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for events the day of Monday, March 26, 2018.

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Monday, March 26, 2018

3:00 pm in 243 Altgeld Hall,Monday, March 26, 2018

Integration of Structure Equations of G-Structures

Ivan Struchiner (University of Sao Paulo)

Abstract: The infinitesimal data attached to a (finite type) class of G-structures with connections are its structure equations. Such structure equations give rise to Lie algebroids endowed with extra geometric information. On the other hand, given such a Lie G-algebroid a natural question is that of the existence of a G-structure with connection which corresponds to the given G-algebroid through differentiation. This integration problem is called Cartan's Realization Problem for G-structures. In this talk I will describe a way of solving the realization problem by integrating the G-algebroid. I will focus on the case of Riemannian metrics (G=O(n)). The talk is based on joint work with Rui Loja Fernandes.

4:00 pm in 245 Altgeld Hall,Monday, March 26, 2018

Universal objects in functional analysis

Timur Oikhburg   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Many classes contain a universal object. For instance, every compact metrizable space is a continuous image of the Cantor set (the Cantor set is projectively universal), and embeds isomorphically into the Hilbert cube (the Hilbert cube is injectively universal). Any separable Banach space is a quotient of $\ell_1$, and embeds isometrically into $C[0,1]$. We discuss the following topics: (1) The existence of (injectively) universal objects that are (almost) homogeneous - that is, any isometry between two finite subsets of such an object extends (or almost extends) to an isometry of the object itself. (2) The existence of universal objects for specific classes of Banach spaces (such as reflexive spaces, or spaces with a separable dual). (3) Universal objects for Banach lattices (based on the recent work with M.-A. Gramcko-Tursi and others).

5:00 pm in 241 Altgeld Hall,Monday, March 26, 2018

$\ell_2$-(co)homology and $\ell_2$-Betti numbers

Anton Bernshteyn (Illinois Math)

Abstract: In this instalment of the Operator Algebra Learning Seminar, we will move on to talking about topological invariants that can by defined using Hilbert $\Gamma$-modules.

Thursday, April 26, 2018

3:00 pm in 345 Altgeld Hall,Thursday, April 26, 2018

Hearing the Limiting Shape of a Hypersurface Configuration

Robert Walker (U. Michigan)

Abstract: There is a niche body of work on limiting shapes, i.e., asymptotic Newton polyhedra, of symbolic generic initial systems considered for polynomial rings in characteristic zero (e.g., by my academic sister Sarah Mayes-Tang, and separately by Dumnicki, Szemberg, Szpond, and Tutaj-Gasinska, a quartet of Polish mathematicians). In particular, in one joint paper the latter four authors compute the limiting shape for ideals defining zero-dimensional star configurations in projective space--star configurations turn out to be a steady source of a lot of interesting "ALGECOM" phenomenology. In this talk, we discuss work-in-progress to generalize their computation to the case of ideals defining zero-dimensional configurations in projective space determined by hypersurfaces of a common fixed degree. Along the way, we draw connections to a 2015 investigation (published in Transactions of the AMS in 2017) of select homological and asymptotic properties of hypersurface and matroidal configurations by Geramita, Harbourne, Migliore, and Nagel. I'll aim to close the talk by indicating how we might see--or "hear"--select asymptotic numerical invariants in the limiting shape: Waldschmidt constants, asymptotic Castelnuovo-Mumford regularity, and resurgences for homogeneous polynomial ideals