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Seminar Calendar
for events the day of Tuesday, October 6, 2015.

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Tuesday, October 6, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, October 6, 2015

Combinatorial rigidity of the arc complex

Valentina Disarlo (Indiana Math)

Abstract: We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. This generalizes a result of Irmak - McCarthy. As an application we deduce some rigidity results for the Fomin-Shapiro-Thurston cluster algebra associated to a surface. Our proofs do not employ any known simplicial rigidity result.

12:30 pm in 464 Loomis Laboratory,Tuesday, October 6, 2015

Universal scaling properties after a Quantum Quench

Damian A. Galanti (Perimeter Institute for Theoretical Physics)

Abstract: We will present the problem of a quantum quench in QFT, i.e., following the real-time evolution of an operator whose coupling is time dependant. Inspired by results in holography, we propose certain universal scaling properties that appear for fast and smooth quenches. We provide evidence for it in free scalar and fermionic field theories and provide a general argument of why the scaling should hold in arbitrary interacting CFTs. We also speculate on universal behaviour emerging for slow quenches through a critical point. Based on 1401.0560, 1411.7710, 1505.05224 and work in progress.

1:00 pm in 345 Altgeld Hall,Tuesday, October 6, 2015

Axiomatizing classes of Banach spaces

Ward Henson   [email] (UIUC Math)

Abstract: The context is first order logic for metric structures, applied to (unit balls of) Banach spaces and their expansions. Using this logic, many classes of Banach lattices are known to be axiomatizable. However, relatively few of the associated classes of Banach spaces have been similarly understood. With Yves Raynaud, we have developed some new techniques for transferring axiomatizability of suitable classes of Banach lattices to their Banach space reducts. The techniques involve facts about disjointness preserving linear isometries (between Banach lattices) from functional analysis, and facts about definability, especially definability of sets, from continuous model theory.

1:00 pm in 241 Altgeld Hall,Tuesday, October 6, 2015

CR Control Geometry on Unbounded Domains in ${\mathbb C}^2$

Aaron Peterson (Northwestern)

Abstract: Let $\Omega\subset \mathbb{C}^2$ be a weakly pseudoconvex finite-type domain with smooth boundary. We will discuss the Carnot-Carath\'eodory control geometry induced on ${\rm b}\Omega$ by the real and imaginary parts of the CR vector field, which has been used to describe the behavior of various objects associated to the $\bar{\partial}$- and $\bar{\partial}_b$-problems. After reviewing the local theory, we will develop a framework for studying this geometry on certain unbounded model domains. We will explore examples of unbounded domains where the global structure of the control geometry is vastly different than the local structure, and discuss a subclass of such examples where the control geometry still describes objects associated to the $\bar{\partial}$- and $\bar{\partial}_b$-problems.

2:00 pm in 347 Altgeld Hall,Tuesday, October 6, 2015

Variance Swaps on Time-Changed Markov Processes

Roger Lee   [email] (University of Chicago)

Abstract: Relationships between, on one hand, expectations of random variables dependent on the path of a process X, and on the other hand, expectations of path-independent random variables that depend only on X_T for a fixed time T, have applications in finance, as they relate the prices of complex derivative contracts to the prices of basic derivative contracts. Consider the case of a "variance swap", in which the path-dependent random variable is quadratic variation of the log of a positive martingale -- given here by a Markov process, time-changed by a general continuous stochastic clock, which is allowed to depend on the driving Markov process, which is allowed to have state-dependent jump distributions. We show that the path-independent random variable can be chosen as G(X_T) if G satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. In some examples, G can be computed explicitly.

3:00 pm in 243 Altgeld Hall,Tuesday, October 6, 2015

Boundary Conditions in QFT and Modules in Category O

Justin Hilburn (University of Oregon)

Abstract: By now it is well known that many features of the representation theory of semisimple Lie algebras can in fact be generalized to to any noncommutative algebra that arises as the quantization of functions on a symplectic cone. In particular for any such algebra there is an analogue of the BGG category O. These varieties seem to appear in "symplectic dual" pairs such that the associated pair of categories O are Koszul dual. In the 90s physicists described all known pairs of dual symplectic cones as the Higgs and Coulomb branches of the moduli space of vacua in 3d N=4 SUSY field theories, but they did not study the associated categories O. In this talk I will describe some work in progress with Bullimore, Dimofte, and Gaiotto to explain symplectic duality by studying boundary conditions in the associated field theory. In particular, I will describe an interesting construction of projective modules in category O for an abelian theory (aka Hypertoric Category O).

3:00 pm in 241 Altgeld Hall,Tuesday, October 6, 2015

Tutorial on the Container Method

Adam Zsolt Wagner   [email] (UIUC Math)

Abstract: Many important theorems and conjectures in combinatorics can be rephrased as problems about counting independent sets in some specific graphs and hypergraphs. The Container Method, whose basic idea can be traced back to Kleitman-Winston, and has recently been further developed by Balogh-Morris-Samotij and Saxton-Thomason, essentially states that hypergraphs satisfying some natural conditions have very few independent sets. In this survey-style talk I will show some recent applications of the method, and try to give an easy recipe containing all the key ideas one needs to know to prove similar results.

4:00 pm in 245 Altgeld Hall,Tuesday, October 6, 2015

The wrong number to plug into the wrong formula to get the right price of a stock option

Roger Lee (University of Chicago)

Abstract: Traders of stock options often quote prices not in dollars and cents, but rather in "implied volatility" -- sometimes described as "the wrong number to plug into the wrong formula to get the right price". We define what this means and explore why this makes sense, in the context of stochastic models of financial asset prices.

4:00 pm in 149 Henry building,Tuesday, October 6, 2015

Number Theory in its various avatars

Amita Malik (UIUC)

Abstract: In this talk, we will discuss certain number theoretic ubiquitous tools and see some examples on how they can be used to attack various problems lying at the interface of number theory and other mathematical areas. We will see how some deceptively simple looking problems are related to certain very well known problems. This talk can be thought of as an $\epsilon$-window into the mind of a number theorist. No prior knowledge will be assumed.