Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, December 4, 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, December 4, 2014

11:00 am in 241 Altgeld Hall,Thursday, December 4, 2014

#### The Bettin-Conrey Reciprocity Theorem

###### Juan Auli (San Francisco State University)

Abstract: In a recent paper, Bettin and Conrey define a family of cotangent sums that generalize the classical notion of Dedekind sum and share with it the property of satisfying a reciprocity law. These sums are naturally linked to the computation of estimates of weighted moments of the Riemann zeta function, which are relevant in the approach of Nyman, Beurling, Baez-Duarte and Vasyunin to the Riemann hypothesis. We study particular instances of these arithmetic sums for which it is possible to obtain a simpler reciprocity using an analytic technique introduced by Rademacher in one of several proofs he gave of the reciprocity law of Dedekind sums.

1:00 pm in Altgeld Hall 243,Thursday, December 4, 2014

#### Higgs bundles, character varieties and cohomological Hall algebras

###### Ben Davison (Lausanne)

Abstract: I'll mainly be talking about the space $M_{g,n}$ of n-dimensional representations of the fundamental group of genus g Riemann surfaces. It has been known (or at least conjectured) for some time that the neatest expression for $p(M_{g,n})$, where p is the mixed Hodge polynomial/Poincare polynomial is best described by considering the partition function $1+p(M_{g,1})z+p(M_{g,2})z^2+...$. An explanation for this is that the cohomology of $M_{g,n}$, which is a moduli space of objects in a 2-CY category (2-CY here is a just a fancy way to package Poincare duality) are related in a natural way to virtual' or critical' cohomology of a moduli space of objects in a related 3-CY category. This in turn brings us to the world of DT theory, where these partition functions are somewhat ubiquitous. I'll explain how this all goes, and finish by describing some extra tools and structures on the cohomology of $M_{g,n}$ that arise when we think of it in terms of 3-CY geometry, and relate the above to the conjectures of Hausel and Rodriguez-Villegas on the mixed Hodge polynomials of twisted character varieties, as well as the P=W conjecture relating the weight filtration on cohomology of character varieties to the perverse filtration on the cohomology of Higgs bundles. View talk at http://youtu.be/IgXfsqyjF1g

1:00 pm in 347 Altgeld Hall,Thursday, December 4, 2014

#### Suslin trees and partition relations

###### Dilip Raghavan (National University of Singapore)

Abstract: We will describe some recent joint work with Stevo Todorcevic on partition calculus. In particular, we will show that the existence of an ${\aleph}_{\omega + 1}$-Suslin tree implies the failure of the ordinary partition relation ${\aleph}_{\omega + 1} \rightarrow ({\aleph}_{\omega + 1}, \omega + 2)^2$.

2:00 pm in 007 Illini Hall,Thursday, December 4, 2014

#### Zeros of combinations of the Riemann $\xi$-function on bounded vertical shifts

###### Nicolas Robles (University of Zurich)

Abstract: We consider a series of bounded vertical shifts of the Riemann $\xi$-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G. H. Hardy and W. L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform. This is a joint work with Dixit, Roy, and Zaharescu.

3:00 pm in 243 Altgeld Hall,Thursday, December 4, 2014

#### Maximal minors and linear powers

###### Winfried Bruns (Universität Osnabrück)

Abstract: We say that an ideal I in a polynomial ring S has linear powers if all the powers of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that all ideals defining rational normal scroll have linear powers. Joint work with Aldo Conca and Matteo Varbarao (J. Reine Angew. Math., to appear).

4:00 pm in 245 Altgeld Hall,Thursday, December 4, 2014

#### Finite generating partitions for continuous actions of countable groups

###### Anush Tserunyan (Department of Mathematics, University of Illinois)

Abstract: Let a countable group $G$ act continuously on a Polish space $X$. A countable Borel partition $\mathcal{P}$ of $X$ is called a \emph{generator} if the set of its translates $\{g P : g \in G, P \in \mathcal{P}\}$ generates the Borel $\sigma$-algebra of $X$. For $G = \mathbb{Z}$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they don't exist in the presence of an invariant probability measure with infinite entropy. It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We show that the answer is positive for an arbitrary countable group $G$ and $\sigma$-compact $X$ (in particular, for locally compact $X$). We also show that finite generators always exist for aperiodic actions in the context of Baire category (allowing ourselves to disregard a meager set), thus answering a question of Alexander Kechris from the mid-90s.