Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 16, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 16, 2014

10:00 am in 143 Altgeld Hall,Thursday, October 16, 2014

#### To Be Announced

###### Seckin Demirbas (UIUC Math)

11:00 am in 241 Altgeld Hall,Thursday, October 16, 2014

#### Congruences for diagonals of rational power series

###### Eric Rowland   [email] (University of Liège)

Abstract: In the past decade several researchers have studied number theoretic properties of fundamental combinatorial sequences such as the Catalan and Motzkin numbers. For example, in 2008 Eu, Liu, and Yeh proved that no Motzkin number is divisible by 8. For the most part, proofs of such statements have used ad hoc methods particular to each sequence. However, by realizing a sequence as the diagonal of a rational power series in multiple variables, we can compute congruence information modulo prime powers completely automatically. This method lets us prove many known and new results in a uniform way, with little human effort. Joint work with Reem Yassawi.

12:30 pm in 464 Loomis Laboratory,Thursday, October 16, 2014

#### Higgs bundles and branes in the A-model and B-model

###### Laura Schaposnik (UIUC Math)

Abstract: Higgs bundles were first introduced in 1987 by N. Hitchin when studying Yang-Mills self duality equations, and have since then found applications in several branches of mathematics and physics. During the talk we shall first introduce Higgs bundles and their basic properties, and then present an overview of different ways in which they can be used to see mirror symmetry (through the SYZ conjecture of string theory), and to study A-branes and B-branes, real integrable systems and representations of 3-manifolds coming from knots or graph complements.

1:00 pm in Altgeld Hall,Thursday, October 16, 2014

#### To Be Announced

2:00 pm in 347 Altgeld Hall,Thursday, October 16, 2014

#### Approximate Monte Carlo methods for many-body quantum problems

###### Lucas Wagner (UIUC Physics)

Abstract: Most materials that are used today are reasonably well described using a weakly interacting approximation; that is, that the interactions between electrons do not affect the properties of the material in a qualitative way. A grand challenge in condensed matter physics is to go beyond the weakly interacting limit and describe phases of electronic matter that qualitatively depend on interactions between electrons. This requires solution of the many-body Schroedinger equation, which is a very high dimensional partial differential equation. I will describe some of the successes and challenges in using Monte Carlo techniques to tackle these problems.

2:00 pm in 243 Altgeld Hall,Thursday, October 16, 2014

#### Square functions, uniform rectifiability and Wolff potentials

###### Vasilis Chousionis (University of Helsinki)

Abstract: We characterize uniform rectifiability via square functions. In particular we show that an Ahlfors-David $n$-dimensional measure $\mu$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at ${{\rm supp}}(\mu)$, $$\int_0^R \int_{x\in B(x_0,R)} \left|\frac{\mu(B(x,r))}{r^n} - \frac{\mu(B(x,2r))}{(2r)^n} \right|^2\,d\mu(x)\,\frac{dr}r \leq c\, R^n.$$ This can be realized as a square functions analogue of Preiss theorem which characterizes rectifiability in terms of the existence of densities. We also discuss how these density square functions are related to Wolff potentials.

2:00 pm in 007 Illini Hall,Thursday, October 16, 2014

#### Hardy's proof of infinitude of zeros of $\zeta(s)$ on the critical line

###### Arindam Roy (UIUC Math)

Abstract: In 1914, Hardy proved that infinitely many zeros of the Riemann zeta-function lie on the vertical line $\sigma =1/2$. In this talk, we discuss his approach to the proof of this result. If time permits, then we will discuss some recent applications of his method.

3:00 pm in 243 Altgeld Hall,Thursday, October 16, 2014

#### Chen ranks and resonance

###### Hal Schenck (UIUC Math)

Abstract: The Chen groups of a group G are the lower central series quotients of the maximal metabelian quotient of G. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of G, a jump locus for the cohomology of G. In the case where G is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugating automorphisms of a finitely generated free group. Commutative algebra plays a key role in the story, via the Bernstein-Gelfand-Gelfand correspondence.

4:00 pm in 245 Altgeld Hall,Thursday, October 16, 2014

#### Symplectic resolutions and Lie algebras

###### Andrei Okounkov (Columbia University)

Abstract: In geometric representation theory, one often links the properties of noncommutative algebras to the geometry of their commutative degenerations, and also often aims to make an algebra act on cohomology or K-theory of another algebraic variety. These points of view are in a certain sense dual, and my goal in these lectures will be to discuss several recent ideas and advances in this direction.

Andrei Okounkov of Columbia University will deliver the Trjitzinsky Lectures on October 14, 15, 16, 2014.