Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 28, 2015.

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Tuesday, April 28, 2015

11:00 am in 243 Altgeld Hall,Tuesday, April 28, 2015

Topological Hochschild homology and Atiyah duality

Cary Malkiewich (UIUC)

Abstract: We describe how some classical theorems on the Hochschild homology of cochains in X lift to theorems about the topological Hochschild homology of the ring spectrum DX. In particular, we focus on the duality between THH(DX) and LX, which is a special case of the Ayala-Francis Poincare/Koszul duality theorem. When X is a smooth manifold, this duality may be interpreted as Atiyah duality for the free loop space LM, which is an infinite-dimensional manifold. We also explain how to prove that this duality is equivariant, using some surprising technical results that have consequences for the general study of orthogonal G-spectra.

1:00 pm in 347 Altgeld Hall,Tuesday, April 28, 2015

The Fuglede Conjecture holds in ${\Bbb Z}_p \times {\Bbb Z}_p$

Alex Iosevich (University of Rochester)

Abstract: We shall prove that if $E \subset {\Bbb Z}_p^2$ possesses an orthogonal basis of characters, then $E$ tiles ${\Bbb Z}_p^2$ by translation. This problem arose in 1974 when Fuglede stated and proved this conjecture in Euclidean space in the case when either the tiling set or the spectrum is a lattice. In 2004 Tao disproved the Fuglede Conjecture in Euclidean space by producing a counter-example initiation constructed in ${\Bbb Z}_3^5$.This led to an interesting question of just how tiling and spectral properties are related in a variety of continuous and arithmetic settings. Our result is a step in this direction. Geometric combinatorics and elementary Galois theory play the key role in the proof. This is joint work with Azita Mayeli and Jonathan Pakianathan.

1:00 pm in Altgeld Hall 243,Tuesday, April 28, 2015

An effective asymptotic result for the Lebesgue measure of the sum-level sets for continued fractions

Byron Heersink (UIUC Math)

Abstract: For every positive integer $n$, let $C_n$ be the set of real numbers in $[0,1]$ whose continued fraction expansion $[a_1,a_2,...]$ satisfies $a_1+...+a_k=n$ for some $k$. Using results from infinite ergodic theory, Kessebohmer and Stratmann proved that the Lebesgue measure of $C_n$ is asymptotically equivalent to $1/\log_2 n$ as $n$ approaches $\infty$. In this talk, we provide a simplified proof of this result, mostly using basic properties of the transfer operator of the Farey map and Karamata's Tauberian theorem, while avoiding most of the ergodic results in the proof of Kessebohmer and Stratmann. Additionally, we obtain an error term by adapting Freud's effective version of Karamata's theorem to this situation.

1:00 pm in 345 Altgeld Hall,Tuesday, April 28, 2015

Oligomorphic groups: representations, property (T), and measure-preserving actions

Todor Tsankov (University of Paris 7)

Abstract: A permutation group G acting on a countable set M is called oligomorphic if its diagonal action on finite powers of M has only finitely many orbits. Such groups arise naturally as automorphism groups of omega-categorical structures. It turns out that in a number of situations, those groups behave in a way similar to compact groups: for example, they all have property (T) and often their actions can be classified. In the talk, I will concentrate on unitary representations and measure-preserving actions. Some of the results I am going to present are joint work with David Evans.

2:00 pm in 347 Altgeld Hall,Tuesday, April 28, 2015

Fractal properties of rough differential equations driven by fractional Brownian motion

Shuwen Lou (UIC)

Abstract: We will introduce fractal properties of rough differential equations driven by frational Brownian motion with Hurst parameter H>1/4. We will first survey some known results on density and tail estimates of such processes. Then we will show the Hausdorff dimension of the sample paths is equal to min(d, 1/H), where d is the dimension of the process. Also we will show that with positive probability, the level sets in the form of {t: X_t=x } has Hausdorff dimension 1-dH when dH<1, and are almost surely empty otherwise.

3:00 pm in 243 Altgeld Hall,Tuesday, April 28, 2015

Normality of Secant Varieties

Brooke Ullery (Michigan)

Abstract: If X is a smooth variety embedded in projective space, we can form a new variety by looking at the closure of the union of all the lines through 2 points on X. This is called the secant variety to X. Similarly, the Hilbert scheme of 2 points on X parametrizes all length 2 zero-dimensional subschemes. I will talk about how these two constructions are related. More specifically, I will show how we can use certain tautological vector bundles on the Hilbert scheme to help us understand the geometry of the secant variety, leading to a proof that for sufficiently positive embeddings of X, the secant variety is a normal variety.

3:00 pm in 241 Altgeld Hall,Tuesday, April 28, 2015

Multicoloring and Paintability

Thomas Mahoney   [email] (UIUC Math)

Abstract: Let $G$ be a graph, and let $f,g:V(G)\to\mathbb N$. A $g$-fold coloring of $G$ assigns to each vertex $v$ a set of $g(v)$ colors such that adjacent vertices receive disjoint sets. We say that $G$ is $(f,g)$-choosable if for any list assignment $L$ satisfying $|L(v)|\ge f(v)$, there exists a $g$-fold coloring where the set of $g(v)$ colors is chosen from $L(v)$. When $f(v)=a$ and $g(v)=b$ for all $v$ and $G$ is $(f,g)$-choosable, we say that $G$ is $(a,b)$-choosable. Schauz and Zhu independently introduced an online version of list coloring called paintability, where the coloring algorithm sees colors presented one at a time. Suppose that on round $i$, the coloring algorithm must decide which vertices will receive color $i$ without knowing which colors will appear later in the lists. This process is modeled by the Lister/Painter game. Each vertex $v$ begins with a positive number $f(v)$ of tokens. In each round, Lister marks a nonempty subset of vertices that have not received $g(v)$ colors; each marked vertex loses one token. Painter colors an independent subset of the marked vertices. Lister wins the game by marking a vertex with no tokens, and Painter wins by creating a $g$-fold coloring of $G$. When Painter has a winning strategy in this game, we say that $G$ is $(f,g)$-paintable. When $f(v)=a$ and $g(v)=b$ for all $v$ and $G$ is $(f,g)$-paintable, we say that $G$ is $(a,b)$-paintable. We show that $C_{2k+1}$ is $(a,b)$-paintable if and only if $a\ge 2b+\left\lceil\frac{b}{k}\right\rceil$. We prove that planar graphs are $(5m,m)$-paintable for all $m$, which strengthens results of Thomassen, Schuaz, and of Tuza and Voigt. Strengthening Brooks' Theorem, we prove that if $G$ is a connected and not a complete graph or odd cycle, then $G$ is $(\Delta(G)m,m)$-paintable for all $m$.

4:00 pm in 243 Altgeld Hall,Tuesday, April 28, 2015

On the zeros of linear combinations of the Riemann zeta-function and its derivatives

Paolina Koutsaki (UIUC Math)

Abstract: In this talk, we begin by presenting some classical results about the number of zeros of the Riemann zeta-function. We then discuss Bruce C. Berndt's paper from 1970 on the number of zeros for the derivatives of the Riemann zeta-function and show how similar techniques can be used to study the zeros of linear combinations of the zeta-function and its derivatives. No prior knowledge of analytic number theory will be assumed. This is joint work with A. Tamazyan and A. Zaharescu.

4:00 pm in 245 Altgeld Hall,Tuesday, April 28, 2015

The hitchhiker's guide to Levy processes

Alexey Kuznetsov (York University)

Abstract: What is the connection between the financial time series and grazing patterns of bacteria, cash flows of insurance companies and fluctuations and transport in plasma, seismic series and spreading of epidemic processes? The answer is: Levy processes, which are used in modeling all of these diverse phenomena. I will begin this talk with a gentle introduction to Levy processes followed by some of their applications. Then I will focus on exponential functionals of Levy processes, and I will describe how they fit in the theory of positive self-similar Markov processes in general and stable processes in particular. After discussing some analytical tools and techniques which are used in studying the exponential functional, I will explain why the Answer to the Great Question of Life, the Universe and Stable Processes is exactly forty-two.