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for events the day of Tuesday, December 9, 2014.

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Tuesday, December 9, 2014

11:00 am in 241 Altgeld Hall,Tuesday, December 9, 2014

Arithmetics on Integral Apollonian-3 Circle Packings

Xin Zhang (Tel Aviv University)

Abstract: Apollonian circle packings have aroused great interest among number theorists and homogeneous dynamicists during the last decade. In this talk, I’ll describe a theorem which generalizes Bourgain-Kontorovich’s Strong Density Theorem on integral Apollonian packings to the setting of Apollonian-3 packings. If time permits, I’ll discuss some other problems on circle packings.

11:00 am in 243 Altgeld Hall,Tuesday, December 9, 2014

The formal category theory of (∞,1)-categories

Emily Riehl   [email] (Harvard University)

Abstract: Theorems in abstract homotopy are occasionally stated in the language of “(∞,1)-categories,” which can be modeled by quasi-categories ("∞-categories”) or by complete Segal spaces. In pioneering work of Joyal and Lurie, ordinary category theory has been extended to quasi-categories, making this model particularly convenient for proving abstract theorems. In joint work with Verity, we show that the category theory of quasi-categories is 2-categorical: new definitions of the basic notions — (co)limits, adjunctions, fibrations — equivalent to the Joyal-Lurie definitions, can be encoded in the homotopy 2-category of quasi-categories. From this new vantage point, the basic categorical theorems become easier to prove; the 2-categorical proofs in the homotopy 2-category mimic the classical ones in the 2-category CAT. Importantly, complete Segal spaces and more general categories of Rezk objects have an analogous homotopy 2-category, so this “formal” approach to the development of the category theory of quasi-categories immediately extends to other models of higher homotopical categories.

1:00 pm in 345 Altgeld Hall,Tuesday, December 9, 2014

Multiplicative subgroups of real closed fields

Erin Caulfield (UIUC)

Abstract: Let $\Gamma$ be a multiplicative subgroup of the unit circle in $\mathbb{C}$ and let $\Delta$ be a multiplicative subgroup of $\mathbb{R}^{>0}$ of the form $a^{\mathbb{Z}}$ for some $a > 1$. Suppose that the product $\Gamma\Delta$ has the Mann property. Let $K,L$ be real closed fields. Let $G,H$ be dense multiplicative subgroups of the unit circles in $K^2$ and $L^2$ respectively, and let $A,B$ be regularly discrete multiplicative subgroups of $K^{>0}$ and $L^{>0}$ respectively. We give conditions for the structures $(K,G,A,(\delta)_{\delta \in \Delta}, (\gamma)_{\gamma \in \Gamma})$ and $(L,H,B,(\delta)_{\delta \in \Delta}, (\gamma)_{\gamma \in \Gamma})$ to be elementarily equivalent. This work is a first step towards classifying expansions of $\mathbb{R}$ by arbitrary finite rank multiplicative subgroups of $\mathbb{C}$.

1:00 pm in 241 Altgeld Hall,Tuesday, December 9, 2014

Billiards, the square root of eleven and a Teichmuller curve of genus one.

Ronen Mukamel (Chicago)

Abstract: A Teichmuller curve is an algebraic curve which is algebraically and isometrically immersed in the moduli space of Riemann surfaces. We will present explicit topological, hyperbolic and algebraic models for particular Teichmuller curves, including examples of positive genus. We will also present evidence drawn from our examples that Teichmuller curves share many of the amazing properties enjoyed by Shimura curves.

2:00 pm in 347 Altgeld Hall,Tuesday, December 9, 2014

Intrinsic scaling properties for nonlocal operators

Ante Mimica   [email] (University of Zagreb Math)

Abstract: We present a probabilistic approach for studying regularity properties of harmonic functions of a class of nonlocal operators that will be understood as generators of Markov processes. This class includes stable-like processes and also processes with jumping kernel comparable to the one of the geometric stable processes. In this approach the order of the operator will represented by a function and not by a number (as in the stable-like case). The proof of the main result will be based on intrinsic scaling properties of the underlying operators and stochastic processes. This is joint work with Moritz Kassmann from University of Bielefeld.

3:00 pm in 241 Altgeld Hall,Tuesday, December 9, 2014

Towards a shellability proof of an identity of Dixon

Ruth Davidson   [email] (UIUC Math)

Abstract: Shellability is a combinatorial property with topological and algebraic consequences that simplicial and other cell complexes may or may not have. For example, shellability can be used to compute the Betti numbers of a complex. We will give a brief overview of the place of shellability in the mathematical ecosystem and highlight some newer applications of shellability. Then we will give some recent results towards a novel proof of a known identity of Dixon reformulated as the Euler-Poincaré relation for a non-pure shellable simplical complex. No prior knowledge of shellability or topological combinatorics will be assumed.

4:00 pm in 245 Altgeld Hall,Tuesday, December 9, 2014

Noncommutative and Poisson geometry in mirror and symplectic duality

Travis Schedler (University of Texas at Austin)

Abstract: Mirror symmetry, arising from two-dimensional physical field theories, has given rise to deep mathematical statements, such as Kontsevich's homological mirror symmetry conjecture. Recently, a totally new mathematical symmetry, called symplectic duality, has arisen which corresponds to three-dimensional physical field theories; it has already given rise to deep conjectures and results, closely related to Koszul duality and representation theory of Lie algebras and their generalizations. I will speak about an approach to these symmetries via the Poisson geometry of symplectic singularities and their quantizations, using the algebraic technique of differential operators (D-modules), and give applications to representation theory, singularity theory, and algebraic geometry. Among our results are new homology theories (such as Poisson-de Rham and Hochschild-de Rham) and new constructions of old theories (cyclic homology and its Gauss-Manin connection), as well as information on existence and topology of symplectic resolutions of singularities.