in general. The term “singularity” is often used in a philosophical sense to describe a frightening or catastrophically situation which is often unknown. Singularity theory in mathematics is a well defined discipline with the aim to tame the “catastrophe”. I will give a general introduction to singularity theory with some examples from real life. Then I consider a special kind of taming a singularity, the normalization, and give an overview of classical and recent results on simultaneous normalization of families of algebraic and analytic varieties. I will also discuss some open problems.

Nov 12 Classification of Singularities in positive characteristic GERT-MARTIN GREUEL (Technische Universitat Kaiserslautern)

The classification of hypersurface singularities aims at writing down a normal form of the defining power series with respect to some equivalence relation, and to give list of normal forms for a distinguished class of singularities. Arnold’s famous ADE-classification of singularities over the complex numbers had an enormous influence on singularity theory and beyond. I will report on some of the impact of his work to other disciplines and to some real-life applications of the classification. Stimulated by Arnold’s work, the classification has been carried on to singularities over fields of positive characteristic, partly with surprising differences. I will report on recent results about this classification and about related problems.

Nov 19 On Benjamini-Schramm convergence Arie Levit (Yale)

Benjamini-Schramm convergence is a probabilistic notion useful in studying the asymptotic behavior of sequences of metric spaces. The goal of this talk is to discuss this notion and some of its applications from various perspectives, e.g. for groups, graphs, hyperbolic manifolds and locally symmetric spaces, emphasizing the distinction between the hyperbolic rank-one case and the rigid high-rank case. Understanding the “sofic” part of the Benjamini-Schramm space, i.e. all limit points of “finitary” objects, will play an important role. From the group-theoretic perspective, I will talk about sofic groups, i.e. groups which admit a probabilistic finitary approximation, as well as a companion notion of permutation stability. Several results and open problems will be discussed.

Nov 26 On bounded continuous solutions of the archetypal equation with rescaling Gregory Defel (BGU)
Dec 3 Cubic Fourfolds: Rationality and Derived Categories Howard Nuer (UIC)

The question of determining if a given algebraic variety is rational is a notoriously difficult problem in algebraic geometry, and attempts to solve rationality problems have often produced powerful new techniques. A well-known open rationality problem is the determination of a criterion for when a cubic hypersurface of five-dimensional projective space is rational. After discussing the history of this problem, I will introduce the two conjectural rationality criteria that have been put forth and then discuss a package of tools I have developed with my collaborators to bring these two conjectures together. Our theory of Relative Bridgeland Stability has a number of other beautiful consequences such as a new proof of the integral Hodge Conjecture for Cubic Fourfolds and the construction of full-dimensional families of projective Hyper Kahler manifolds. Time permitting I’ll discuss a few of the many applications of the theory of relative stability conditions to problems other than cubic fourfolds.

Dec 10 Geometry of integral vectors Uri Shapira (Technion)

Given an integral vector, there are several geometric and arithmetic objects one can attach to it. For example, its direction (as a point on the unit sphere), the lattice obtained by projecting the integers to the othonormal hyperplane to the vector, and the vector of residues modulo a prime p to name a few. In this talk I will discuss results pertaining to the statistical properties of these objects as we let the integral vector vary in natural ways.

Dec 17 Harmonic Analysis on $GL(n)$ over Finite Fields. Shamgar Gurevitch (University of Wisconsin - Madison)

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently (, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms” (P-of-CF), which is (since the 60’s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).

Dec 24 Matrix convexity, Arveson boundaries and Tsirelson problems Adam Dor On (University of Copenhagen)

Following work of Evert, Helton, Klep and McCullough on free linear matrix inequality domains, we ask when a matrix convex set is the closed convex hull of its (absolute) extreme points. This is a finite-dimensional version of Arveson’s non-commutative Krein-Milman theorem, which may generally fail completely since some matrix convex sets have no (absolute) extreme points. In this talk we will explain why the Arveson-Krein-Milman property for a given matrix convex set is difficult to determine. More precisely, we show that this property for certain commuting tensor products of matrix convex sets is equivalent to a weak version of Tsirelson’s problem from quantum information. This weak variant of Tsirelson’s problem was shown, by a combination of results of Kirchberg, Junge et. al., Fritz and Ozawa, to be equivalent to Connes’ embedding conjecture; considered to be one of the most important open problems in operator algebras. We do more than just provide another equivalent formulation of Connes’ embedding conjecture. Our approach provides new matrix-geometric variants of weak Tsirelson type problems for pairs of convex polytopes, which may be easier to rule out than the original weak Tsirelson problem.

Based on joint work with Roy Araiza and Thomas Sinclair

Dec 31 Flavors of bicycle mathematics Sergei Tabachnikov (Penn State University)

This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon:

1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences.

2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation.

3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam’s problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.

Jan 7 Universal models in ergodic theory and topological dynamics Tom Meyerovitch (BGU)

A number of of important results in modern mathematics involve an understanding the space of invariant probability measures for a homeomorphism, a flow, or group of homeomorphisms.

In this talk we will focus on finding situations where the space of invariant probability measures is essentially ``as big as possible’’: A topological dynamical system is $(X,S)$ \emph{universal} in the ergodic sense if any measure preserving system $(Y,T,\mu)$, there exists an S-invariant probability measure $\nu$ so that $(X,S,\nu)$ is isomorphic to $(Y,T,\mu)$ as measure preserving systems, assuming that the entropy of (Y,T,\mu) is strictly lower than the topological entropy of $(X,S)$. Krieger’s generator theorem (1970) states that the shift map on the space bi-infinite of $N$-letter sequences is universal. Lind and Thouvenot (1977) used Kreiger’s theorem to prove that Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations. Recent conditions for universality of Soo-Quas (2016) and David Burguet (2019) imply that any ergodic automorphism of a compact group is universal. Together with Nishant Chandgotia we recently established a new and more general sufficient condition for ergodic universality.

Some new consequences include: - A generic homeomorphism of a compact manifold (having dimension at least 2) can model any aperiodic measure preserving transformation. - Any aperiodic measure preserving transformation can be modeled by a homeomorphism of the 2-torus which preserves Lebesgue measure. - The space of 3-colorings of the standard Cayley graph of $\mathbb{Z}^d$, with $\mathbb{Z}^d$ acting by translations is universal.
In this talk I will discus and explain some of the older and newer results. No specific background in ergodic theory will be assumed.

Jan 14 Enumerative geometry and Lie (super)algebras Michael Polyak (Technion)

One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety.

We explain this counting procedure as a construction of certain cycles on moduli of rigid tropical curves. Cycles on these moduli turn out to be closely related to Lie algebras.

In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Witten descendants) are similarly related to the super-Lie structure on the quantum torus.

[No preliminary knowledge of tropical geometry or the quantum torus algebra is expected.]