# Colloquia/Fall18

# Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, **unless otherwise indicated**.

## Fall 2015

Go to next semester, Spring 2016.

## Abstracts

### September 4: Isaac Goldbring (UIC)

Title: On Kirchberg's embedding problem

Abstract: In his seminal work on the classification program for nuclear C*-algebras, Kirchberg showed that a particular C*-algebra, the Cuntz algebra O2, plays a seminal role. Subsequent work with Chris Phillips showed that O2 also plays a prominent role in regards to the wider class of exact C*-algebras, and this led Kirchberg to conjecture that every C*-algebra is finitely representable in O2, that is, is embeddable in an ultrapower of O2. The main goal of this talk is to sketch a proof of a local finitary reformulation of this conjecture of Kirchberg. The proof uses model theory and in particular the notion of model-theoretic forcing. No knowledge of C*-algebras or model theory will be assumed. This is joint work with Thomas Sinclair.

### September 11: Doron Puder (IAS)

Title: Word-Measures on Groups.

Abstract: Let w be a word in the free group on k generators, and let G be a finite (compact) group. The word w induces a measure on G by substituting the letters of w with k independent uniformly (Haar) chosen random elements of G and evaluating the product. Questions about word-measures on groups attracted attention in recent years both for their own sake and as a tool to analyze random walks on groups.

We will explain some properties of word-measure, give examples and state conjectures. We will also talk about recent results regarding word-measures on symmetric groups and word-measures on unitary groups.

### September 18: Izzet Coskun (UIC)

Title: The geometry of points in the plane

Abstract: Grothendieck's Hilbert scheme of points is a smooth compactification of the configuration space of points in the plane. It has close connections with combinatorics, representation theory, mathematical physics and algebraic geometry. In this talk, I will survey some of the basic properties of this beautiful space. If time permits, I will discuss joint work with Arcara, Bertram and Huizenga on codimension one subvarieties of the Hilbert scheme.

### September 25: Ourmazd (UW-Milwaukee)

Title: Structure and Dynamics from Random Observations

Abstract: At weddings, the bridal photo is taken under bright lights, with the happy couple holding still. Traditionally in science, the “best” observations are those with the largest signal from the most tightly controlled system. Like bridal photos, the results are not always exciting. In a wide range of phenomena – from the dance of proteins during their function, to the breaking of molecular bonds on the femtosecond scale – tight control is neither possible, nor desirable. Modern data-analytical techniques extract far more information from random sightings than usually obtained from set-piece experiments. I will describe on-going efforts to extract structural and dynamical information from noisy, random snapshots. Examples will include YouTube videos, the structure and conformations of molecular machines such as the ribosome, and the ultrafast dynamics of bond-breaking in small molecules like nitrogen.

#### October 9: Chanwoo Kim

Title: Coercivity in the Boltzmann equation

Abstract: The Boltzmann equation is a fundamental equation of rarefied gas. Around the natural steady state, so called Maxwellian, a linearized operator is degenerated coercive. In this talk we will see how to recover this degenerated part so that the linearized operator is coercive effectively.

### October 16: Hadi Salmasian (Ottawa)

Title: The Capelli problem and spectrum of invariant differential operators

Abstract: The Capelli identity is a mysterious result in classical invariant theory with a long history. It was demystified by Roger Howe, who used it in an ingenious and elegant fashion in the modern theory of representations of real reductive groups. In this talk, I will introduce the Capelli identity, and exhibit the relationship between an extension of this identity with certain polynomials which describe the spectrum of invariant differential operators on symmetric superspaces. These polynomials are analogs of the Jack and Knop-Sahi/Okounkov-Olshanski polynomials. This talk is based on a joint project with Siddhartha Sahi.

### October 23: Lu Wang

Title: Singularities of Mean Curvature Flow

Abstract: Mean curvature flow (MCF) of hypersurfaces is the gradient flow of volume functional, which decreases the volume in its steepest way. Any compact MCF will develop singularities in finite time, which are modeled by self-shrinkers, a special class of solutions of MCF. Recently, Colding-Minicozzi proposed a dynamical approach to study the singularities formation of MCF. In this talk, I will survey some progress in the classification of self-shrinkers (from different point views) as well as some major open problems. Part of the work is joint with Jacob Bernstein.

### October 30: Ruth Charney (Brandeis)

Title: Finding hyperbolic behavior in non-hyperbolic spaces

Abstract: In the early 90’s, Gromov introduced a notion of hyperbolicity for geodesic metric spaces. The study of groups of isometries of such spaces has been an underlying theme of much of the work in geometric group theory since that time. Many geodesic metric spaces, while not hyperbolic in the sense of Gromov, nonetheless display some hyperbolic-like behavior. I will discuss a new invariant, the Morse boundary of a space, designed to capture this behavior. This is joint work with Harold Sultan, together with recent work of my students Matt Cordes and Devin Murray.

### November 6: Chris Rycroft (Harvard)

Title: Interfacial dynamics of dissolving objects in fluid flow

Abstract: An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow will be presented. By taking advantage of conformal invariance of the model, a numerical method will be introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of several dissolving objects will be shown, all of which show collapse to a single point in finite time. The simulations reveal a surprising connection between the position of the collapse point and the initial Laurent coefficients, which was subsequently derived analytically using residue calculus.

### November 13: David Fisher (Indiana)

Title: Rigidity of quasi-isometric embeddings

Abstract: In geometric group theory, one defines a metric on finitely generated groups and then asks when algebraic properties are related to metric ones. The most famous example is Gromov's theorem on polynomial growth which state that a group has polynomial growth iff it has a finite index subgroup which is nilpotent. In this talk I will focus on when a geometric mapping is in fact algebraic and so an isomorphism or homomorphism. For the case of isomorphisms, this phenomena was first discovered by Schwartz, with other examples followed in work of Farb-Schwartz and Eskin. I will talk about the first results of this kind for homomorphisms that are not onto; this is joint work with Thang Nguyen. The talk will be accessible to graduate students.

### November 20: Avy Soffer

Nonlinear Long Range Scattering and Normal Form Analysis"

Abstract

First I will describe the source and nature of long range dynamics in general. This fundamental effect is responsible to the change in the asymptotic behavior of the system at large times. It is present in Coulomb and Gravitational dynamics, in theories with mass-less particles (gauge theories) and in low power nonlinear dispersive and hyperbolic equations. Then, I will describe new results and new Normal Form techniques to deal with the nonlinear Klein-Gordon equation in one dimension, with quadratic and variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. We prove global existence and (in L-infinity) scattering as well as a certain kind of strong smoothness for the solution at time-like infinity; it is based on several new classes of normal-form transformations. The analysis also shows the limited smoothness of the solution, in the presence of the resonances. In particular we observe the phenomena of growth of some Invariant Sobolev norm of high order. This seems to be generic for such nonlinear systems.