Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 12, 2016.

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Tuesday, April 12, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 12, 2016

A new approach to etale homotopy theory

David Carchedi (George Mason)

Abstract: Etale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a pro-object in the homotopy category of spaces, and can be used as a tool to extract topological invariants of the scheme in question. It is a celebrated theorem of theirs that, after profinite completion, the etale homotopy type of an algebraic variety of finite type over the complex numbers agrees with the homotopy type of its underlying topological space equipped with the analytic topology. We will present work of ours which offers a refinement of this construction which produces a pro-object in the infinity-category of spaces (rather than its homotopy category) and applies to a much broader class of objects, including all algebraic stacks. We will also present a generalization of the previously mentioned theorem of Artin-Mazur, which holds in much greater generality than the original result.

12:00 pm in Altgeld Hall 243,Tuesday, April 12, 2016

Subset currents on surface groups

Dounnu Sasaki (Waseda University, Tokyo)

Abstract: Subset currents on free groups and on surface groups, which were introduced by I. Kapovich and T. Nagnibeda in 2013, are the generalization of geodesic currents on surface groups originally studied by Bonahon in late 1980s. The space of geodesic currents on the fundamental group $G$ of a closed hyperbolic surface $\Sigma$ can be thought of as a measure-theoretic completion of the set of all conjugacy classes in $G$. Similarly, subset currents are measure-theoretic generalizations of conjugacy classes of finitely generated subgroups of $G$. Geodesic currents have been successfully used in the study of the Teichm\”uller space and the mapping class group of $\Sigma$. In this talk I will discuss some new results on subset currents on surface groups. To every nontrivial finitely generated subgroup $H\le G=\pi_1(\Sigma)$ we associate a "counting subset current" $\eta_H$. We prove that the set of all scalar multiples of counting currents is dense in the space of all subset currents on $\Sigma$, generalizing a result of Kapovich and Nagnibeda in the free group case. We extend Bonahon's "geometric intersection number" between geodesic currents to an intersection number between subset currents. We also construct a continuous linear "Euler characteristic" functional on the space of subset currents on $\Sigma$ such that, when evaluated on a counting current $\eta_H$, this functional computes the Euler characteristic of the surface corresponding to the core of the cover of $\Sigma$ corresponding to $H$.

1:00 pm in 345 Altgeld Hall,Tuesday, April 12, 2016

The Set of Distal Points of a Flow

Robert Kaufman (UIUC)

Abstract: Let M be a compact metric space, and G a group of homeomorphisms of M. [In the example below, G is cyclic.] For each pair of points (x,y) in M, we define r(x,y) = inf d(gx,gy), (g in G). Then the pair is "proximal" if r(x,y)=0, otherwise "distal". And x is distal if it is proximal only to itself. By elementary arguments, the set D of distal points is co-analytic. There is almost a converse to this: every co-analytic set in M is the set of distal points of a flow on a slightly larger space M' containing M. All of this depends on elementary methods, but a slight variation leads to a much more difficult problem: what happens if the flow is minimal?

2:00 pm in 347 Altgeld Hall,Tuesday, April 12, 2016

Random Walk with Site-Based Feedback

Nick Travers (Indiana University, Bloomington)

Abstract: We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$. We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either $R=1$ or $L=1$ and when $R=L=2$. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed. Our model is related to the excited random walk model introduced by Benjamini and Wilson, and some extensions thereof. Many of the same techniques from the study of excited random walks are used in our analysis as well. Joint work with Ross Pinsky.

3:00 pm in 243 Altgeld Hall,Tuesday, April 12, 2016

Test Ideals in F-regular rings

Kevin Tucker (UIC)

Abstract: In this talk, I will discuss an answer to a question of Mustata-Yoshida -- showing that every ideal in a strongly F-regular ring can be realized as a test ideal. I will also parallel the story in characteristic zero for multiplier ideals, and highlight some open questions in this direction.

3:00 pm in 241 Altgeld Hall,Tuesday, April 12, 2016

On solution-free sets of integers

Andrew Treglown (University of Birmingham)

Abstract: Given a linear equation L, a set A, which is a subset of [n], is L-free if A does not contain any 'non-trivial’ solutions to L. We consider the following three general questions: (i) What is the size of the largest L-free subset of [n]? (ii) How many L-free subsets of [n] are there? (iii) How many maximal L-free subsets of [n] are there? We obtain (exact) results for all three questions for a number of three-variable linear equations L. To attack (iii) we use an approach developed by Balogh and his coauthors which in turn makes use of container and removal lemmas of Green. This is joint work with Robert Hancock.

4:00 pm in 243 Altgeld Hall,Tuesday, April 12, 2016

An intro to the theory of Stöhr-Voloch with applications

Dane Skabelund (UIUC Math)

Abstract: In their 1985 paper "Weierstrass Points and Curves over Finite Fields", Stöhr and Voloch associate to a projective embedding of an algebraic curve some simple invariants measuring the interaction between the Frobenius map and the generic osculating spaces to the curve. They then use these invariants to give a very pretty proof of the Riemann hypothesis for curves over finite fields. This talk will be mostly an exposition of that paper, along with discussion of a few applications for the study so-called "maximal curves".

4:00 pm in 245 Altgeld Hall,Tuesday, April 12, 2016

CANCELLED

Marc Goovaerts (Katholieke Universiteit Leuven)

5:00 pm in Illini Union Ballroom,Tuesday, April 12, 2016

Department Awards Banquet

Abstract: The Department of Mathematics Awards Banquet will be held from 5-8 pm in the Illini Union Ballroom.