Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, January 21, 2014.

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Tuesday, January 21, 2014

3:00 pm in 241 Altgeld Hall,Tuesday, January 21, 2014

The typical structure of sparse Kr+1-free graphs

Jozsef Balogh   [email] (UIUC Math)

Abstract: Many important theorems and conjectures in combinatorics, such as the theorem of Szemeredi on arithmetic progressions and the Erdos-Stone The- orem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called `sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the meth- ods used are very diferent from one another and have diferent strengths and weaknesses. In this talk, we explain a third, completely diferent approach to proving extremal and structural results in sparse random sets that also yields their natural `counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs. In this talk we focus on a specic application of the method, proving the following theorem: For every r we determine the best possible mr(n) that for every $c > 0$ for $m > mr(n)$ almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite. Joint work with Robert Morris, Wojciech Samotij and Lutz Warnke.

4:00 pm in 245 Altgeld Hall,Tuesday, January 21, 2014

Division algebras and representations of Lie groups

Benjamin Antieau (University of Washington)

Abstract: I will give a basic introduction to principal bundles over different kinds of spaces, and I will discuss the various spaces that classify these bundles. A key insight is that even topologically these bundles can be classified by maps to complex algebraic varieties. This led Atiyah and Hirzebruch to the first counterexamples to the original, integral Hodge conjecture. I will discuss how a different approach to these algebraic classifying spaces, due to Totaro, led Ben Williams and myself to a method for answering concrete questions about division algebras over fields, and I will explain how these ideas resulted in our answer to a question in algebra from 1960.