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for events the day of Thursday, February 4, 2016.

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Thursday, February 4, 2016

11:00 am in 241 Altgeld Hall,Thursday, February 4, 2016

Local deformation rings when $\ell \ne p$

Jack Shotton (University of Chicago)

Abstract: Given a mod $p$ representation of the absolute Galois group of $\mathbb{Q}_\ell$ , consider the universal framed deformation ring $R$ parametrising its lifts. When $\ell$ and $p$ are distinct I will explain a relation between the mod $p$ geometry of $R$ and the mod $p$ representation theory of $\mathrm{GL}_n(\mathbb{Z}_\ell)$, that is parallel to the Breuil-Mézard conjecture in the $\ell = p$ case. I will give examples and say something about the proof, which uses automorphy lifting techniques.

12:00 pm in Altgeld Hall 243,Thursday, February 4, 2016

Flattening equations for Chromatic Polynomials

Frank Bernhart (Rochester Institute of Technology)

Abstract: In the 1940s a long treatment of “planar” chromatic polynomials by G.D. Birkhoff and D.C. Lewis led to the discovery of mysterious identities concerning “chromials” obtained by fixing the configuration and varying the boundary coloring pattern. In the 1970s W. T. Tutte at the University of Waterloo (Canada) fruitfully suggested a distinction between “planar” boundary colorings and, the others. Joint research, never fully published, showed that chromials with planar boumndary patterns give a linear basis, and dimension numbers form a sequence related to Motzkin and Catalan numbers, which I had previously called Riordan numbers. Given a circular arrangement of vertices, planar partitions may be defined. The unrestricted count is Catalan. To get the numbers R(n) make the additional restriction that vertices adjacent on the circle cannot be in the same part.

2:00 pm in 243 Altgeld Hall,Thursday, February 4, 2016

A characterization result for BLD-mappings and applications

Rami Luisto (University of Helsinki)

Abstract: Bounded Length Distortion mappings are a class of mapping that can be seen either as non-homeomorphic generalizations of bilipschitz mappings or as a more rigid subclass of quasiregular mappings (which, in turn, are a generalization of complex analytic mappings to higher dimensions). In this talk we discuss a theorem characterizing BLD-mappings as so called discrete Lipschitz quotient mappings in a metric setting. As an application we obtain a new proof for a limit theorem of BLD-mappings. Martio and Väisälä showed in 1988 that between Euclidean domains the limit of L-BLD-mappings is L-BLD. This was generalized by Heinonen and Rickman to show that between generalized manifolds of type A the limits of L-BLD-mappings are K-BLD with a quantitative constant K. With the new characterization result we can show that in fact K = L.

2:00 pm in 241 Altgeld Hall,Thursday, February 4, 2016

Roth's Theorem Part 2

George Shakan (UIUC Math)

Abstract: I will continue with the proof of Roth's theorem, in particular we will start by assuming the existence of a large non-zero Fourier coefficient of the characteristic function of A. This will lead to a density increment in a subset of A with respect to a new progression, as outlined in the talk. The key input is a lemma from exponential sums that allows one to break $Z/NZ$ into $N^{3/4} + O(N^{1/2})$ progressions each of length $N^{1/4} + O(1)$ such that $x \mapsto e(ax/N)$ is roughly constant (roughly constant means $|e(ax/N)-e(ax'/N)| = o(1)$ for any $x,x'$ in the same progression) on each progression. Feel free to try to prove such a lemma on your own, the proof I will present uses only elementary tools. Also, a well written explanation of the construction I briefly mentioned at the end of the talk can be found here: . Three digit monetary awards will be awarded to participants who are able to significantly improve upon this construction.

4:00 pm in 245 Altgeld Hall,Thursday, February 4, 2016

Approximating freeness under constraints

Sorin Popa (UCLA)

Abstract: I will discuss a method for constructing Haar unitaries $u$ in a subalgebra $B$ of a von Neumann II$_1$ factor $M$ that are "as free-independent as possible" (approximately) with respect to a given finite set of elements in $M$. This technique had most surprising applications over the years: to Kadison-Singer type problems, to proving vanishing cohomology results for II$_1$ factors (e.g., compact valued derivations and the Connes-Shlyakhtenko 1-cohomology), as well as in subfactor theory (notably, to the discovery of the proper axiomatisation of the group-like objects arising from subfactors). After explaining this method, which I call incremental patching, I will comment on all these applications and its potential for future use.