Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 20, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 20, 2015

11:00 am in 243 Altgeld Hall,Tuesday, October 20, 2015

#### Weight structures and the algebraic K-theory of stable $\infty$-categories

###### Ernie Fontes (UT Austin)

Abstract: Algebraic K-theory is a spectral invariant of module categories with applications to number theory and manifold geometry. Recently, various people have used the technology of $\infty$-categories to establish universal characterizations for K-theory. Many of the basic structural results about K-theory have been elevated to apply in the $\infty$-categorical context. I will describe Waldhausen's sphere theorem, a new analogous result for the algebraic K-theory of stable $\infty$-categories, and some applications of the new theorem.

12:00 pm in Altgeld Hall 345,Tuesday, October 20, 2015

#### Gap Distributions in Circle Packings

###### Xin Zhang (UIUC Math)

Abstract: Given a configuration of finitely many tangent circles, one can form a packing of infinitely many circles by Möbius inversions. Fixing one circle from such a packing, we study the distribution of tangencies on this circle via the spectral theory of automorphic forms. Specifically, we will use Anton Good's theorem to show that these tangencies are uniformly distributed when naturally ordered by a growing parameter, and the limiting gap distribution exists, which is conformally invariant. This is a joint work with Zeev Rudnick.

1:00 pm in 345 Altgeld Hall,Tuesday, October 20, 2015

#### Some new logical zero-one laws

###### Caroline Terry   [email] (UIC Math)

Abstract: Suppose $\mathcal{L}$ is a finite first-order language and for each integer $n$, suppose $F(n)$ is a set of $\mathcal{L}$-structures with underlying set $\{1,\ldots, n\}$. We say the family $F=\bigcup_{n \in \mathbb{N}} F(n)$ has a zero-one law if for every first order sentence $\phi$, the proportion of elements in $F(n)$ which satisfy $\phi$ goes to zero or one as $n\rightarrow \infty$. In this talk we give a brief overview of the history of this topic, then present some new examples of families with zero-one laws. This is joint work with Dhruv Mubayi.

2:00 pm in 347 Altgeld Hall,Tuesday, October 20, 2015

#### Double Roots of Random Littlewood Polynomials

###### Arnab Sen   [email] (University of Minnesota)

Abstract: We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions. This is joint work with Ohad Feldheim, Ron Peled and Ofer Zeitouni.

3:00 pm in 243 Altgeld Hall,Tuesday, October 20, 2015

#### Equivariant motivic cohomology

###### Jeremiah Heller (UIUC Math)

Abstract: Motivic cohomology is an important invariant of smooth varieties and a fundamental tool for understanding algebraic K-theory. Joint with Mircea Voineagu and Paul Arne Ostvaer we have construced an equivariant version of motivic cohomology, for a finite group G, using equivariant cycle complexes a la Suslin-Voevodsky. These are naturally graded by the representations of G and are in general different than Edidin-Graham's higher Chow groups. The motivating case is G=Z/2, where these are expected to be related to Hermitian algebraic K-theory of rings with involution. In this talk I'll discuss these invariants, their properties, and how these compare to topological invariants for complex varieties with involution.

3:00 pm in 241 Altgeld Hall,Tuesday, October 20, 2015

#### Adding edges to increase the chromatic number of a graph

###### Alexandr Kostochka   [email] (UIUC Math)

Abstract: For a positive integer $k$ and a connected graph $G$ with at least $k+1$ vertices, let $f(G,k)$ denote the minimum number $m$ of edges such that after adding $m$ edges (and any number of vertices) to $G$ we obtain a graph with chromatic number at least $k+1$. Since $G$ is connected, we can add at most $\binom{k}{2}$ to some subtree of $G$ with $k$ edges so that the resulting graph contains the complete graph $K_{k+1}$. Thus $f(G,k)\leq \binom{k}{2}$ for every connected graph $G$ with at least $k+1$ vertices. One may expect that if in addition $G$ is $k$-chromatic, then $f(G,k)< \binom{k}{2}$. However, in the seventies, Bollob\' as suggested that for every $k\geq 3$ there exists a $k$-chromatic connected graph $G_k$ with at least $k+1$ vertices such that $f(G_k,k)= \binom{k}{2}$. The goal of this talk is to prove this suggestion. This is joint work with J. Nešetřil.

4:00 pm in 245 Altgeld Hall,Tuesday, October 20, 2015

#### Patterns of Phase-Shift Synchrony: From Animal Gaits to Binocular Rivalry

###### Martin Golubitsky (Ohio State University)

Abstract: This talk will review previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system. We will also discuss how rigid phase-shift synchrony is related to symmetry.