# Calendar

## Yesterday

### Automorphic Forms and Representation Theory Seminar, Professor Chung Pang Mok, Purdue University, BRNG 1260

Thursday, Dec 1 1:30 pm - 2:30 pm

## Speculations on the Langlands Program

Abstract: In this talk we give a selected overview of the major results on the Langlands correspondence, and suggest some open questions as possibilities for future development.## Wellposedness of a Nonlocal Nonlinear Diﬀusion Equation of Image Processing

Abstract: In this talk, we will establish the wellposedness of a degenerate regularization of the well-known Perona-Malik equation in noise reduction for discontinuous initial data. We will also show the (exponential) asymptotic stability of stationary solutions.## Today

### Geometric Analysis Reading Seminar, Qinfeng Li, Purdue University, MATH 731

Friday, Dec 2 10:00 am - 11:00 am

## A Regularity Theory for Harmonic Maps I

Abstract: In this first talk, I'll introduce the notion of harmonic map, and then I'll present the fundamental paper "a regularity theory for harmonic maps" by Schoen and Uhlenbeck.## Topics in Low Re flow: From Microfluidic Fluid-Structure Interactions to Proppant Transport in Hydraulic Fractures

Abstract: In this talk, I will summarize some recent results from and new research directions for my research group. The first topic is from the field of microfluidics, in which rectangular channels with deformable walls are used as one of the simplest models for lab-on-a-chip devices. Experimentally, these devices are found to deform into a non-rectangular cross-section due to fluid--structure interactions. These deformations result in a non-linear relationship between the volumetric flow rate and the pressure drop, which we seek to predict. Via perturbative calculations, the flow rate--pressure drop relation can be obtained by analyzing a coupled system of Stokes ($Re=0$) flow in a three-dimensional (3D) rectangular channel with a top wall that is linearly elastic, specifically a Kirchhoff--Love plate. We have recently benchmarked and verified the theoretical predictions by 3D numerical simulations, calibrated with experimental pressure drop--flow rate data, using the commercial software suite ANSYS. The second topic addresses some new ideas about controlling particle migration using geometry and hydrodynamics. Particle migration in flows is a phenomenon common to many areas of the engineering sciences, specifically in predicting how proppants are deposited into hydraulic fractures. Fracking involves the use of not just clear fluids but also fluids bearing proppants, which are particulate materials meant to settle into cracks to prop them open, prevent crack localization instabilities and increase fracture conductivity. There still remain fundamental aspects to particle migration in flows that are not fully understood. I will discuss some open problems, specifically related to formulating a transport problem for proppants and understanding how nonuniform conduit shapes and nonuniform hydrodynamic forcing could be exploited to preferentially control particle migration.### Special Department of Mathematics Colloquium, Professor Lise-Marie Imbert-Gerard, Post-doctoral Researcher, Cathleen Morawetz Post-doctoral Fellow, Courant Institute, NYU, BRNG B222

Friday, Dec 2 4:30 pm - 5:30 pm

## Variable Coefficients and Numerical Methods for Electromagnetic Waves

Abstract: In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis. In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computating the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.Post-doctoral Researcher, Cathleen Morawetz Post-doctoral Fellow

Research Area: Mathematical and Numerical Methods

## Next Week

### Geometric Analysis Seminar, Prof. Liz Vivas, The Ohio State University, MATH 731

Monday, Dec 5 3:30 pm - 4:30 pm

## Parabolic Skew-Products and Parametrization

Abstract: It is a classical result that parametrization of unstable manifolds of hyperbolic holomorphic maps can be obtained by a limit of iterates of the map composed with an appropriate inverse action.In this talk I will generalize this result for skew-product invariant holomorphic maps that are parabolic. I will first give an overview of the results known in one and several complex dimensions.

## Discontinuous Galerkin Methods for Relativistic Vlasov-Maxwell System

Abstract: The relativistic Vlasov-Maxwell (RVM) system is a kinetic model that describes the dynamics of plasma when the charged particles move in the relativistic regime and their collisions are not important. In this paper, we formulate and investigate discontinuous Galerkin (DG) methods to solve the RVM system. When standard piecewise polynomial functions are used to define trial and test spaces, the methods conserve mass as expected. However the energy conservation does not hold due to the specific form of the total energy of the system. In order to obtain provable mass and energy conservation, we take advantage of the flexibility of DG discretizations and enrich the discrete spaces with some non-polynomial function. For the semi-discrete DG methods with standard and enriched spaces, stability and error estimates are established together with their properties in conservation. In actual implementation with the enriched space, special care is needed to reduce the loss of significance for better numerical stability. Numerical experiments, including streaming Weibel instability and wakefield acceleration, are presented to demonstrate the performance of the methods. Positivity-preserving limiter is also used in simulating wakefield acceleration to obtain physically more relevant solutions.### Special Department of Mathematics Colloquium, Professor Martina Hofmanová , Technical University Berlin, BRNG B222

Monday, Dec 5 4:30 pm - 5:30 pm

## Randomness in Convection-Diffusion Problems

Abstract: In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.Junior Professor

Research area: Stochastic and rough PDEs, stochastic analysis, rough paths

### Basic Skills Seminar, Dave Zwicky and Natasha Johnson, Math Library Lounge

Tuesday, Dec 6 1:00 pm - 2:00 pm

## EndNote Workshop

Abstract: EndNote citation management software is a clever tool to store, organize, and manipulate your citations. With EndNote, users are able to build a personal library of citations that can be used to create in-text citations and bibliographies for documents, proposals, dissertations, and journal submissions. Additionally, EndNote has powerful sharing capabilities that make working with a group easy. In this session, we will discuss: Importing citations, exporting citations, Cite While You Write feature, and sharing with a group. If possible, please bring your laptop.### Probability Seminar, Lluís Quer, Universitat Autònoma de Barcelona, BRNG 1242

Tuesday, Dec 6 3:30 pm - 4:30 pm

## The Hyperbolic Anderson Model with Rough Noise in Space

Abstract: We consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index $H∈(14,12)H∈(14,12)$. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the pp-th moment of the solution, for any $p≥2p≥2$. The condition $H>14H>14$ turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one interpreted in the Itô sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent. The talk is based on a joint work with Raluca Balan (Univ. of Ottawa) and Maria Jolis (Autonomous Univ. of Barcelona).### Special Department of Mathematics Colloquium, Professor Anton Zeitlin, Columbia University, BRNG B222

Tuesday, Dec 6 4:30 pm - 5:30 pm

## Generalizations of Teichmueller Space

Abstract: Teichmueller space, which parameterizes surfaces, is a fundamental space that is important in many areas of mathematics and physics. In recent times generalizations of this space have been intensely studied. An example of such higher Teichmueller spaces is given by so-called super-Teichmueller spaces. These appear when studying a combinatorial approach to spin structures on Riemann surfaces and the generalization to supermanifolds. Super means that the structure sheaf is Z/2Z graded and contains odd or anti-commuting coordinates. The super-Teichmueller spaces arise naturally as higher Teichmueller spaces corresponding to supergroups, which play an important role in geometric topology, algebraic geometry and mathematical physics. There the anti-commuting variables correspond to Fermions. After the introduction of these spaces, I will provide the solution of a long-standing problem of describing the analogue of Penner coordinates on super-Teichmueller space and its generalizations. The importance of these coordinates is justified by two remarkable properties: the action of the mapping class group is rational and the Weil-Petersson form is given by a simple explicit formula. I will end outlining some of the many emerging applications of this theory.Assistant Professor

Research Area: Representation theory with applications to geometry, topology and mathematical physics.

### Spectral and Scattering Theory Seminar, Jeffrey Galkowski, McGill University, MATH 731

Wednesday, Dec 7 1:30 pm - 2:30 pm

## Pointwise Bounds for Steklov Eigenfunctions

Abstract: Let $(\Omega,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial \Omega$. The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary and proving a conjecture of Hislop and Lutzer. The estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations $Pu=0$ near the characteristic set $\{\sigma(P)=0\}$. This talk is based on joint work with John Toth.### Commutative Algebra Seminar, Christina Jamroz, Purdue University, UNIV 219

Wednesday, Dec 7 1:30 pm - 2:30 pm

## Betti Numbers of Piecewise Lex Ideals

Abstract: In this talk, we extend a result of Caviglia and Sbarra to a polynomial ring with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function, and Betti numbers at least as large as those of the original ideal. This is joint work with Gabriel Sosa.### Special Department of Mathematics Colloquium, Giulia Saccá, Stony Brook University, BRNG 2290

Wednesday, Dec 7 4:30 pm - 5:30 pm

## Compact Hyperkahler Manifolds in Algebraic Geometry

Abstract: Hyperkahler (HK) manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Compact HK manifolds are one of the building blocks for algebraic varieties with trivial first Chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this talk I will give an overview of the theory of compact HK manifolds and then focus on some of my work, including a recent joint work with R. Laza and C. Voisin.James H. Simons Instructor

Research area: Algebraic geometry

## Coassembly for Representation Spaces

Abstract: I'll discuss models for a coassembly map from representation spaces to topological K-theory. At its most basic, this map carries a representation of G to the K-theory class of its associated vector bundle over BG. This map can be realized in at least two ways as a map of ring spectra, leading to various geometric consequences. In particular, I'll discuss applications to groups with property (T) and to flat connections over the 3-dimensional Heisenberg manifold.### Automorphic Forms and Representation Theory Seminar, Dr. Bin Xu, University of Calgary, BRNG 1260

Thursday, Dec 8 1:30 pm - 2:20 pm

## On The Combinatorial Structure of Arthur Packets: p-adic Symplectic and Orthogonal Groups

Abstract: The irreducible smooth representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, which form a subset of the usual Langlands parameters. The set of irreducible representations associated with a single Arthur parameter is called an Arthur packet. Following Arthur's classification theory of automorphic representations of symplectic and orthogonal groups, the Arthur packets are now known in these cases. On the other hand, Moeglin independently constructed these packets in the p-adic case by using very different methods. In this talk, I would like to describe a combinatorial procedure to study the structure of the Arthur packets following the works of Moeglin. As an application, we show the size of Arthur packets in these cases can be given by counting integral (or half-integral) points in certain polytopes.## A Maximum Principle Technique for Fully Nonlinear Elliptic PDEs

Abstract: I will describe a geometric technique, particularly well suited to Crandall-Lions viscosity solutions of nonlinear elliptic equations. This technique provides a common approach to several known results, including the Bardi-Da Lio strong maximum principle for solutions of nonlinear elliptic pdes and the strong maximum principle of Yu for the gradient of solutions of the infinity Laplacian. Combined with a novel approximation result, it also proves the strong maximum principle for the difference of two solutions of a nonlinear, uniformly elliptic pde. As far as I can determine, this is a new result for the general fully nonlinear, uniformly elliptic case.### Number Theory Seminar, Mr. Abhishek Parab, Purdue University, MATH 731

Thursday, Dec 8 4:30 pm - 5:30 pm

## Duality Between $S_k$ and $GL_n (\Bbb C)$

Abstract: We discuss Chapter 34 of Dan Bump’s Lie Groups (Second Edition).## 2017

### Department of Mathematics Colloquium, Xiang Tang, Washington University, Saint Louis, MATH 175

Tuesday, Feb 7 4:30 pm - 5:30 pm

### Department of Mathematics Colloquium, Thomas Koberda, University of Virginia, MATH 175

Tuesday, Feb 14 4:30 pm - 5:30 pm

## TBA

Refreshments will be served in the Math Library Lounge at 4:00 p.m.### Department of Mathematics Colloquium, Professor Andrew Putman, University of Notre Dame, MATH 175

Tuesday, Feb 21 4:30 pm - 5:30 pm

## TBA

Refreshments will be served in the Math Library Lounge at 4:00 p.m.### CCAM Seminar, Professor Christian Klingenberg, Wurzburg University, TBD

Monday, Mar 6 4:30 pm - 5:30 pm

### Department of Mathematics Colloquium, Aaron Naber, Northwestern University, MATH 175

Tuesday, Mar 7 4:30 pm - 5:30 pm

## TBA

Refreshments will be served in the Math Library Lounge at 4:00 p.m.### Department of Mathematics Colloquium, Christian Klingenberg, Wurzburg University, TBD

Wednesday, Mar 8 4:30 pm - 5:30 pm

## TBA

### Department of Mathematics Colloquium, Professor Dietmar Bisch, Vanderbilt University, MATH 175

Tuesday, Mar 21 4:30 pm - 5:30 pm

## TBA

Refreshments will be served in the Math Library Lounge at 4:00 p.m.### Department of Mathematics Colloquium, Tom Church, Stanford University, MATH 175

Tuesday, Apr 4 4:30 pm - 5:30 pm