Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, September 8, 2015.

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Tuesday, September 8, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 8, 2015

Group spectra and twisting structures

Marc Stephan (U Chicago)

Abstract: Group spectra are the group objects in Kan's category of semisimplicial spectra. They provide an algebraic, combinatorial model for the stable homotopy category. After introducing Kan's category of spectra, we will construct the analogues of Kan's loop group functor, of its right adjoint Wbar and corresponding classifying bundles such that the category of semisimplicial spectra becomes a twisted homotopical category in the sense of Farjoun and Hess. This will enable us to formulate and study an up-to-homotopy notion of normal subgroup spectra.

12:00 pm in 345 Altgeld Hall,Tuesday, September 8, 2015

Arithmetic progressions in the primitive length spectrum

Nicholas Miller (Purdue University)

Abstract: The length spectrum, i.e. the collection of all lengths of closed geodesics on a hyperbolic manifold, has drawn much attention over the last few decades. Of particular interest has been the question of whether the length spectrum determines the commensurability class of such a manifold. There have also been a host of prime geodesic theorems displaying a surprising analogy between the behavior of primitive, closed geodesics on hyperbolic manifolds and the behavior of the prime numbers in the integers. For instance, just as the prime number theorem dictates the asymptotic growth of primes less than n, there is an analogous asymptotic for primitive, closed geodesics of length less n. In this talk, I will review some basics on the length spectrum and survey some existing results exhibiting this connection. I will then go on to discuss some recent work on arithmetic progressions in the primitive length spectrum extending this relationship. View talk at https://youtu.be/ymQbUWDyPZo

1:00 pm in 241 Altgeld Hall,Tuesday, September 8, 2015

Non-embeddability of certain strongly pseudoconvex algebraic hypersurfaces

Ming Xiao (UIUC Math)

1:00 pm in 345 Altgeld Hall,Tuesday, September 8, 2015

On "Baire measurable paradoxical decompositions via matchings" by A. Marks and S. Unger (continued)

Anton Bernshteyn (UIUC Math)

Abstract: The Banach--Tarski paradox states that the unit ball in $\mathbb{R}^3$ is equidecomposable with two copies of itself. Of course, there can be no such equidecomposition where each piece is measurable. Thus a natural question (first asked by Marczewski) is whether there exists such an equidecomposition where each piece has the Baire property. The answer is positive, as demonstrated by an intricate construction due to Dougherty and Foreman. This paper provides an alternative (short) proof of this result. In fact, a more general result is established, namely if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. The key ingredient of the proof is a Borel analogue of Hall's celebrated marriage theorem from graph theory. In this series of talks we will go over the proofs and enjoy the elegant transition back and forth between the original problem and its combinatorial counterpart.

3:00 pm in 241 Altgeld Hall,Tuesday, September 8, 2015

Hypergraphs without exponents

Zoltan Furedi   [email] (Renyi Institute of Mathematics, Budapest)

Abstract: Let $\mathcal H$ be a family of k-graphs (k-uniform hypergraphs). Let $ex(n,\mathcal H)$ denote their Turan number, i.e., the maximum number of k-sets avoiding all members of $\mathcal H$. Very recently Bukh, Conlon and Fox showed that given any rational number r, 1< r< 2, there is a finite set of graphs $\mathcal H$ with $ex(n,\mathcal H)= \Theta (n^r)$. Similar statement was proved for k-uniform hypergraphs by Frankl 30 years ago. Here we give a new short concise proof for the following result of Frankl and the speaker from 1987: There exists a k-uniform hypergaph H (for $k\geq 5$) without exponent, i.e., when the Turan function is not polynomial in n. More precisely, we have $ex(n,H)=o(n^{k-1})$ but it exceeds $n^{k-1-c}$ for any positive c for $n> n_0(k,c)$.

3:00 pm in 243 Altgeld Hall,Tuesday, September 8, 2015

Asymptotic Syzygies

Lawrence Ein (UIC)

Abstract: I'll discuss some of my joint work with Rob Lazarsfeld on asymptotic syzygies of algebraic varieties.

4:00 pm in 149 Henry Building,Tuesday, September 8, 2015

Groups, geometry and dynamics: Train-tracks and beyond

Caglar Uyanik (UIUC)

Abstract: This will be an introductory talk about the "Mapping Class Groups" which is a topic that lies at the intersection of geometry, topology, group theory and dynamics. I will provide plenty of pictures to illustrate what particular examples of mapping classes look like. The goal of this talk is to show how all of the above areas of mathematics are connected to each other by studying curves on surfaces.

5:00 pm in 245 Altgeld Hall,Tuesday, September 8, 2015

IGL Kickoff Meeting

Abstract: Fall 2015 organizational meeting for the Illinois Geometry Lab