Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 7, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 7, 2014

8:00 am in Altgeld Hall,Tuesday, October 7, 2014

Department of Mathematics External Review

Abstract: The Department of Mathematics External Review will be October 7-9, 2014.

11:00 am in 243 Altgeld Hall,Tuesday, October 7, 2014

The Hecke algebra action on Morava E-theory of height 2

Yifei Zhu   [email] (Northwestern)

Abstract: Using Bousfield-Kuhn functors, Rezk constructs logarithmic cohomology operations that naturally act on the units of any strictly commutative ring spectrum. In particular, given a Morava E-theory associated to a formal group of height n over a perfect field of characteristic p, Rezk writes down a formula for its "logarithm" in terms of power operations. For height n = 2 and all primes p, we show that any modular form of level prime to p, with zeros and poles only at the cusps, gives an element in the kernel of this logarithm. This is based on an explicitation for an algebra of Hecke operators acting on the E-theory, which in turn builds on calculations with elliptic curves for power operations at the primes 2, 3 and 5.

1:00 pm in 345 Altgeld Hall,Tuesday, October 7, 2014

The Asymptotic Couple of the Field of Logarithmic Transseries

Allen Gehret (UIUC Math)

Abstract: We will define the differential field of logarithmic transseries and discuss its value group $\Gamma$. The value group $\Gamma$ can be given the additional structure of a map $\psi:\Gamma\to\Gamma$ which is induced by the field derivation. The structure $(\Gamma,\psi)$ is the asymptotic couple of the field of logarithmic transseries. We will discuss properties of abstract asymptotic couples (i.e., ordered abelian groups with an additional map that satisfies certain axioms). We will present a quantifier elimination result for the theory of the asymptotic couple $(\Gamma,\psi)$ in an appropriate first-order language and discuss various other things (definable functions on a certain discrete set, a stable embedding result, maybe NIP?).

1:00 pm in Altgeld Hall 243,Tuesday, October 7, 2014

Effective separability for hyperbolic surface and 3-manifold groups

Priyam Patel (Purdue University)

Abstract: The fundamental groups of hyperbolic surfaces and 3-manifolds, referred to as surface groups and 3-manifold groups, respectively, have various algebraic finiteness properties. Two of these properties, residual finiteness and subgroup separability, have played an important role in the recent resolution of some outstanding conjectures in 3-manifold theory. To begin this talk, we will define residual finiteness and subgroup separability. We will then explain how effective proofs of these properties can help us "quantify" separability and discuss the topological implications of such quantifications. We focus on the surface case and discuss related work on lifting geodesics in hyperbolic surfaces to embedded ones in finite covers. View talk at http://youtu.be/4h3sXUj9hws

1:00 pm in 347 Altgeld Hall,Tuesday, October 7, 2014

Spectral instability of characteristic boundary layer flows

Toan Nguyen (Penn State)

Abstract: We study the unstable spectrum of linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $Re \to \infty$. We show that there is always a range of wave numbers and Reynolds number, in which unstable eigenvalues exists. Notably, the profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. Our approach avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators. This is a joint work with E. Grenier and Y. Guo.

3:00 pm in 241 Altgeld Hall,Tuesday, October 7, 2014

The Local Action Lemma

Anton Bernshteyn (UIUC Math)

Abstract: The Lovász Local Lemma is a very important and powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL actually provides more than just pure existence results: there is an effective randomized algorithm that can be used to find the desired object. In order to analyze this algorithm Moser and Tardos developed the so-called entropy compression method. It was discovered lately (and somewhat unexpectedly) that one can obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealing to the LLL. We provide a general and purely probabilistic statement that implies both these new combinatorial results and the LLL itself.