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for events the day of Tuesday, April 19, 2016.

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Tuesday, April 19, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 19, 2016

A^1-homotopical classification of principal G-bundles

Marc Hoyois (MIT)

Abstract: Let k be an infinite field and G an isotropic reductive k-group. If X is a smooth affine k-variety, then locally trivial G-torsors over X are classified by maps X → BG in the A^1-homotopy category. I will discuss the proof of this statement and some applications. This is joint work with Aravind Asok and Matthias Wendt.

12:00 pm in Altgeld Hall 243,Tuesday, April 19, 2016

Separated nets in the Heisenberg group

Anton Lukyanenko (University of Michigan)

Abstract: A co-compact lattice is a standard example of a separated net, but other nets also arise in applications (most famously the quasi-crystals in chemistry), and one would like to know whether they are simply perturbations of lattices. In Euclidean space, a criterion of Laczkovich allows one to easily make nets that are not a bounded-distance perturbation of any lattice (not BD rectifiable), and an intricate construction due to McMullen and Burago-Kleiner provides a net that is not even bi-Lipschitz to any lattice (not BL rectifiable). We study nets and quasi-crystals in the Heisenberg group and more generally (rational) Carnot groups. Lattices in these groups are quite tame, and by a theorem of Malcev may even be viewed as the integer points in appropriate coordinates. We show that a generic net need not be well-behaved: in addition to nets that are not BD or BL rectifiable, there exist BD-rectifiable ``exotic nets'' that are neither coarsely dense nor uniformly discrete in Malcev coordinates. On the other hand, in applications, a natural construction of quasi-crystals yields easily-understood nets whose BD rectifiability is based on a certain Diophantine condition, showing that almost every Heisenberg quasi-crystal is a BD perturbation of a lattice.

2:00 pm in 347 Altgeld Hall,Tuesday, April 19, 2016

Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

Jessica Lin (UW Madison)

Abstract: I will present optimal quantitative error estimates in the stochastic homogenization for uniformly elliptic equations in nondivergence form. From the point of view of probability theory, stochastic homogenization is equivalent to identifying a quenched invariance principle for random walks in a balanced random environment. Under strong independence assumptions on the environment, the main argument relies on establishing an exponential version of the Efron-Stein inequality. As an artifact of the optimal error estimates, we obtain a regularity theory down to microscopic scale, which implies estimates on the local integrability of the invariant measure associated to the process. This talk is based on joint work with Scott Armstrong.

3:00 pm in Altgeld Hall,Tuesday, April 19, 2016

The number of union-free families

Adam Zsolt Wagner (Illinois Math)

Abstract: A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved Erdős' conjecture, stating that every union-free family has size at most roughly the middle layer. Later, Burosch--Demetrovics-Katona-Kleitman-Sapozhenko asked for the number α(n) of such families, they proved a non-trivial upper bound and conjectured that log α(n) should be asymptotic to the size of the middle layer. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.

4:00 pm in 243 Altgeld Hall,Tuesday, April 19, 2016

An introduction to Quantum Cohomology of Toric Varieties

Joseph Pruitt (UIUC Math)

Abstract: The goal of this talk is to give an introduction to understanding the quantum cohomology ring of a toric variety through the data of its associated fan. While the quantum cohomology ring of several classes of toric varieties has been understood in terms of generators and relations, using the techniques of Localization and Equivariant Cohomology we can further understand the multiplication structure of the ring. The talk will be introductory and won't require any prior knowledge of toric varieties.