Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 13, 2018.

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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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Tuesday, March 13, 2018

11:00 am in 241 Altgeld Hall,Tuesday, March 13, 2018

#### Monotonicity properties of L-functions

###### Paulina Koutsaki (UIUC Math)

Abstract: In this talk, we discuss some monotonicity results for a class of Dirichlet series. The fact that $\zeta'(s)$ is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Moreover, we will see how our methods give rise to another formulation of the Riemann Hypothesis for the L-function associated to the Ramanujan-tau function. Based on joint work with S. Chaubey and A. Zaharescu.

12:00 pm in 243 Altgeld Hall,Tuesday, March 13, 2018

#### Group actions on quiver varieties and applications

###### Vicky Hoskins (Freie Universität Berlin)

Abstract: We study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the field of complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes. This is joint work with Florent Schaffhauser.

1:00 pm in 347 Altgeld Hall,Tuesday, March 13, 2018

#### Low Regularity Global Existence for the Periodic Zakharov System

###### Erin Compaan (MIT)

Abstract: In this talk, we present a low-regularity global existence result for the periodic Zakharov system. This is a dispersive model for the motion of ionized plasma. Its dynamics have been extensively studied, and existence of solutions is in known for data in the Sobolev space $H^\frac12 \times L^2$. We present a global existence result which holds for even rougher data, in a class of Fourier Lebesgue spaces. It is obtained by combining the high-low decomposition method of Bourgain with an almost-conserved energy result of Kishimoto. Combining these two tools allows us to obtain a low-regularity result which was out of reach of either method alone.

1:00 pm in 345 Altgeld Hall,Tuesday, March 13, 2018

#### Weak containment in ergodic theory and representation theory

###### Peter Burton (U Texas at Austin)

Abstract: The relation of weak containment for unitary representations of locally compact groups is a very useful tool in comparing such representations. Recently Kechris introduced an analogous definition of weak containment for measure-preserving actions of countable discrete groups. We will discuss the relationship between these concepts, and present a result showing that weak containment of measure-preserving actions is an essentially stronger notion than weak containment of the corresponding Koopman representations.

2:00 pm in 347 Altgeld Hall,Tuesday, March 13, 2018

#### The size of the boundary in first-passage percolation

###### Wai-Kit Lam (Indiana University Bloomington)

Abstract: First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_e)$ on the nearest-neighbor edges of $\mathbb{Z}^d$. An initial infection occupies the origin and spreads along the edges, taking time $t_e$ to cross the edge $e$. In this talk, we study the size of the boundary of the infected region at time $t$, $B(t)$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+\alpha}$ for some $\alpha > 0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ (edges touching the unbounded component of the complement of $B(t)$) is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\partial B(t) \leq (log t)^C t^{d-1}$ for all large t. This is a joint work with Michael Damron and Jack Hanson.

3:00 pm in 243 Altgeld Hall,Tuesday, March 13, 2018

#### Constructible 1-motives

###### Simon Pepin Lehalleur (Freie Universität Berlin)

Abstract: Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise, we have at our disposal a mature theory of triangulated categories of mixed motivic sheaves with rational coefficients over general base schemes, with a "six operations" formalism and the expected relationship with algebraic cycles and algebraic K-theory. A parallel development has taken place in the study of Voevodsky's category of mixed motives over a perfect field, where the subcategory of 1-motives (i.e., generated by motives of curves) has been completely described by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to combine these two sets of ideas to study the triangulated category of 1-motivic sheaves over a base. Our main results are the definition of the motivic t-structure for constructible 1-motivic sheaves, a precise relation with Deligne 1-motives, and the extraction of the "1-motivic part" of a general motivic sheaves via a "motivic Picard functor".

3:00 pm in 241 Altgeld Hall,Tuesday, March 13, 2018

#### On large bipartite subgraphs in dense H-free graphs

###### Bernard Lidicky (Iowa state University)

Abstract: A long-standing conjecture of Erdős states that any n-vertex triangle-free graph can be made bipartite by deleting at most n^2/25 edges. In this talk, we study how many edges need to be removed from an H-free graph for a general graph H. By generalizing a result of Sudakov for 4-colorable graphs H, we show that if H is 6-colorable then G can be made bipartite by deleting at most 4n^2/25 edges. Moreover, this amount is needed only in the case G is a complete 5-partite graph with balanced parts. As one of the steps in the proof, we use a strengthening of a result of Füredi on stable version of Turán's theorem. This is a joint work with P. Hu,T. Martins-Lopez, S. Norin and J. Volec.

4:00 pm in 245 Altgeld Hall,Tuesday, March 13, 2018

#### Fields Medal Confidential: Behind the scenes of mathematicians’ most famous prize, 1936-1966

###### Michael J. Barany (Dartmouth College)

Abstract: First presented in 1936, the Fields Medal quickly became one of mathematicians' most prestigious, famous, and in some cases notorious prizes. Because its deliberations are confidential, we know very little about the early Fields Medals: how winners were selected, who else was considered, what values and priorities were debated---all these have remained locked in hidden correspondence. Until now. My talk will analyze newly discovered letters from the 1950 and 1958 Fields Medal committees, which I claim demand a significant change to our understanding of the first three decades of medals. I will show, in particular, that the award was not considered a prize for the very best mathematicians, or even for the very best young mathematicians. Debates from those years also shed new light on how the age limit of 40 came about, and what consequences this had for the Medal and for the mathematics profession. I argue that 1966 was the turning point that set the course for the Fields Medal's more recent meaning.

4:00 pm in 1 Illini Hall,Tuesday, March 13, 2018

#### Algebraic Morse theory from GIT

###### Jesse Huang (UIUC)

Abstract: Birational geometry is closely tied to GIT quotients and variations. In this episode of GIT series, I will apply the machinery to a countable set of basic examples, through which we shall see how the change of linearization produces elementary birational transformations.