Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 15, 2013.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 15, 2013

11:00 am in 243 Altgeld Hall,Tuesday, October 15, 2013

Additivity and Formality for $E_n$ operads

Nick Rozenblyum (Northwestern)

Abstract: Let $E_n$ be the little disks operad. It is well known that for $n>1$, the rational homology of $E_n$ is $P_n$, the $(n-1)$ shifted Poisson operad. More generally, for all $n$, $E_n$ has a filtration whose associated graded operad is $P_n$. Dunn's additivity theorem states that the Boardman-Vogt tensor product of $E_k$ and $E_l$ is $E_{k+l}$. We will show that this equivalence is compatible with the filtration and, time permitting, explain the generalization of this statement to factorization algebras. This fact has a number of remarkable consequences. An immediate corollary is the formality theorem for $n>2$ by induction starting with formality for $E_2$. Furthermore, the factorization algebra version of the result provides a local-to-global version of the BV-AKSZ formalism in quantum field theory, and sheds new light on the problem of quantization.

11:00 am in 347 Altgeld Hall,Tuesday, October 15, 2013

Orthogonality Properties of Complex Harmonic Polynomials

Zhenghui Huo (UIUC)

1:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2013

Properties of minimizers of the Lawrence-Doniach energy with perpendicular magnetic fields

Guanying Peng (Purdue Math)

Abstract: We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder. For an applied magnetic field in the intermediate regime that is perpendicular to the layers, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as the reciprocal of the Ginzburg-Landau parameter and the interlayer distance tend to zero. Under an appropriate assumption on the relationship between these two parameters, we establish comparison results between the minimum Lawrence-Doniach energy and the minimum 3D anisotropic Ginzburg-Landau energy.

2:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2013

Scaling limits for the exit problem for conditioned diffusions via Hamilton-Jacobi equations

Yuri Bakhtin (Georgia Tech, Mathematics)

Abstract: The classical Freidlin--Wentzell theory on small random perturbations of dynamical systems operates mainly at the level of large deviation estimates. It would be interesting and useful to supplement those with central limit theorem type results. We are able to describe a class of situations where a Gaussian scaling limit for the exit point of conditioned diffusions holds. Our main tools are Doob's h-transform and new gradient estimates for Hamilton--Jacobi equations. Joint work with Andrzej Swiech.

2:00 pm in 345 Altgeld Hall,Tuesday, October 15, 2013

Faces and maximizer subsets of highest weight modules

Apoorva Khare (Stanford)

Abstract: Verma modules over a complex semisimple Lie algebra, as well as their simple quotients are important and well-studied objects in representation theory. We present three formulas to compute the set of weights of all such simple highest weight modules (and others) over a complex semisimple Lie algebra $\mathfrak{g}$. These formulas are direct and do not involve cancellations. Our results extend the notion of the Weyl polytope to general highest weight $\mathfrak{g}$-modules $V^\mu$. We also show that for all such simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic subgroup $W_J$. We compute its vertices, faces, and symmetries - more generally, we do so for all parabolic Verma modules, and for all modules $V^\mu$ with $\mu$ not on a simple root hyperplane. Our techniques also enable us to completely classify inclusions between "weak faces" of arbitrary $V^\mu$, in the process extending results of Vinberg, Chari, Cellini, and others from finite-dimensional modules to all highest weight modules.

3:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2013

The effect of long-range interactions in DNA melting

Aaron Santos (Simpson College, Physics)

Abstract: A theoretical understanding of the DNA melting transition may provide insight into the biological mechanisms of transcription and replication. If this process occurs via nucleation, it should exhibit several key features: metastability, rapid spontaneous growth, and droplet formation. In this talk, I describe the results of recent computational and theoretical studies on nearest-neighbor and long-range DNA models. While the models exhibit some characteristics of classical nucleation when the interaction range is short, they may undergo spinodal nucleation when the interaction range is long. In contrast to classical nucleation droplets, which are compact, spinodal critical droplets are diffuse, fractal-like, and similar to the metastable state. These results have clear implications for transcription and replication in biological DNA.

3:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2013

Genera and derived algebraic geometry

Nick Rozenblyum (Northwestern)

Abstract: We will describe an approach, motivated by quantum field theory, to describe invariants of algebraic varieties using derived algebraic geometry. In particular, we will describe a version of non-abelian duality that can be used to produce volume forms on derived mapping spaces. Integration of these volume forms produces interesting invariants such as the Todd genus, the Witten genus and the B-model operations on Hochschild homology.

3:00 pm in Altgeld Hall,Tuesday, October 15, 2013

The 1-2-3 Conjecture and its relatives on graphs and hypergraphs

Florian Pfender   [email] (UC Denver)

Abstract: The 1-2-3 Conjecture from 2004 by Karonski, Luczak and Thomason states that you can weigh the edges of any connected graph on at least $3$ vertices with weights from the set $\{ 1,2,3\}$ such that the weighted vertex degrees induce a proper vertex coloring. This conjecture has spurred a lot of activity in recent years, but it looks like we are still ways away from solving it completely. In this talk I will survey some results on this and related conjectures. Further, I will present some new results on the equivalent question for hypergraphs. Surprisingly, we can give sharp bounds for large classes of hypergraphs. Joint work with M. Kalkowski and M. Karonski.

4:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2013

The Geometry of Filtered Quiver Varieties

Mee Seong Im (UIUC Math)

Abstract: Invariant theory has connections to many areas of mathematics: to name a few, Higgs bundles, David Mumford's geometric invariant theory and Hilbert schemes in algebraic geometry, Nakajima's quiver variety in representation theory, the Hamiltonian reduction construction in symplectic geometry, combinatorics, graph theory, coding theory, DNA strand configuration, and fingerprint technology. Around 1990's, Aidan Schofield and a number of other mathematicians introduced and extended the study of classical invariant theory to quiver varieties. I will discuss the evolution of invariant theory, invariant theory in geometric representation theory, some results and conjectures, and interesting applications.