Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 8, 2013.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 8, 2013

11:00 am in 243 Altgeld Hall,Tuesday, October 8, 2013

Refined intersection homology

Pierre Albin (UIUC Math)

Abstract: I will discuss the Hodge cohomology and a refinement of the intersection homology of stratified spaces. Because of the singularities of the space, Hodge cohomology requires imposing `ideal boundary conditions', which depend on a consistent system of flat vector bundles over the singular set. On the other hand this system can be used to define a sheaf `refining' the sheaves of middle perversity intersection homology. The resulting cohomology theories coincide. This is joint work with Eric Leichtnam, Paolo Piazza, Rafe Mazzeo, and Markus Banagl.

2:00 pm in Altgeld Hall 347,Tuesday, October 8, 2013

Some estimates for stochastic differential equations driven by fractional Brownian motions

Cheng Ouyang (UIC Math)

Abstract: Study of stochastic differential equations (SDE) driven by fractional Brownian motions has been an active area of current research for a while, especially when considered as a specific application of the rough path theory. In this talk, I will give a brief survey of some recent results on the estimates of density functions of solutions to such SDEs. In particular, I will present a sharp upper bound for those density functions.

3:00 pm in 241 Altgeld Hall,Tuesday, October 8, 2013

Some recent developments on the structure of graphs with large treewidth

Chandra Chekuri   [email] (UIUC CS)

Abstract: The seminal work of Robertson and Seymour on graph minors introduced tree decompositions and treewidth. One of their key results is the Excluded Grid Theorem which states that every graph $G$ with treewidth at least k contains a $f(k) \times f(k)$ grid-minor for some (slowly growing) function $f$. Treewidth has since become a fundamental tool for structural and algorithmic results on graphs. In this talk we will discuss some recent developments on the structure of graphs with large treewidth. These development were motivated by the study of polynomial-time approximation algorithms for the maximum disjoint paths problem, culminating in a breakthrough by Chuzhoy in 2011. Subsequent work, building upon some of her ideas and other prior tools, has led to several new results. A highlight is a polynomial relationship between treewidth of a graph and the size of its largest grid-minor. A key technical tool is a theorem stating that a graph with treewidth $k$ can be partitioned into $h$ disjoint subgraphs each with treewidth at least $r$ as long $k \ge poly(h,r) polylog(k)$. This tool by itself gives several interesting applications to Erdos-Posa type theorems and fixed parameter tractable algorithms. The main purpose of the talk is to state and explain the main results and highlight some applications. No prior background on treewidth will be assumed. The talk is primarily based on two papers joint with Julia Chuzhoy.

3:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2013

Stacky Resolutions of Singularities

Matthew Satriano (University of Michigan)

Abstract: We will discuss a technique which allows one to approximate singular varieties by smooth spaces called stacks. As an application, we will address the following question, as well as some generalizations: given a linear action of a group G on complex n-space C^n, when is the quotient C^n/G a singular variety? We will also mention some applications to Hodge theory and to derived equivalences.

4:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2013

Introduction to Grothendieck Topologies

Juan S. Villeta-Garcia (UIUC Math)

Abstract: We will introduce Grothendieck topologies, sites, sheaves on them, and their cohomology. Examples will be taken from scheme theory and commutative algebras. The exposition will be basic and aimed at beginners (such as the speaker). Professors are welcome to attend.