Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 10, 2015.

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Tuesday, February 10, 2015

11:00 am in 243 Altgeld Hall,Tuesday, February 10, 2015

The K-Theory of Varieties

Jonathan Campbell (UT Austin)

Abstract: The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all Grothendieck rings, one may hope that it arises as $\pi_0$ of a $K$-theory spectrum, $K(Var_k)$. Using her formalism of assemblers, Zahkarevich showed that this is in fact that case. I'll present an alternate construction of the spectrum that allows us to quickly see the $E_\infty$-structure on $K(Var_k)$ and produce various character maps out of $K(Var_k)$. I'll end with a conjecture about $K(Var_k)$ and iterated $K$-theory.

1:00 pm in 345 Altgeld Hall,Tuesday, February 10, 2015

Distance Structures for Generalized Metric Spaces

Gabriel Conant (UIC)

Abstract: We study the model theory of generalized metric spaces as combinatorial structures in a relational language. In particular, we focus on the Urysohn space over R, denoted U(R), where R is a countable positively ordered commutative monoid. We state several results which illustrate that, if the theory of U(R) has quantifier elimination, then the model theoretic complexity of this theory can be characterized by straightforward algebraic properties of R. In order to understand when quantifier elimination holds for U(R), we first define a nonstandard extension R* of R, with the property that arbitrary models of U(R) can be canonically considered as generalized metric spaces over R*. We then show that quantifier elimination for the theory of U(R) can be characterized by a natural continuity property in R*.

1:00 pm in Altgeld Hall 243,Tuesday, February 10, 2015

Around Bochner-Krall problem

Boris Shapiro (Stockholm University)

Abstract: A linear differential operator $T=\sum_{i=1}^k Q_i(x) \frac{d^i}{dx^i}$ with polynomials coefficients is called {\it exactly solvable} if the degree of each $Q_i(x)$ is at most $i$ and there exists all least one value of $i$ for which the equality holds. One can easily see that such operators are have one eigenpolynomial in every sufficiently large degree. Already in 1929 S.Bochner asked for which exactly solvable operators the corresponding sequence of eigenpolynomials consists of orthogonal polynomials. This problem was considered in large numbers of publications over many decades. I will present modern results about the root asymptotics for sequences of eigenpolynomials of exactly solvable operators, several conjectures and the relation of the latter root asymptotics to quadratic and higher order differentials in the complex plane. View talk at http://youtu.be/zjag_IS6Gos

2:00 pm in 347 Altgeld Hall,Tuesday, February 10, 2015

Stochastic hybrid systems in and out of equilibrium: moment closure, boundedness, and explosions

Lee DeVille (UIUC Math)

Abstract: We present a variety of results analyzing the behavior of a class of stochastic processes — referred to as Stochastic Hybrid Systems — in or near equilibrium, and determine general conditions on when the moments of the process will, or will not, be well-behaved. We also study the potential for finite-time blowups for these processes, and exhibit a set of random ecurrence relations that govern the behavior for long times. This work is joint with Sairaj Dhople, Alejandro Dominguez-Garcia, and Jiangmeng Zhang.

3:00 pm in 243 Altgeld Hall,Tuesday, February 10, 2015

Reductive moduli problems, stratifications, and applications

Daniel Halpern-Leistner (IAS and Columbia)

Abstract: Many moduli problems in algebraic geometry are "too big" to possibly be parameterized by a quasiprojective scheme. Nevertheless one can find a stratification of the moduli problem for which the large open stratum has a good moduli space, and the remaining strata have nice modular interpretations as well. I will introduce a framework for generalizing and analyzing stratifications of this kind arising in geometric invariant theory and in moduli problems for objects in derived categories of coherent sheaves, and I will discuss some applications of these stratifications to understanding the geometry of these moduli problems. This framework leads to the notion of a "reductive moduli problem" (which generalizes the notion of a reductive group) -- these are the moduli problems for which the results of geometric invariant theory generalize in a nice way.

3:00 pm in 241 Altgeld Hall,Tuesday, February 10, 2015

$3$-regular subgraphs of $4$-regular pseudographs and $(3,1)$-colorings

Anton Bernshteyn (UIUC Math)

Abstract: An old conjecture of Berge that every simple $4$-regular graph contains a $3$-regular subgraph (not necessarily spanning), was proved by Tashkinov. The analogous statement for multigraphs (and, more generally, pseudographs) is not true, and the problem of classifying all $4$-regular pseudographs without $3$-regular subgraphs is still open. In particular, it is unknown whether there exist $4$-regular pseudographs without multiple edges (but with loops, at most one at each vertex) that do not contain $3$-regular subgraphs. In this talk we will give an overview of the problem and discuss a particular class of edge colorings ($(3,1)$-colorings), the properties of which turn out to be closely related to the presence of $3$-regular subgraphs.

4:00 pm in 243 Altgeld Hall,Tuesday, February 10, 2015

What are Quasiconformal Mappings?

Colleen Ackermann

Abstract: This will be a light introduction to quasiconformal mappings including various, properties, definitions and examples. I may also discuss some of the major theorems in the field.