# Calendar

## Yesterday

### Automorphic Forms Seminar, Prof. Yi Ouyang, University of Science and Technology of China, REC 302

Thursday, April 17, 2014, 1:30 - 2:30 PM EDT

## On the Cohomology of Semi-stable p-adic Galois Representations

Abstract: Let K be a field of characteristic 0 complete with respect to a nontrivial discrete valuation with perfect residue field k of characteristic p>0. Let V be a $p$-adic representation of the absolute Galois group of K. We compute explicitly Kato's filtration on the continuous cohomology group $H^1(K,V)$. When k is finite, we give a simple proof of Hyodo's celebrated result $H^1_g(K,V)=H^1_{st}(K,V)$ for V a potentially semi-stable Galois representation.### Learning Seminar on Perfectoid Space, Professor Donu Arapura, MATH 731

Thursday, April 17, 2014, 3:30 - 4:30 PM EDT

## Toric Varieties and Perfectoid Spaces

Abstract: I will do a quick overview of toric geometry, and explain the connection to perfectoid spaces as in section 8 of Scholze's paper.### Student Colloquium, Ms. Nancy Hernandez Ceron, Purdue University, REC 309

Thursday, April 17, 2014, 3:30 - 4:30 PM EDT

## Probability Meets Epidemiology: Math Models to Study Epidemics

Abstract: Mathematical models, both deterministic and stochastic, can be used to describe and study epidemics. In this talk we will focus on the latter. After a quick overview on Branching process, I will explain how this ideas and results can be used to compute parameters that are important for measuring the impact of an epidemic. To finish, I will present some of my current work, which focusses on the choice of distribution for the (random) infectious period.### Commutative Algebra Seminar, Mr. Youngsu Kim, Purdue University, UNIV 301

Thursday, April 17, 2014, 4:30 - 5:30 PM EDT

## Tangent Cones of Some Determinantal Rings

## Today

### CCAM Lunch Seminar, Professor Aditya Viswanathan, Michigan State University, BRNG 1222

Friday, April 18, 2014, 11:30 - 12:30 PM EDT

## Phase Retrieval Using Bandlimited Window Functions

Abstract: Phase Retrieval refers to the problem of reconstructing a signal from its intensity measurements. It arises in many applications such as x-ray crystallography and diffraction imaging, where the detectors only capture intensity information about the underlying physical process. As is well known, the phase encapsulates vital information about the underlying signal, which makes signal recovery from such measurements extremely challenging. In this talk, we introduce a novel and efficient computational framework for reconstructing signals from their Fourier magnitude measurements. Through the use of window functions (or masks), we estimate phase differences between pairs of Fourier coefficients. The band-limited design of these window functions allows us to restrict this computation to a small subset of all possible phase difference pairs. We then solve an angular synchronization problem to recover the unknown phases. Theoretical and numerical results demonstrating the accuracy and efficiency of this reconstruction framework will be presented.### Computational & Applied Mathematics Seminar, Professor Fabio Milner, Arizona State University, REC 316

Friday, April 18, 2014, 3:30 - 4:30 PM EDT

## A Model for Hypertumors

Abstract: : In a 2004 paper Nagy developed a model for a heterogenous primary neoplasm by tracking the mass of two different parenchyma cell types, the mass of the vascular endothelial cells and the total length of micro vessels, based on a system ordinary differential equations. We modified Nagy's ODE system to incorporate spatial structure describing advection and diffusion for the two different phenotypes and for the endothelial cells. The model consists of a system of PDEs with a free boundary representing the edge of the tumor. A local existence result is obtained for a simplified case, and a finite difference method is used to approximate solutions. Results of simulations corresponding to qualitatively different dynamics will be presented and interpreted.### Basic Notions Seminar, Justin Tittelfitz, Purdue University, BRNG 1268

Friday, April 18, 2014, 4:30 - 5:30 PM EDT

## A Brief Tour of Inverse Problems

Abstract: Classically, the study of the application of partial differential equations to mathematical physics is concerned with the question "Given a model for a physical system (i.e. some parameters that describe its properties) and a governing PDE, can we (quantitatively and/or qualitatively) describe the solutions?" In contrast to these so-called forward problems are inverse problems, which are (very) roughly "Given the PDE and some information about solutions, can we recover the model parameters?" For instance, if I convince you to let me shoot x-rays through you and we measure the attenuation (loss in intensity), what can we say about the density of the tissue inside your body? Or, if an earthquake occurs at an unknown location and time, but we can measure the resulting waves as they arrive at some sensing stations, can we determine that initial location and time? In this talk, I will overview some famous inverse problems and their applications: the Radon transform (the CAT scan described above), Calder\'on's problem (Electrical Impedance Tomography), inverse source problems for the wave equation (the seismology problem above), and if there is time, a case where the answer to the question is "No", but in a good way (making things invisible!).## Next Week

## Homotopy of Moduli Spaces

Abstract: We will discuss recent advances in understanding the homotopy groups, covering spaces, and the universal cover of moduli spaces of representations### Bridge to Research Seminar, Dr. Marius Dadarlat, Purdue University, BRNG B222

Monday, April 21, 2014, 3:30 - 4:30 PM EDT

## Infinite Groups as Geometric Objects and Large Scale Geometry

Abstract: Gromov had the insight that some of the most important properties of an infinite discrete group G are of geometric nature and they can be read off from the Cayley graph of G. I plan to give a light introduction to this circle of ideas and discuss the problem of drawing the Cayley graph of a group in a Hilbert space without excessive distortion.## The a posteriori error estimation in finite element method for the H(curl) problems

Committee: Z. Cai (Chair), J. Xia, J. Shen, P. Li## C^1-continuous Spectral ElementsSub-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

## Creating and Customizing Your Purdue Homepage

### Operator Algebras Seminar, Professor Bogdan Udrea, University of Illinois at Urbana, MATH 731

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

TBA

### 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.

## Two Weeks

### 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.