# Calendar

## Tomorrow

### CCAM Seminar, Professor Mohsen Zayernouri, Michigan State University

Monday, Feb 15 3:30 pm - 4:30 pm

## Spectral and High-Order Methods For Fractional PDEs

Abstract: Fractional PDE models generalize the standard (integer-order) calculus and PDEs to any differential form of fractional orders. In this talk, we first present two recent theories on fractional and tempered fractional Sturm-Liouville eigen-problems subject to some nonlocal boundary conditions. They provide explicit (non-polynomial) eigenfunctions, namely as (tempered) Jacobi poly-fractonomials, which enjoy attractive properties. These eigenfunctions extend the well-known family of Jacobi polynomials to their fractional counterparts. Then, we present several Petrov-Galerkin spectral methods for solving fixed-order, variable-order, and distributed-order fractional PDEs. Of particular interest, we present a unified Petrov-Galerkin spectral method for solving parabolic, elliptic, and hyperbolic (time- and space-) fractional PDEs in (d+1)-dimensions in a hypercube. At last, we present a novel variation of fractional Adams-Bashforth and fractional Adams-Multon methods for solving fractional ODEs. This new framework allows us to develop high-order splitting methods for nonlinear time- and space-fractional PDEs.## J-holomorphic Subvarieties

Abstract: In this talk, we discuss the J-holomorphic subvarieties in a 4-dimensional symplectic manifold. We will start by showing that a subvariety of a complex rational surface in an exceptional rational curve class could have higher genus components. On the other hand, such exotic phenomenon won't happen for ruled surfaces.We will then show a general cone theorem as in Mori theory and calculate the moduli space of a spherical class in a rational or ruled surface, which could be decomposed as linear systems.

## Next Week

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Feb 16 10:30 am - 12:00 pm

Intro to Hypermatrix Computations Seminar, Edinah Gnang, Purdue University, MATH 431 Tuesdays, Jan 26 10:30AM - 12:00 PM The themes of the seminar discusses computational and combinatorial aspects of the algebra of matrices and hypermatrices. The seminar will introduce participants to free open source Math typesetting tools such as TeX, LaTeX, SageTeX and LyX. The seminar will also introduce participants to programming using the free open source Python-based mathematical software called Sage.

## Can One Hear The Shape of a Random Walk?

Abstract: We consider a Gibbs distribution over random walk paths on the square lattice, proportional to the cardinality of the path's boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plain to the sum of their principle eigenvalue and perimeter (with respect to some norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.### Department of Mathematics Colloquium, Professor Kiran Kedlaya, UCSD, UNIV 001

Tuesday, Feb 16 4:00 pm - 5:00 pm

## Equidistribution of Frobenius Eigenvalues

Abstract: Consider a system of polynomial equations with integer coefficients. For each prime number p, one can count the solutions of these equations in the integers modulo p; while the structure of these counts is a rather deep topic in number theory, one can pose statistical questions about these counts for which the answers are expected to be somewhat simpler (although still deep). We discuss several variations on this theme, including the Chebotarev density theorem, the Sato-Tate conjecture for elliptic curves, a general but imprecise conjecture of Serre, and a precise form of Serre's conjecture for genus 2 curves due to Fite-Kedlaya-Rotger-Sutherland.Refreshments will be served in the Math Library Lounge at 3:30 P.M.

NOTE: SPECIAL TIME AND LOCATION

### Informal Algebraic Geometry Seminar, Laszlo Lempert, Purdue University, MATH 731

Wednesday, Feb 17 9:30 am - 10:30 am

## Complete Deformations of Complex Manifolds and Their Existence

### Commutative Algebra Seminar, Dr. Arindam Banerjee, Purdue University, UNIV 301

Wednesday, Feb 17 1:30 pm - 2:30 pm

## Algebra of Powers of Edge Ideals of Bipartite Graphs

### AG Seminar, Professor Thomas Reichelt, Universitat Heidelberg, MATH 731

Wednesday, Feb 17 3:30 pm - 5:00 pm

## Hodge Theory of GKZ Systems

Abstract: I will explain a close relationship between GKZ hypergeometric systems after Gelfand, Kapranov and Zelevinsky, Gauss-Maninsystems of families of Laurent polynomials and the intersection cohomology of hyperplane sections in a toric variety. As an application, I will show how to construct non-affine Landau-Ginzburg models which are mirror partners for complete intersections in toric varieties with a numerical effective anti-canonical divisor.## Flow-based methods for Community Detection

Abstract: A common problem in network analysis is to find "communities" in graphs--sets of vertices that are better connected to each other than they are to the rest of the graph. In this talk we'll introduced a class of methods for finding communities based on maximum flow calculations. After proving the min-cut/max-flow theorem, we'll take a survey of these flow-based methods and illustrate their performance on a few graphs.### Automorphic Forms and Representation Theory Seminar, Professor Eyal Kaplan, Ohio State University, REC 316

Thursday, Feb 18 1:30 pm - 2:30 pm

## A Godement-Jacquet Type Integral and the Metaplectic Shalika Model

Abstract: In a joint work with Jan Mollers, we present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.### Numerical Linear Algebra Seminar, Nicole Eikmeier, Purdue University, HAAS 101

Thursday, Feb 18 2:00 pm - 3:00 pm

## An Introduction to Information Based Complexity Theory

Abstract: Information Based Complexity Theory aims to define a minimal cost in computing an approximate solution to a problem, where cost is the number of matrix-vector products. I will give the background needed to discuss optimality in this context, specifically for the problem of solving linear systems. We will see that the Krylov Information has almost optimal cost among all possible matrix-vector information.### Operator Algebras Seminar, Isaac Goldbring, University of Illinois at Chicago, MATH 731

Thursday, Feb 18 2:30 pm - 3:30 pm

## Revisiting the First-order Theories of McDuff's $II_1$ factors

Abstract: McDuff was the first to provide a family of continuum many pairwise-nonisomorphic separable $II_1$ factors. In a recent preprint, Boutonnet, Chifan, and Ioana proved that any ultrapowers of any two distinct McDuff examples are also nonisomorphic. As a result, this shows that McDuff's examples are also pairwise non-elementarily equivalent, thus settling the question of how many first-order theories of $II_1$ factors there are. From the model-theoretic point of view, this resolution of the question is not satisfying as we do not see an explicit family of sentences that distinguish the McDuff examples. In this talk, I will present a partial resolution to this problem by discussing the following result: If $M_\alpha$ and $M_\beta$ are two of McDuff's examples, where $\alpha,\beta \in 2^{\omega}$ are such that $\alpha|k=\beta|k$ but $\alpha(k\not=\beta(k)$, then there must exist a formula of quantifier-complexity at most $6k+3$ on which they disagree! The proof uses Ehrenfeucht-Fraisse games. The talk represents joint work with Bradd Hart.## Limit Mixed Hodge Structures and Log Deformations

Abstract: This talk will be the continuation of last talk. I will finish up monodromy filtration and give an example of degeneration of curves. Then I will do Steenbrink's construction on limiting mixed hodge structure.## Two Weeks

### Computational & Applied Mathematics Seminar, Nate Roberts, Argonne National Laboratory, REC 225

Monday, Feb 22 3:30 pm - 4:30 pm

## Geometric Multigrid Preconditioners for DPG Systems in Camellia

Abstract: The discontinuous Petrov-Galerkin finite element methodology of Demkowicz and Gopalakrishnan (DPG) [1, 2] offers a host of appealing features, including automatic stability and minimization of the residual in a user-controllable energy norm. DPG is, moreover, well-suited for high-performance computing, in that the extra work required by the method is embarrassingly parallel; the use of a discontinuous test space allows the computation of optimal test functions to be done element-wise. Additionally, the approach gives almost total freedom in the choice of basis functions, so that high-order discretizations can be employed to increase computational intensity (the number of floating point operations per unit of communication). Finally, since the method is stable even on a coarse mesh and comes with a built-in error measurement, it enables robust adaptivity which in turn means less human involvement in the solution process, a desirable feature when running large-scale computations.Camellia [3] is a software framework for DPG with the aim of enabling rapid development of DPG solvers both for running on a laptop and at scale. Camellia supports spatial meshes in 1D through 3D; initial support for space-time elements is also available. Camellia supports h- and p-adaptivity, and offers distributed computation of essentially all the algorithmic components of a DPG solve. (One exception, which we plan to address, is the generation and storage of the mesh geometry; at present, this happens redundantly on each MPI rank.) Camellia supports static condensation for reduction of the global problem, and has a robust, flexible interface for using third-party direct and iterative solvers for the global solve.

Until recently, we have almost always solved the global DPG system matrix using parallel direct solvers such as SuperLU Dist. This is not a scalable strategy, particularly for 3D and space-time meshes. Both memory and time costs therefore motivate the present work, an exploration of iterative solvers in the context of Poisson and Stokes problems. Since Camellia's adaptive mesh hierarchy provides us with rich geometric information, we focus on hp-geometric multigrid preconditioners with additive Schwarz smoothers of minimal or small overlap. Preconditioning a conjugate gradient solve using such preconditioners, we are able to solve much larger problems within the same memory footprint.

References [1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation. Comput. Methods Appl. Mech. Engrg., 199:1558-1572, 2010. See also ICES Report 2009-12. [2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions. Numer. Meth. Part. D. E., 27(1):70-105, January 2011. [3] N. V. Roberts. Camellia: A software framework for discontinuous Petrov-Galerkin methods. Computers & Mathematics with Applications, 68(11):1581-1604, December 2014.

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Feb 23 10:30 am - 12:00 pm

Intro to Hypermatrix Computations Seminar, Edinah Gnang, Purdue University, MATH 431 Tuesdays, Jan 26 10:30AM - 12:00 PM The themes of the seminar discusses computational and combinatorial aspects of the algebra of matrices and hypermatrices. The seminar will introduce participants to free open source Math typesetting tools such as TeX, LaTeX, SageTeX and LyX. The seminar will also introduce participants to programming using the free open source Python-based mathematical software called Sage.

### Department of Mathematics Colloquium, Noah Snyder, Indiana University, MATH 175

Tuesday, Feb 23 4:30 pm - 5:30 pm

## Subfactors and Their Classification

Abstract: Subfactors are inclusions of von Neumann factors, and play a similar role in operator algebras that Galois theory plays in ring theory. Each subfactor has an index, analogous to the degree of a field extension, but these indices do not need to be integers. The celebrated Jones index theorem says that among subfactors of index below 4, only a discrete sequence of index values can happen. This suggests that there may be some hope of classifying subfactors. Such a classification splits into two steps, one largely algebraic and the other largely analytic. The algebraic step is to classify certain "quantum group"-like objects that play the role of Galois groups. The analytic step is to then understand how many ways each of these "quantum groups" can act on a particular factor. I'll mainly focus on the algebraic part of the classification, which is now known up to index 5.25, and the examples that appear in this classification. I'll also briefly discuss what's known about the analytic part of the classification in the case of the hyperfinite II_1 factor, based on deep results of Ocneanu and Popa.## Refreshments will be served in the Math Library Lounge at 4:00 p.m.

### CCAM Lunch Seminar, Shenghao Li, University of Cincinnati, REC 226

Friday, Feb 26 11:30 am - 12:30 pm

## A Non-homogeneous Boundary Value Problem of the Sixth Order Boussinesq Equation in a Quarter Plane

Abstract: In this talk, we will discuss the initial-boundary-value problem (IBVP) for the sixth order Boussiensq equation $$ u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0\text{ where }\beta=\pm 1 $$ with non-homogeneous boundary conditions. It will be shown that the IBVP is locally well-posed on certain Sobolev spaces. The ideas used for the well-posedness is based upon the estimates of the corresponding boundary integrals for the non- homogeneous boundary data as well as the results from the initial value problems. Moreover, an related global well-posedness result will be presented.## Three Weeks

### Computational & Applied Mathematics Seminar, Professor Tao Lin, Virginia Tech, REC 225

Monday, Feb 29 3:30 pm - 4:30 pm

## Immersed Finite Element Functions Defined with the Actual Interface

Abstract: Interface problems often appear in numerical simulations over domains consisting of multiple materials which lead to discontinuous coefficients in the involved partial differential equations whose solutions are inevitably less smooth around the material interfaces. This deficiency of global regularity in the exact solution requires traditional finite element (FE) methods to use body‐fitting meshes in which each element essentially contains one of the materials; otherwise, their convergence cannot be guaranteed. Having to use unstructured body‐fitting meshes can hinder efficient applications of FE methods in some applications. The recently developed immersed finite element (IFE) methods are non‐traditional that can utilize interface‐independent meshes; hence, they can solve problems with nontrivial interfaces by structured/Cartesian meshes. However, most IFE methods developed so far employ polynomials whose approximation capability are of $O(h^2)$. Locally on each element, an IFE function in these methods is a macro polynomial piecewisely defined according to a line approximating the actual material interface curve. In this talk, we will start with a brief survey of IFE methods and a few motivations for developing IFE methods whose convergence rates are better than linear polynomial’s $O(h^2)$ rate. We will explain the need to construct IFE functions as piecewise polynomials defined with subelements formed by partitioning each interface element with the actual interface curve instead of its linear approximation. We will then discuss a particular IFE space to demonstrate some analysis ideas for dealing with new challenges generated by the fact that the restriction of a polynomial on an arbitrary curve is generally not a polynomials anymore. We will conclude the talk with numerical examples to show features of new IFE methods defined with actual interface.### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Mar 1 10:30 am - 12:00 pm

Intro to Hypermatrix Computations Seminar, Edinah Gnang, Purdue University, MATH 431 Tuesdays, Jan 26 10:30AM - 12:00 PM The themes of the seminar discusses computational and combinatorial aspects of the algebra of matrices and hypermatrices. The seminar will introduce participants to free open source Math typesetting tools such as TeX, LaTeX, SageTeX and LyX. The seminar will also introduce participants to programming using the free open source Python-based mathematical software called Sage.

### Department of Mathematics Colloquium, Professor David Nualart, University of Kansas, MATH 175

Tuesday, Mar 1 4:30 pm - 5:30 pm

## TBA

## Refreshments will be served in the Math Library Lounge at 4:00 p.m.

## March

### Computational & Applied Mathematics Seminar, Professor Li-Shi Luo, Old Dominion University, REC 225

Monday, Mar 7 3:30 pm - 4:30 pm

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Mar 8 10:30 am - 12:00 pm

### Department of Mathematics Colloquium, Professor Kazumasa Kuwada, Toyko Institute of Technology, MATH 175

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

## Maximal Diameter Theorem and Heat Flow

Abstract: It is well-known that Riemannian manifolds with uniformly positive Ricci curvature have Bonnet-Myers’ diameter upper bound, and that the equality is attained if and only if the space is isometric to a sphere. The latter assertion is called S. Y. Cheng’s maximal diameter theorem. In this talk, we give two generalizations of the maximal diameter theorem. The first one is concerned with the Bakry-Emery Ricci tensor, which is naturally associated with a (possibly non-symmetric) diffusion generator on manifold. The second one is in the framework of (possibly singular) metric measure spaces with “uniform positive lower Ricci curvature bound” in a certain sense. In each theorem, a canonical heat flow on these spaces plays a fundamental role.## Refreshments will be served in the Math Library Lounge at 4:00 p.m.

### Computational & Applied Mathematics Seminar, Professor Andrew Noymer, UC Irvine

Wednesday, Mar 9 3:30 pm - 4:30 pm

TBA

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Mar 15 10:30 am - 12:00 pm

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Mar 22 10:30 am - 12:00 pm

### Grad Student Invited Colloquium, Professor Tara Holm, Cornell University, MATH 175

Tuesday, Mar 22 4:30 pm - 5:30 pm

### CCAM Lunch Seminar, Oscar Bruno, California Institute of Technology, REC 226

Friday, Mar 25 11:30 am - 12:30 pm

## Fourier Continuation, Resolution of the Gibbs Phenomenon, and Application to Numerical Analysis and Computation

Abstract: Fourier expansions possess excellent properties of approximation of smooth and periodic functions--which make them ideal elements for simulation of periodic systems. This talk presents a new approach which extends use of rapidly convergent Fourier series to general problems in computational science. We will demonstrate these ideas with results of recent applications to numerical solution of linear and non-linear Partial Differential Equations in complex three dimensional geometries, including general solution of PDEs in the time domain such as the fluid-dynamics and elastic wave equations (where virtual dispersionlessness is demonstrated) and, using integral equation methods and related fast highly-accurate algorithms, solution of Laplace eigenvalue problems and problems of acoustics and electromagnetism at high frequencies.### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Mar 29 10:30 am - 12:00 pm

### Math Is Key Colloquium, Professor Jim Gates, University of Maryland, MATH 175

Tuesday, Mar 29 4:30 pm - 5:30 pm

## When The Cosmos Does The Wave It Does Wave Gravity

ABSTRACT: This presentation is a non-technical introduction to the subject of gravitation waves. The talk begins by using illustrative means to unveil the mathematics that Einstein used to reach the conclusion that waves of gravity must exist just as do waves of light. CGI will be essential to show the accessibility to Einstein's suggestion that has recently been seen by the LIGO (laser interferometric gravity-wave observatory) facilty.## Refreshments will be served in the Math Library Lounge at 4:00 p.m.

## April

### Computational & Applied Mathematics Seminar, Professor Chao Yang , Lawrence Berkeley Lab, REC 225

Monday, Apr 4 3:30 pm - 4:30 pm

## TBA

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Apr 5 10:30 am - 12:00 pm

### Department of Mathematics Colloquium, Professor Gilles Frankfort, Universite Paris-Nord, MATH 175

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

## TBA

## Refreshments will be served in the Math Library Lounge at 4:00 p.m.

### Computational & Applied Mathematics Seminar, David Bindel, Cornell University, REC 225

Monday, Apr 11 3:30 pm - 4:30 pm

## TBA

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Apr 12 10:30 am - 12:00 pm

### Department of Mathematics Colloquium, Professor Magdalena Musat, University of Copenhagen, MATH 175

Tuesday, Apr 12 4:30 pm - 5:30 pm

### Computational & Applied Mathematics Seminar, Professor Luoding Zhu, IUPUI, REC 225

Monday, Apr 18 3:30 pm - 4:30 pm

TBA

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Apr 19 10:30 am - 12:00 pm

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, Apr 26 10:30 am - 12:00 pm

### Probability Seminar, Alex Fribergh, Université de Montreal, BRNG 1254

Tuesday, Apr 26 3:30 pm - 4:30 pm

## TBA

## May

### CCAM Seminar, Professor Ching-Shan Chou, Ohio State University, REC 225

Monday, May 2 3:30 pm - 4:30 pm

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, May 3 10:30 am - 12:00 pm

### Hypermatrix and Computation Seminar, Edinah Gnang, Purdue University, MATH 431

Tuesday, May 10 10:30 am - 12:00 pm

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