Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 1, 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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Tuesday, April 1, 2014

2:00 pm in 241 Altgeld Hall,Tuesday, April 1, 2014

#### Centralizers of generic measure-preserving automorphisms (part 1)

Abstract: We present a proof by Melleray and Tsankov of the fact that for the generic measure-preserving automorphism $T$ of a standard probability space $(X,\mu)$, the centralizer of $T$ inside $\text{Aut}(X,\mu)$ is as small as possible, i.e. is equal to the closure of the group generated by $T$.

3:00 pm in 241 Altgeld Hall,Tuesday, April 1, 2014

#### Packing $k$-partite $k$-graphs

###### Richard Mycroft   [email] (University of Birmingham)

Abstract: Let $G$ and $H$ be graphs or hypergraphs. A perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. In the simplest case, where $H$ is the graph consisting of a single edge, a perfect $H$-packing in $G$ is simply a perfect matching in $G$; Dirac's theorem tells us that such a packing must exist if $G$ has minimum degree at least $n/2$ (where $n$ is the number of vertices of $G$). The problem of what minimum degree is needed to ensure a perfect $H$-packing in $G$ for general graphs $H$ was then tackled by many researchers, before K\"uhn and Osthus finally established the correct threshold for all graphs $H$ (up to an additive constant). However, for $k$-uniform hypergraphs (or $k$-graphs) much less is known. The case of a perfect matching has been well-studied, but apart from this there were previously no known asymptotically correct results on the minimum degree needed to ensure a perfect $H$-packing in $G$ for $k > 4$ (for any of the various common generalisations of the notion of degree to the $k$-graph setting). In this talk I will demonstrate, for any complete $k$-partite $k$-graph $H$, the asymptotically best-possible minimum codegree condition for a $k$-graph $G$ which ensures that $G$ contains a perfect $H$-packing. This condition depends on the sizes of the vertex classes of $H$, and whether these sizes, or their differences, share any common factors greater than one.

3:00 pm in 243 Altgeld Hall,Tuesday, April 1, 2014

#### Cohomological characterization of products of theta-divisors

###### Sofia Tirabassi (University of Utah)

Abstract: We present a joint work with J. Jiang and M. Lahoz in which it is proven that any smooth complex projective variety of maximal Albenese dimension, with Euler characteristic 1 and Albanese image normal and of general type is a product of theta-divisors. We also generalize in higher dimension Hacon--Pardini classification of surfaces of maximal Albanese dimension with genus and irregularity equal 3. The techniques we use are based on Green--Lazarsfeld generic vanishing theorems and on the use of integral transforms.

4:00 pm in Altgeld Hall,Tuesday, April 1, 2014

#### Asymptotics of certain families of Higgs bundles

###### Brian Collier (UIUC Math)

Abstract: Higgs bundles are algebro-geometric objects that live over a Kahler manifold. Through nonabelian Hodge theory, the moduli space of Higgs bundles is homeomorphic to the space of reductive representations of the fundamental group (or a central extension) of the manifold. To get this homeomorphism one goes through two deep, nonconstructive existence theorems. In this talk I will sketch this correspondence, then consider a family of Higgs bundles of particular geometric interest, and talk about some new results on the asymptotics of certain families of Higgs bundles.