Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 5, 2016.

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Tuesday, April 5, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 5, 2016

Lie algebras and v_n-periodic spaces

Gijs Heuts (Copenhagen)

Abstract: We use the Goodwillie tower of the category of pointed spaces to relate the telescopic homotopy theory of spaces (in the sense of Bousfield) to the homotopy theories of Lie algebras and commutative coalgebras in T(n)-local spectra, in analogy with rational homotopy theory. As a consequence, one can determine the Goodwillie tower of the Bousfield-Kuhn functor in terms of topological Andre-Quillen homology, giving a different perspective on recent work of Behrens and Rezk.

12:00 pm in Altgeld Hall 243,Tuesday, April 5, 2016

Homology and volume for hyperbolic 3-orbifolds, and enumeration of arithmetic groups

Peter Shalen (University of Illinois at Chicago)

Abstract: A theorem of Borel's asserts that for any positive real number $V$, there are at most finitely many arithmetic lattices in ${\rm PSL}_2({\mathbb C})$ of covolume at most $V$, or equivalently at most finitely many arithmetic hyperbolic $3$-orbifolds of volume at most $V$. Determining all of these for a given $V$ is algorithmically possible for a given $V$ thanks to work by Chinburg and Friedman, but appears to be impractical except for very small values of $V$, say $V=0.41$. (The smallest covolume of a hyperbolic $3$-orbifold is about $0.39$.) It turns out that the difficulty in the computation for a larger value of $V$ can be dealt with if one can find a good bound on $\dim H_1(O,\mathbb Z/2 \mathbb Z)$, where $O$ is a hyperbolic $3$-orbifold of volume at most $V$. In the case of a hyperbolic $3$-manifold $M$, not necessarily arithmetic, joint work of mine with Marc Culler and others gives good bounds on the dimension of $H_1(M,\mathbb Z/2 \mathbb Z)$ in the presence of a suitable bound on the volume of $M$. In this talk I will discuss some analogous results for hyperbolic $3$-orbifolds, and the prospects for applying results of this kind to the enumeration of arithmetic lattices. A feature of the work that I find intriguing is that while it builds on my geometric work with Culler, the new ingredients involve primarily purely topological arguments about manifolds---the underlying spaces of the orbifolds in question---and have a classical, combinatorial flavor. At this point it appears that I can prove the following statement: If $\Omega$ is a hyperbolic 3-orbifold of volume at most $1.72$, having a link as singular set and containing no embedded turnovers, then $$\dim H_1(\Omega;\mathbb Z_2)\le 1+ \max\bigg(3,7\bigg\lfloor\frac{10}{3}{\rm vol}(\Omega)\bigg\rfloor\bigg)+ \max\bigg(3, 7\bigg\lfloor\frac{5}{3}{\rm vol}(\Omega)\bigg\rfloor\bigg). $$ In particular, $\dim H_1(\Omega;\mathbb Z_2)\le50$. Various stronger bounds on $\dim H_1(\Omega;\mathbb Z_2)$ follow from stronger bounds on the volume of $\Omega$. The restriction on turnovers is not an obstruction to applying the results to the enumeration of arithmetic groups. The assumption that the singular set is a link is more serious, but as it is used only in a mild way in this work, the methods seem promising for the prospective application.

1:00 pm in UIC,Tuesday, April 5, 2016

MidWest Model Theory Day at UIC

Abstract: The MidWest Model Theory Day will be on Tuesday, April 5th, 2016 at UIC. The speakers are Steffen Lempp, Gabriel Conant and H. Jerome Keisler. For details, see http://www.math.wisc.edu/~andrews/MWMTD9.html

2:00 pm in 347 Altgeld Hall,Tuesday, April 5, 2016

Moderate deviation principles for stochastic differential equations.

Arnab Ganguly (Louisiana State University)

Abstract: Moderate and large deviation principles involve estimating the probabilities of rare events. In particular, they often help to assess the quality of approximating models obtained through law of large number-type results. The talk will focus on a weak convergence based approach to moderate deviation principles for stochastic differential equations (SDEs) with jumps. The SDEs are driven by Poisson random measure (PRM), and the proofs use suitable variational representations for expected values of positive functionals of a PRM.

4:00 pm in 245 Altgeld Hall,Tuesday, April 5, 2016

Homotopy theory and modular surfaces

Tyler Lawson (University of Minnesota)

Abstract: Computations in algebraic topology can effectively be divided up into rational information together with information at each prime. Refinements of this led to a division of homotopy theory according to primes. I'll discuss how Quillen connected homotopy theory to formal group laws, leading to effective invariants of stable homotopy groups in terms of Bernoulli numbers and modular forms. We'll then move on to discussing ongoing work connecting this to work of Picard on modular functions in two variables, as well as modular forms with level structure.

4:00 pm in 243 Altgeld Hall,Tuesday, April 5, 2016

Introduction to supermanifolds

Mi Young Jang (UIUC Math)

Abstract: Supermanifold is a generalization of ordinary manifolds to sheaf of $\mathbb{Z}_2$-graded algebras. In this talk, I'll go through definitions about supermanifolds/superschemes, projectedness and splitness. After that, I will describe the Hilbert superscheme of constant Hilbert polynomials.