Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 14, 2016.

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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, April 14, 2016

11:00 am in 241 Altgeld Hall,Thursday, April 14, 2016

#### Statistics of abelian varieties over finite fields

###### Michael Lipnowski (Duke University)

Abstract: Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.

12:00 pm in Altgeld Hall 243,Thursday, April 14, 2016

#### (Hyper-)Kähler geometry of character varieties

###### Brice Loustau (Rutgers - Newark)

Abstract: I will present a construction due to Andy Sanders of a (hyper-)Kähler metric in the character variety of a closed surface group which generalizes the Weil-Petersson metric on Teichmüller space as well as Hitchin's metric in the moduli space of Higgs bundles.

1:00 pm in 347 Altgeld Hall,Thursday, April 14, 2016

#### On the Hodge conjecture for q-complete manifolds

###### Alexander Sukhov (University of Lille, France)

Abstract: A complex n-dimensional manifold X is called q-complete if it admits a smooth exhaustion function whose Levi form has at least $n-q+1$ positive eigenvalues. For example, 1-complete manifolds are exactly Stein manifolds. Such a manifold is always non-compact and its a priori highest-dimensional non-trivial cohomology group is $H^{n+q-1}(X,Z)$. In the case where $n+q-1$ is even we prove that every class from this group is Poincaré-Lefschetz dual to an analytic cycle (relative to the boundary) consisting of proper holomorphic images of the ball. This is a joint work with Franc Forstneric and Jaka Smrekar, Geometry & Topology 2016.

2:00 pm in 243 Altgeld Hall,Thursday, April 14, 2016

#### Walk around the maximum principle for vector Riesz potentials

Abstract: The recent remarkable progress in the theory of analytic capacity (and more general, of Calderon-Zygmund capacities) is based on the study of the relation between boundedness of the vector Riesz operator $R_\mu^s: L^2(\mu)\to L^2(\mu)$ in $\mathbb R^d$, and geometric properties of a measure $\mu$. Here $$R_\mu^s f(x)=\int\frac{y-x}{|y-x|^{s+1}}f(y)\,d\mu(y).$$ The key role in the proofs of several results plays the relation $$\max_{x\in{\mathbb R}^d}|R_\mu^s{\bf1}(x)|\le C\max_{x\in\text{supp}\,\mu}|R_\mu^s{\bf1}(x)|,\quad C=C(d,s),$$ which for all $s\in (0,d)$ and even for measures with smooth densities is still an open problem. We give a survey of related results and indicate the methods used for different $s$.

2:00 pm in Altgeld Hall 241 ,Thursday, April 14, 2016

#### Dynamics of Apollonian Circle Packings

###### Sneha Chaubey (UIUC Math)

Abstract: In this talk we construct a dynamical system on the unit sphere which in particular allows one to read off the word corresponding to any Descartes quadruple. A Descartes quadruple is a quadruple of curvatures ( inverse radii ) of any collection of four circles which are pairwise mutually tangent, with disjoint interiors. The motivation to study this comes from works of Romik, and others on dynamics on a tree of Pythagorean triples.

4:00 pm in 245 Altgeld Hall,Thursday, April 14, 2016

#### Structure of the zero set of monochromatic random waves

###### Yaiza Canzani (Harvard)

Abstract: There are several questions about the zero set of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the size of the zero set, the study of the number of connected components, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions (known as monochromatic random waves). In this talk I will present some recent results in this direction related to the study of the topology and the nesting of the components of the zero sets of these monochromatic random waves. The results I'll present are based on joint works with Boris Hanin and Peter Sarnak.