Mathematics

No seminar

On Singularity Formation in General Relativity

Xinliang An (U Toronto Math)

Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, with three different approaches coming from hyperbolic PDE, quasilinear elliptic PDE and dynamical system, I will present results on four physical questions: i) Can “black holes” form dynamically in the vacuum? ii) To form a “black hole”, what is the least size of initial data? iii) Can we find the boundary of a “black hole” region? Can we show that a “black hole region” is emerging from a point? iv) For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse?

Fractal Properties of Additive Lévy Processes

Haihua Shi (NanJing University of Science and Technology & UIUC)

Abstract: Additive Lévy processes first arose to simplify the study of Lévy sheets. They also arise in the theory of intersection and self-intersection of Lévy processes. Moreover, additive Lévy processes have a rich and interesting structure on their own. We focus on the fractal properties of this process , and get the exact Hausdorff measure function of the range of additive stable subordinator.

List Coloring Cartesian Products of Graphs: Criticality and the List Color Function

Hemanshu Kaul (Illinois Institute of Technology)

Abstract: The list chromatic number of the Cartesian product of graphs is not well understood. The best result is by Borowiecki, Jendrol, Kral, & Miskuf (2006) who proved that the list chromatic number of the Cartesian product of two graphs can be bounded in terms of the list chromatic number and the coloring number of the factors, implying a bound exponential in the list chromatic number of the factors. We show how the knowledge of the list color function (list coloring analogue of the chromatic polynomial) can be applied to list coloring of Cartesian products whose one factor is a strong k-chromatic choosable graph. We introduce the notion of strongly chromatic choosable graphs, that includes odd cycles, cliques, many more infinite families of graphs, and the join of a clique with any other such graph, as a strict generalization of the notion of strong critical graphs (Stiebitz, Tuza & Voigt, 2008). This leads to improved bounds on choosability of Cartesian product of certain large classes of graphs and to classes of chromatic-choosable Cartesian products of graphs. This is joint work with Jeffrey Mudrock.

A Tate duality theorem for local Galois symbols

Evangelia Gazaki (University of Michigan)

Abstract: Let $K$ be a $p$-adic field and $M$ a finite continuous $Gal(\overline{K}/K)$-module annihilated by a positive integer $n$. Local Tate duality is a perfect duality between the Galois cohomology of $M$ and the Galois cohomology of its dual module, $Hom(M,\mu_n)$. In the special case when $M=A[n]$ is the module of the $n$-torsion points of an abelian variety, Tate has a finer result. In this case the group $H^1(K,A[n])$ has a "significant subgroup", namely there is a map $A(K)/n\rightarrow H^1(K,A[n])$ induced by the Kummer sequence on $A$. Tate showed that under the perfect pairing for $H^1$, the orthogonal complement of $A(K)/n$ is the corresponding part, $A^\star(K)/n$, that comes from the points of the dual abelian variety $A^\star$ of $A$. The goal of this talk will be to present an analogue of this classical result for $H^2$. We will see that the "significant subgroup" in this case is given by the image of a cycle map from zero cycles on abelian varieties to Galois cohomology, while the orthogonal complement under Tate duality is given by an object of integral $p$-adic Hodge theory. We will then discuss how this computation fits with the expectations of the Bloch-Beilinson conjectures for abelian varieties defined over algebraic number fields.

Results of the department climate survey

Matthew Ando (Illinois)

Abstract: Matt Ando will talk about the results of the department climate survey.