Department of

Mathematics

Seminar Calendar
for events the day of Tuesday, September 23, 2014.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, September 23, 2014

1:00 pm in 347 Altgeld Hall,Tuesday, September 23, 2014

Sobolev stability of plane wave solutions to the nonlinear Schrodinger equation

Bobby Wilson (U Chicago Math)

Abstract: We will discuss the question of Sobolev stability of certain solutions to the nonlinear Schr¨odinger equation on the d-dimensional torus. In particular, we will discuss results concerning the arbitrarily long-time orbital stability of plane wave solutions under generic perturbations

1:00 pm in Altgeld Hall 243,Tuesday, September 23, 2014

Trajectories on surfaces and moduli spaces: a survey in honor of Maryam Mirzakhani’s Fields medal.

Abstract: In this talk, we survey a small portion of the work of Maryam Mirzakhani. In particular, we focus on her results on counting simple geodesics on surfaces, and her recent joint work with Alex Eskin and its application to billiard flows. http://youtu.be/hqu0ru8iWvM

1:00 pm in 345 Altgeld Hall,Tuesday, September 23, 2014

Model theory of transseries

Lou van den Dries (UIUC Math)

Abstract: Last spring, Matthias Aschenbrenner, Joris van der Hoeven, and LvdD finished a twenty year quest by finding and proving the key model-theoretic and algebraic facts about the differential field of transseries (in the spirit of Tarski's classical results about the logical properties of the field of real numbers). In two semnar talks on this work, I will sketch our results, and discuss some problems about the differential algebra and model theory of transseries that are still open. - This is the second talk. The first talk was given in the Model Theory and Descriptive Set Theory Seminar on Friday September 19th.

2:00 pm in Altgeld Hall 347,Tuesday, September 23, 2014

Parabolic Harnack inequalities for a family of time-dependent non-symmetric Dirichlet forms

Janna Lierl (UIUC Math)

Abstract: Moser iteration is a method that is used to prove mean value estimates, which can then be applied to obtain a parabolic Harnack inequality. Aronson and Serrin applied this technique to a wide class of non-symmetric operators on Euclidean space. On complete Riemannian manifolds, it is known from the works of A. Grigor'yan and L. Saloff-Coste that the parabolic Harnack inequality is equivalent to Poincare inequality together with volume doubling, as well as to two-sided heat kernel bounds. Some part of these results was extended to time-dependent non-symmetric Dirichlet spaces by K.-T. Sturm. I will talk about some recent work on applying parabolic Moser iteration in the context of (non-symmetric) time-dependent forms. This is joint work with L. Saloff-Coste.

3:00 pm in 243 Altgeld Hall,Tuesday, September 23, 2014

Kernels of numerical pushforwards

Mihai Fulger (Princeton)

Abstract: If $\pi:X\to Y$ is a morphism of projective varieties over an algebraically closed field, and Z is an effective k-cycle on X, then $\pi_*Z=0$ iff Z is a combination of subvarieties of X that are contracted by $\pi$. When working not with cycles, but with cycle classes (modulo numerical equivalence), it is natural to ask when can we expect a similar geometric conclusion given the vanishing of a class $\pi_*\alpha$. I will present progress on this question, in particular leading to new cases of two conjectures essentially due to Debarre, Jiang, and Voisin. This is joint work with B. Lehmann.

3:00 pm in 241 Altgeld Hall,Tuesday, September 23, 2014

On a Conjecture of Erdős, Gallai, and Tuza

Gregory J. Puleo   [email] (Coordinated Science Lab, UIUC)

Abstract: Erdős, Gallai, and Tuza posed the following problem: given an n-vertex graph $G$, let $τ_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $α_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $α_1(G) + τ_1(G) ≤ n^2/4$? A positive answer would generalize Mantel's Theorem, which states that the largest possible number of edges in a triangle-free graph is $n^2/4$. In this talk, we show three main results. We first obtain the upper bound $α_1(G) + τ_1(G) ≤ 5n^2/16$, as a partial result towards the Erdős--Gallai--Tuza conjecture. We then study the properties of a minimal counterexample to the conjecture, showing that any minimal counterexample has "dense edge cuts" and in particular has minimum degree greater than $n/2$. This reconciles the two different formulations of the conjecture found in the literature, since it implies that the Erdős--Gallai--Tuza conjecture holds for all graphs if and only if it holds for graphs for which every edge lies in a triangle. Finally, we show that the conjecture holds for all graphs which contain no induced subgraph isomorphic to $K_4^-$, the graph obtained from $K_4$ by removing an edge.

4:00 pm in 245 Altgeld Hall,Tuesday, September 23, 2014

Fall Department Faculty Meeting

Abstract: The Fall Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception.

4:00 pm in 2 Illini Hall,Tuesday, September 23, 2014

Options Valuation with a Transform Approach

Liming Feng (UIUC Industrial and Enterprise Systems Engineering)

Abstract: Transform methods have been widely used for options valuation in models with explicit characteristic functions. We explore the analyticity of the characteristic functions and propose Hilbert transform based methods for the valuation of European, American and path dependent options and Monte Carlo simulation from such characteristic functions. The schemes are easy to implement. Despite the simplicity, they are very accurate, with exponentially decaying errors. Liming Feng is an associate professor in the Department of Industrial and Enterprise Systems Engineering at the University of Illinois at Urbana-Champaign. He obtained his Ph.D. in Industrial Engineering and Management Sciences from Northwestern University in 2006. His main research interests are in quantitative finance. He is interested in developing theory and efficient computational methods for solving various quantitative finance problems. He is affiliated with the Master of Science in Financial Engineering program at the University of Illinois at Urbana-Champaign.