UIUC Dept. of Mathematics Seminar Calendar

      June 2012              July 2012             August 2012     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2    1  2  3  4  5  6  7             1  2  3  4
  3  4  5  6  7  8  9    8  9 10 11 12 13 14    5  6  7  8  9 10 11
 10 11 12 13 14 15 16   15 16 17 18 19 20 21   12 13 14 15 16 17 18
 17 18 19 20 21 22 23   22 23 24 25 26 27 28   19 20 21 22 23 24 25
 24 25 26 27 28 29 30   29 30 31               26 27 28 29 30 31

Friday, February 3, 2012

Model Theory and Descriptive Set Theory Seminar
4:00 pm in 347 Altgeld Hall, Friday, February 3, 2012

Del

Edit

Copy

Submitted by phierony.

Ward Henson (UIUC)
Continuous model theory and Gurarij's universal homogeneous separable Banach space
Abstract: Gurarij’s Banach space was constructed in the 1960s using a metric version of a Fraïssé construction; it is universal isometrically (for separable Banach spaces) and homogeneous in an almost-isometric sense relative to its finite dimensional subspaces. It is the analogue (for Banach spaces) of such structures as the random graph and Urysohn’s metric space. General results in Banach space theory from the 1960s show that its dual space is of the form $L^1(\mu)$ for some measure $\mu$, so it falls into the important class of ``classical Banach spaces,’’ a fact that is far from obvious based on the original construction. Wolfgang Lusky showed in the 1970s that the Gurarij space is isometrically unique, a surprising result. He also indicated that the set of smooth points of norm 1 is an orbit of its automorphism group. In this talk it will be shown how these results can be seen and improved using continuous model theory. In particular, the class of separable Gurarij spaces can be realized as the class of separable models of a certain continuous theory $T$ (of unit balls of Banach spaces); this theory has quantifier elimination and is the model completion of the theory of all Banach spaces. An optimal amalgamation result due to the speaker yields a simple formula for the induced metric on the type spaces of $T$ over sets of parameters, which is the key to the applications that will be discussed in this talk. A highlight of recent research, proved in joint work with Itaï Ben Yaacov, is the following: let $X$ be Gurarij's space and let $E$ be a finite dimensional space whose unit ball is polyhedral (i.e., the convex hull of a finite set). There is an isometric linear embedding $S$ of $E$ into $X$ such that $S(E)$ has the unique Hahn-Banach extension property in $X$; moreover, the set of all such embeddings forms a full orbit under the action of the automorphism group of $X$. Model-theoretically this situation is equivalent to saying that $(X,a)_{a \in S(E)}$ is an atomic model of its theory.

Friday, February 24, 2012

Model Theory and Descriptive Set Theory Seminar
4:00 pm in 347 Altgeld Hall, Friday, February 24, 2012

Del

Edit

Copy

Submitted by phierony.

Slawomir Solecki (UIUC)
The pseudo-arc and its homeomorphism group
Abstract: The pseudo-arc can be constructed as a natural quotient of a projective Fraisse limit P. I will outline this construction and indicate its possible connections with random walk. I will prove an anti-Ramsey theorem (i.e., find an appropriate coloring) for a certain type of structures, which will show, via a dualization of the Kechris--Pestov--Todorcevic theory, that the group of automorphisms of P fixing a given point is not extremely amenable. The theme above will be continued in the next two talks by Ola Kwiatkowska on the work of Oppenheim.

Friday, March 9, 2012

Model Theory and Descriptive Set Theory Seminar
4:00 pm in 347 Altgeld Hall, Friday, March 9, 2012

Del

Edit

Copy

Submitted by phierony.

Lou van den Dries (Department of Mathematics, University of Illinois at Urbana-Champaign)
The structure of approximate groups according to Breuillard, Green, Tao.
Abstract: Roughly speaking, an approximate group is a finite symmetric subset A of a group such that AA can be covered by a small number of left-translates of A. Last year the authors mentioned in the title established a conjecture of H. Helfgott and E. Lindenstrauss to the effect that approximate groups are ``finite-by-nilpotent''. This may be viewed as a sweeping generalisation of both the Freiman-Ruzsa theorem on sets of small doubling in the additive group of integers, and of Gromov's characterization of groups of polynomial growth. Among the applications of the main result are a finitary refinement of Gromov's theorem and a generalized Margulis lemma conjectured by Gromov. Prior work by Hrushovski on approximate groups is fundamental in the approach taken by the authors. They were able to reduce the role of logic to elementary arguments with ultra products. The point is that an ultraproduct of approximate groups can be modeled in a useful way by a neighborhood of the identity in a Lie group. This allows arguments by induction on the dimension of the Lie group. I will give two talks: the one on Tuesday (1pm in 345 AH) will describe the main results, and the sequel on Friday (4pm in 347 AH) will try to give a rough idea of the proofs.

Friday, March 16, 2012

Friday, April 6, 2012

Friday, April 13, 2012

Friday, September 14, 2012

Friday, September 21, 2012

Friday, September 28, 2012

Friday, October 5, 2012

Friday, October 19, 2012

Friday, October 26, 2012

Friday, November 2, 2012

Friday, November 9, 2012

Friday, November 16, 2012

Friday, December 7, 2012