Department of Mathematics



Ph.D. Thesis Defense, Jing Wang, BRNG B212

Wednesday, April 23, 2014, 1:00 - 2:00 PM EDT

Sub-Riemannian Heat Kernels on Model Spaces and Curvature-dimension Inequalities on Contact Manifolds

Committee: F. Baudoin (Chair), R. Banuelos, D. Danielli, N. Garofalo

AWM Basic Skills Workshop, Chapman Flack and Kelly Beranger, Purdue University, REC 121

Wednesday, April 23, 2014, 1:30 - 2:30 PM EDT

How to Build Your Math Department Webpage

Operator Algebras Seminar, Professor Bogdan Udrea, University of Illinois at Urbana, BRNG B222

Wednesday, April 23, 2014, 1:30 - 2:30 PM EDT

Some Rigidity Results for Generalized q-Gaussian Algebras

Abstract: For any $H \subset G$ countable discrete groups with $H$ abelian and $G$ acting on $H$ by automorphisms, we define the generalized q-gaussian algebras $A \rtimes \Gamma_q(G,K)$, where $A=L(H)$ and $K$ is a separable Hilbert space. We then prove that if the group inclusions $H \subset G$ and $H' \subset G'$ satisfy a certain "rigidity" assumption and $G, G'$ have the Haagerup property then $A \rtimes \Gamma_q(G,K) = B \rtimes \Gamma_q(G',K')$ implies that $A$ and $B$ are unitarily conjugate inside $M=A \rtimes \Gamma_q(G,K)$ and $\mathcal{R}_G \cong \mathcal{R}_{G'}$, where $\mathcal{R}_G$, $\mathcal{R}_{G'}$ are the countable, p.m.p. equivalence relations implemented by the actions of $G$, respectively $G'$ on $A=L(H)$. This is joint work with Marius Junge (UIUC) and Stephen Longfield (UIUC).


Automorphic Forms Seminar, Prof. Tong Liu, Purdue University, REC 302

Thursday, April 24, 2014, 1:30 - 2:30 PM EDT

The Weight Part of Serre's Conjecture for GL(2)

Abstract: Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the p-adics. As a consequence, we establish (under the usual Taylor-Wiles hypothesis) the weight part of Serre's conjecture for GL(2) over arbitrary totally real fields.

Commutative Algebra Seminar, Mr. Jeff Madsen, University of Notre Dame, UNIV 101

Thursday, April 24, 2014, 3:00 - 4:00 PM EDT

Rees Algebras of Parameterized Plane Curves

Abstract: If C is a rational plane curve of degree d, the bihomogeneous coordinate ring of the graph of its parametrization is given by the Rees algebra of an almost complete intersection ideal in k[x,y]. The Rees algebra can be viewed as the quotient of the symmetric algebra by its torsion ideal A. Finding a minimal generating set of A is largely an open problem, though it has been solved, for instance, for d at most 6 by the work of Busé and of Kustin, Polini, and Ulrich. I will present results that can be used to find all possible bidegrees of the minimal generators of A when d=7, and show how these degrees correspond to the types of singularities of C. NOTE: UNUSUAL TIME AND LOCATION

Student Colloquium, Mr. Byeongho Lee, Purdue University, REC 309

Thursday, April 24, 2014, 3:30 - 4:30 PM EDT

A Braided Differentiation Rule

Abstract: We all know how to differentiate a polynomial. It is defined as a certain limit of a function that is built out of the given polynomial. But we can just forget about the meaning of it, and remember only the formal aspect of the formula. By doing so we can "differentiate" things that we could not apply the original recipe to. For example, let us consider the symmetric n-linear forms. We can identify these with polynomials using well known formulas. Then we transfer the rules of differentiation using these formula to differentiate the symmetric n-forms. A similar but a little bit different notion is that of braided tensors, which arise in the context of orbifold Gromov-Witten theory. Could we differentiate these things? Let us find out. If you know how to differentiate a polynomial, take tensor products of vector spaces, and know what a finite group is, then you should be fine.

Learning Seminar on Perfectoid Space, Professor Tong Liu, Purdue University, MATH 731

Thursday, April 24, 2014, 3:30 - 4:30 PM EDT

Basic Properties of Perfectoid Space

Abstract: We discuss the proof of the main theorem in Scholze's IHES paper: Weight-Monodromy conejcture in many cases.

PDE Seminar, Professor Hung Tran, University of Chicago, REC 316

Thursday, April 24, 2014, 3:30 - 4:30 PM EDT

Stochastic Homogenization of a Nonconvex Hamilton-Jacobi Equation

Abstract: We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton-Jacobi equation. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality. This method applies for all generic 1D cases. Joint work with Scott Armstrong, and Yifeng Yu.

Probability Seminar, Khalifa Es-Sebaiy Cadi Ayyad University (Morocco), REC 226

Thursday, April 24, 2014, 3:30 - 4:30 PM EDT



Computational & Applied Mathematics Colloquium, Professor Leszek Demkowicz,The University of Texas at Austin, REC 316

Friday, April 25, 2014, 3:30 - 4:30 PM EDT

Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions Fundamentals

Abstract: The coming June will mark the fifth anniversary of the first two papers in which Jay Gopalakrishnan and I proposed a novel Finite Element (FE) technology based on what we called the ``ultra-weak variational formulation'' and the idea of computing (approximately) optimal test functions on the fly [1,2]. We called it the ``Discontinuous Petrov Galerkin Method''. Shortly afterward we learned that we owned neither the concept of the ultra-weak formulation nor the name of the DPG method, both introduced in a series of papers by colleagues from Milano: C. L. Bottasso, S. Micheletti, P. Causin and R. Sacco, several years earlier. The name ``ultra-weak'' was stolen from O. Cessenat and B. Despres. But the idea of computing optimal test functions was new... From the very beginning we were aware of the fact that the Petrov-Galerkin formulation is equivalent to a Minimum Residual Method (generalized Least Squares) in which the (minimized) residual is measured in a dual norm, the idea pursued much earlier by colleagues from Texas A&M: J. Bramble, R. Lazarov and J. Pasciak. Jay and I were lucky; a few months after putting [1,2] on line, Wolfgang Dahmen and Chris Schwab presented essentially the same approach pointing to a connection with mixed methods and the fact that the use of discontinuous test functions is not necessary. The lecture will focus on fundamentals of the DPG method. We will discuss the equivalence of several formulations: Petrov-Galerkin method with optimal test functions, minimum residual formulation and a mixed formulation. We will summarize well-posedness results for formulations with broken test functions: the ultra-weak formulation based on first order systems and the formulation derived from standard second order equations. Standard model problems: Poisson, linear elasticity, Stokes, linear acoustics and Maxwell equations, will be used to illustrate the methodology with h-, p-, and hp-convergence tests. The DPG method comes with a posteriori-error evaluator (not estimator...) built in which provides a natural framework for adaptivity. Take home message: The DPG method guarantees stability for any well-posed linear problem. [1] L. Demkowicz and J. Gopalakrishnan, ``A class of discontinuous Petrov-Galerkin methods. PartI: The transport equation,'' CMAME: 199, 23-24, 1558-1572, 2010. [2] L. Demkowicz and J. Gopalakrishnan, ``A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions,'' Num. Meth. Part. D.E.:27, 70-105, 2011.
Refreshments will be served in the Math Library Lounge at 3:00 PM.

Next Week

Geometry Seminar, Samuel Taylor (University of Texas at Austin), MATH 731

Monday, April 28, 2014, 3:30 - 4:30 PM EDT

Convex Cocompactness and Stability in Mapping Class Groups

Abstract: Convex cocompact subgroups of mapping class groups were introduced by Farb and Mosher and have important connections to the geometry of Teichmüller space, the curve complex, and surface group extensions. In this talk, I will discuss a new characterization of such subgroups that involves only the geometry of the mapping class group. This characterization involves a strong notion of quasiconvexity, which we call stability, and captures the intuition that convex cocompact subgroups are "highly hyperbolic" subgroups of mapping class groups. This is joint work with Matt Durham.

Probability Seminar, Fabrice Baudoin, Purdue University, REC 226

Thursday, May 1, 2014, 3:30 - 4:30 PM EDT

Stochastic Analysis on Sub-Riemannian Manifolds with Transverse Symmetries Abstract: We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenböck type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions. The results are new even in the case of the Heisenberg group which is the simplest example of a sub-Riemannian manifold with transverse symmetries.

CCAM Lunch Seminar, Ms. Stephanie Friedhoff, Tufts University, BRNG 1222

Friday, May 2, 2014, 11:30 - 12:30 PM EDT


Computational and Applied Mathematics Seminar, Professor Shari Moskow, Drexel University, REC 316

Friday, May 2, 2014, 3:30 - 4:30 PM EDT

Inverse Born Series for the Calderon Problem and Related Inverse Problems

Abstract: We propose a direct reconstruction method for the Calderon problem and other related inverse problems based on inversion of the Born series. We characterize the convergence, stability and approximation error of the method and illustrate its use in numerical reconstructions. This is a joint work with S. Arridge, K. Kilgore and J. C. Schotland.

Student PDE Seminar, Mr. Thomas Backing, Purdue University, UNIV 319

Friday, May 2, 2014, 3:30 - 4:30 PM EDT

Regularity of Lipschitz Free Boundaries for an Elliptic Problem

Abstract: In this talk I will introduce the idea of a free boundary problem and the concept of a viscosity solution to a simple elliptic free boundary problem. I will also sketch the main ideas behind Caffarelli's proof that Lipschitz free boundaries for this problem are in fact smooth.

Two Weeks

Spectral and Scattering Theory Seminar, Kiril Datchev, MIT and Purdue, REC 114

Monday, May 5, 2014, 1:30 - 2:30 PM EDT

Quantitative Limiting Absorption Principle in the Semiclassical Limit

Abstract: We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schrodinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential. NOTE: UNUSUAL TIME AND LOCATION


Ph.D. Thesis Defense, Tamas Darvas, MATH 731

Tuesday, May 27, 2014, 1:30 - 2:30 PM EDT

Geometry in the Space of Kahler Potentials

Committee: L. Lempert (Chair), D. Catlin, S.K. Yeung, B. McReynolds

Ph.D. Thesis Defense, Vitezslav Kala, REC 103

Wednesday, May 28, 2014, 2:00 - 3:00 PM EDT

Density of Self-dual Automorphic Representations of GL_n(A_Q)

Committee: F. Shahidi (Chair), B. McReynolds, D. Goldberg, T. Liu