Mathematics

Coassembly in algebraic K-theory

Cary Malkiewich   [email] (UIUC)

Abstract: The coassembly map allows us to approximate any contravariant homotopy-invariant functor by an excisive functor, i.e. one that behaves like a cohomology theory. We apply this construction to a contravariant form of Waldhausen's algebraic K-theory of spaces, and its corresponding THH functor. The results are somewhat surprising: a certain dual form of the A-theory Novikov conjecture is false, but when the space in question is the classifying space BG of a finite p-group, coassembly on THH is split surjective after p-completion. The method of proof suggests new conjectures about both the assembly and coassembly maps for the A-theory of BG. If there is time, we will also discuss related work on the equivariant structure of THH.

Hitting time statistics for observations and application to random dynamical systems

Jerome Rousseau (Universidade Federal da Bahia/Illinois)

Abstract: We study the distribution of hitting and return times for observations of dynamical systems. We apply this results to get an exponential law for the distribution of hitting and return time for rapidly mixing random dynamical systems. For random subshifts of finite type, we analyze the distribution of hitting times with respect to the sample measures and prove that under fast mixing assumptions one can get an exponential law. View lecture at http://youtu.be/tN0jpVRCkfU

Tameness in Abstract Elementary Classes

Will Boney (UIC)

Abstract: Tameness is a locality property of Galois types in AECs. Since its isolation by Grossberg and VanDieren 10 years ago, it has been used to prove new results (upward categoricity transfer, stability transfer) and replace set-theoretic hypotheses (existence of independence notions). In this talk, we will outline the basic definitions, summarize some key results, and discuss some open questions related to tameness.

A distributional equality for suprema of spectrally positive Levy processes

Zoran Vondracek (UIUC Math and University of Zagreb)

Abstract: Let Y be a spectrally positive Levy process with strictly negative expectation, C an independent subordinator with finite expectation, and X=Y+C. A curious distributional equality proved some ten years ago states that if the expectation of X is strictly negative, then the overall supremum of Y and the supremum of X just before the first time its new supremum is reached by a jump of C have the same distribution. In this talk I will give an alternative proof of an extension of this result and offer an explanation why it is true.

On the strong chromatic index of graphs

Michael Santana   [email] (UIUC Math)

Abstract: A strong edge-coloring of a graph $G$ is a proper edge-coloring with the additional property that each color class forms an induced matching in $G$. The strong chromatic index of $G$ is the minimum $k$ for which $G$ has a strong edge-coloring using $k$ colors. Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has strong chromatic index at most $\frac{5}{4}\Delta^2$ if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. If true, both cases are best possible. In 1990, Faudree, Gyárfás, Schelp, and Tuza revised this conjecture of Erdős and Nešetřil for planar graphs with maximum degree at most 3, stating that such graphs should have strong chromatic index at most 9. We verify this conjecture, which is best possible, and extend it to loopless multigraphs. In addition to our result, I will present several unresolved conjectures and areas for further research. This is joint work with A.V. Kostochka, X. Li, W. Ruksasakchai, T. Wang, and G. Yu.

Higgs bundles, spectral data, and fiber products of curves

Steve Bradlow (UIUC Math)

Abstract: I will discuss some interesting relations among Higgs bundles, from the point of view of spectral data, that result from isogenies among low dimensional Lie groups. This will be a report on work in progress with Laura Schaposnik. The secret goal of the talk is to describe the various components that enter in the relations clearly enough so that the algebraic geometry experts in the audience can help me and Laura make sense of it all.

Variable Annuities: The Good, Bad and The Ugly

Dr. Kannoo Ravindran (Annuity Systems Inc.)

Abstract: The first version of Variable Annuities (VAs) appeared in 1952 by the courtesy of TIAA-CREF. Since then, the features inherent in VAs have morphed as insurance companies’ attitudes towards risks, profitability, product customization, market share, required capital etc. changed. In this talk, I will touch on this development and describe the lay of the land as it relates to VAs and Fixed Indexed Annuities (FIAs) as it stands today. In addition to this, I will also discuss the convergence in the thinking between the capital markets and the actuarial community as it related to valuing the risks underlying these products. The talk will conclude with an overview of some of the theoretically challenging problems that practitioners still struggle with on a daily basis.