Department of Mathematics

Calendar


Tomorrow

Student Colloquium, Abishek Parab, Purdue University, UNIV 119

Monday, April 20, 2015, 1:30 - 2:30 PM EDT

A Proof of the Quillen-Suslin Theorem

Abstract: In his famous 1955 FAC paper, Serre asked if projective modules over polynomial rings are free. This is the algebraic interpretation of a geometric statement which came to be known as Serre's conjecture. It was finally proven in 1977 independently by Quillen and Suslin. In the talk, we shall see a simpler proof by Vaserstein, which uses easy commutative and homological algebra.

Computational & Applied Mathematics Seminar, Professor Pengtao Yue, Virginia Tech, REC 108

Monday, April 20, 2015, 3:30 - 4:30 PM EDT

ALE-Phase-Field Simulations of Moving Contact Lines on Moving Particles

Abstract: In this talk, I will present a hybrid Arbitrary-Lagrangian-Eulerian(ALE)-Phase-Field method for the direct numerical simulation of multiphase flows where fluid interfaces, moving rigid particles, and moving contact lines coexist. Practical applications include Pickering emulsions, froth flotation, and biolocomotion at fluid interface. An ALE algorithm based on the finite element method and an adaptive moving mesh is used to track the moving boundaries of rigid particles. A phase-field method based on the same moving mesh is used to capture the fluid interfaces; meanwhile, the Cahn-Hilliard diffusion automatically takes care of the stress singularity at the moving contact line when a fluid interface intersects a solid surface. To fully resolve the diffuse interface, mesh is locally refined at the fluid interface. All the governing equations, i.e., equations for fluids, interfaces, and particles, are solved implicitly in a unified variational framework. As a result, the hydrodynamic forces and moments on particles do not appear explicitly in the formulation and an energy law holds for the whole system. I will show that the three-phase flow is essentially free of parasitic currents if the surface tension term is properly formulated. In the end I will present some recent results on the water entry problem and the capillary interaction between floating particles (a.k.a. the Cheerios effect), with a focus on the effect of contact-line dynamics.

Geometry Seminar, Nicholas Miller, Purdue University, MATH 731

Monday, April 20, 2015, 3:30 - 4:30 PM EDT

Arithmetic Progressions in the Primitive Length Spectrum

Abstract: Since the 1980s, there have been a host of prime geodesic theorems displaying an anaology between primitive, closed geodesics on hyperbolic manifolds and prime numbers in the integers. For instance, there is an analogous asymptotic for primitive, closed geodesics as the asymptotic for primes governed by the prime number theorem. In this talk, I will survey some existing results exhibiting such a connection. I will then go on to discuss some recent work on arithmetic progressions in the primitive length spectrum extending this relationship.


Next Week

Junior Analysis Seminar, Dr. Verónica Quítalo, Purdue University, MATH 731

Tuesday, April 21, 2015, 2:30 - 3:30 PM EDT

Non degeneracy, Lipschitz regularity and monotonicity formula in free boundary problems

Abstract: We will review the basic tools in the study of free boundary problems for harmonic functions with the goal of a deeper understanding.

Department of Mathematics Colloquium, Professor Dihua Jiang, University of Minnesota, MATH 175

Tuesday, April 21, 2015, 4:30 - 5:30 PM EDT

Howe Duality and Endoscopy Correspondence

Abstract: In classical invariant theory, one consider an algebraic group $G$ acting on an affine space or variety $X$. It is important to understand the decomposition of the space of regular functions over $X$ as a $G$-module, in particular, the algebraic invariants. When $G$ is reductive, such a decomposition is a direct sum. There are interesting applications for the pair $(G,X)$ when the decomposition is of multiplicity free. On the other hand, the classical duality theory is to understand the decomposition when it is not of multiplicity free. We first discuss the general approach to such a general case with multiplicities in terms of representations of Weyl algebras and in terms of Howe duality. Then we discuss how to extend such an idea to construct endoscopy correspondences for cuspidal automorphic representations of classical groups. Refreshments will be served in the Math Library Lounge at 4:00 p.m.

PhD Thesis Defense, Yongheng Zhang, AR 102A

Wednesday, April 22, 2015, 2:00 - 4:00 PM EDT

Permutohedra, Configuration Spaces and Spineless Cacti. Committee: R. Kaufmann (Chair), McClure, McReynolds, and Gepner

Working Seminar in PDEs, Changyou Wang, Purdue University, BRNG B247

Wednesday, April 22, 2015, 2:30 - 3:30 PM EDT

Introduction of Navier-Stokes Equations, V

Abstract: In a series of lectures, I plan to introduce a few important theorems and related techniques on analysis of the incompressible NSE. This includes, among other things, (1) Leray’s construction of local classical solutions; (2) Kato’s theorem on mild solutions; (3) Beale-Kato-Majda (BKM) criterion on blow-up of NSE; (4) Caffarelli-Kohn-Nirenberg’s regularity theory; (5) Escauriaza-Seregin-Sverak’s regularity.

Algebraic Geometry Seminar, Prof. Yih Sung, Purdue University, MATH 731

Wednesday, April 22, 2015, 3:30 - 4:30 PM EDT

Rational Points and Hypergeometric Function

Abstract: This is a joint work with professor Sai-Kee Yeung. In this work we investigate the relation between the number of rational points over a finite field $\fin_{p^n}$ on a family of higher genus curves and their periods in terms of hypergeometric functions. For the case of $y^\ell=x(x-1)(x-\lambda)$ we find a closed form in terms of hypergeometric functions associated to the periods on the curve. For the general situation $y^\ell=x^{a_1}(x-1)^{a_2}(x-\lambda)^{a_3}$ we show that the number is a linear combination of hypergemoetric series corresponding to periods again, and give an algorithm to determine the coefficients involved.

PhD Thesis Defense, Bumsik Kim, REC 309

Wednesday, April 22, 2015, 4:00 - 6:00 PM EDT

Functional inequalities, and the curvature dimension inequality on totally geodesic foliations. Committee: Baudoin (Chair), Banuelos, Lempert, and Yeung.

Arithmetic and Topology Seminar, Artur Jackson, Purdue University, REC 108

Wednesday, April 22, 2015, 4:30 - 5:30 PM EDT

A New Approach to Arakelov Geometry II

Abstract: Using ideas from descent, we will construct objects over a compactification of Spec Z.

Commutative Algebra Seminar, Dr. Linquan Ma, Purdue University, MATH 215

Wednesday, April 22, 2015, 4:30 - 5:30 PM EDT

The Category of F-modules has Finite Global Dimension

Abstract: We discuss some homological properties of Lyubeznik's F-modules. We prove that, under very mild conditions, the category of F-modules has global dimension one more than the dimension of the ring.

Automorphic Forms and Representation Theory Seminar, Professor Dihua Jiang, University of Minnesota, UNIV 217

Thursday, April 23, 2015, 1:30 - 2:30 PM EDT

Cuspidality of Certain Global Arthur Packets for Classical Groups

ABSTRACT: Following the endoscopic classification of Arthur, automorphic representations of classical groups in the discrete spectrum are assigned to be in certain sets, called global Arthur packets. It is important to say which global Arthur packets contains only cuspidal automorphic representations and which global Arthur packets contains only non-cuspidal discrete series automorphic representations. By using the structure of Fourier coefficients of automorphic representations, Baiying Liu and I are able to make progress towards those questions. Historically, those questions are closely related to the theory of singular automorphic forms and has been investigated by many people, including the work of Roger Howe and his students from the representation-theoretic point of view.

Basic Skills Workshop, Dr. Andrew Hill, Centers for Disease Control and Prevention, BRNG 1254

Thursday, April 23, 2015, 1:30 - 2:20 PM EDT

Research at CDC

Abstract: Dr. Andrew Hill will share with us some of his experiences working with the Centers for Disease Control and Prevention (CDC), including how he came to work for the CDC and how graduate students can get involved.

PhD Thesis Defense, Nancy Hernandez Ceron, UNIV 019

Thursday, April 23, 2015, 3:00 - 4:30 PM EDT

Discrete epidemic models with arbitrarily distributed disease stages Committee: Feng (Chair), Buzzard, Yip, and A. Hill.

Probability Seminar, Jonathon Peterson, Purdue University, REC 308

Thursday, April 23, 2015, 3:30 - 4:30 PM EDT

Excited Random Walks in Cookie Environments with Markovian Cookie Stacks

Abstract: Excited random walks (also called cookie random walks) are a model of a self-interacting random motion where the transition probabilities depend on the past behavior of the walk through the local time at the present site. More specifically, given a collection {ω x (j)} x∈Z,j≥1 ∈(0,1) Z×N , upon the j -th visit to the site x the random walk steps to the right (resp. left) with probability ω x (j) (resp. 1−ω x (j) ). Most of the results known for excited random walks assume either • non-negative cookies: ω x (j)≥1/2 for all x,j •or boundedly many cookies per site: there is an M<∞ such that ω x (j)=1/2 for all x∈Z and j>M . Until very recently, very little was known in the case when at each site x there were infinitely many j with ω x (j)>1/2 and ω x (j)<1/2 . In this direction, Kozma, Orenshtein, and Shinkar studied the case of periodic cookie sequences; that is, there is a fixed periodic sequence {p(j)} j≥1 with ω x (j)=p(j) for all x,j . Under this assumption they proved an explicit criterion for recurrence/transience of the excited random walk. In this talk we will consider a different model where for each x∈Z the sequence {ω x (j)} j≥1 comes from an independent copy of a finite state Markov chain. This model generalizes both the case of periodic cookie sequences and many instances of boundedly many cookies per site. We are able to extend many of the known results from the boundedly many cookies case to our setup, including a criterion for recurrence/transience, a law of large numbers with an explicit criterion for non-zero speed, and limiting distributions in the transient case.

PDE Seminar, Professor Andrew Lorent, University of Cincinnati, BRNG B261

Thursday, April 23, 2015, 3:30 - 4:30 PM EDT

Rigidity of Pairs of Quasiregular Mappings Whose Symmetric Part of Gradient are Close

Abstract: Consider two $C^1$ mappings $u$, $v$ whose gradients have the property that for every $z$, $Du(z)$ and $Dv(z)$ map the unit ball to a pair of ellipses that are the same after rotation $R(z)$. If $u$ is globally invertible its an exercise to see this implies that $u=l_R \circ v$ where $l_R$ is an affine map whose gradient is a rotation, in particular this implies that $R(z)$ is a constant rotation. So the local \rm geometric property that on gradients implies a global property. We consider the questions of to what extent is this true for pairs of arbitrary Sobolev maps (where neither of them is invertible). This question leads us to the theory of Quasiregular mappings, the Beltrami equation, Stoilow decompositions and the Quantitative Liouville theorem of Friesecke, Muller and James. We will survey these topics and their connections before describing some recent results. One of these results can be though of as a new line of generalization of Friesecke-Muller-James theorem. Specifically F-M-J theorem can be thought of as an optimal quantitative global relation between the pairs of mappings $Id$, $u$ whose symmetric part of gradients are close in $L^p$ norm. A recent theorem is of the same form expect is a suboptimal quantitative global relation between an pair of quasiregular mappings $v$, $u$.

Topology Seminar, Prasit Bhattacharya, Indiana University Bloomington, UNIV 319

Thursday, April 23, 2015, 4:00 - 5:00 PM EDT

Higher Associativity of Moore Spectra

Abstract: Not much is known about homotopy coherent ring structures of the Moore spectrum $M_p(i)$ (the cofiber of $p^i$ self-map on the spherespectrum $S^0$), especially when $i > 1$. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called $A_n$ structures. The only known results are that $M_p(1)$ is $A_{p-1}$ and not $A_p$ and that $M_2(i)$ are at least $A_3$ for $i>1$. In this talk, techniques will be developed to get estimates of `higher associativity' structures on $M_p(i)$.

CCAM Lunch Seminar, Mr, Tao Luo, Hong Kong University of Science and Technology, REC 226

Friday, April 24, 2015, 11:30 - 12:30 PM EDT

Analysis of Epitaxial Growth with Elasticity

Abstract: In epitaxial growth with elasticity on vicinal surfaces, step bunching instability leading to some self-organization phenomenon, is widely believed to be crucial in the fabrication of nanostructures. However, difficulties appear in the modelling and analysis of these growth process due to the nonlocal effects of elasticity and the interaction between different length scales. In this talk, we will first introduce some models for epitaxial growth with elasticity, including both discrete and continuum models. After that, an example of a discrete model will be analyzed rigorously. We will show the energy scaling law, one bunch structure and provide sharp bounds for bunch sizes. Similar results for the continuum models will also be briefly mentioned. This is a joint work with Prof. Yang Xiang and Prof. Aaron Yip.

Secret Seminar, Jonathan Montano, Purdue University, MATH 731

Friday, April 24, 2015, 11:30 - 1:30 PM EDT

Some Computations of Generalized Multiplicities

Abstract: The $j$-multiplicity plays an important role in the intersection theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the $\epsilon$-multiplicity and equisingularity theory. These multiplicities generalize the notion of Hilbert-Samuel multiplicity to non $m$-primary ideals. In this talk, I will show how to compute these multiplicities for some classes of ideals. For monomial ideals, we show that the $j$- and $\epsilon$ multiplicities are equal to normalized volumes of certain regions. This result is an extension of Teissier's volume interpretation of the Hilbert-Samuel multiplicity for $m$-primary monomial ideals. For ideals defining rational normal scrolls, we provide a closed formula for the $j$-multiplicity by establishing a relation with the degree of certain blowup algebras. For ideals defining determinantal varieties, we are able to express these multiplicities as the integral of a polynomial over a region. This talk is based on joint work with Jack Jeffries and Matteo Varbaro.

Special Mathematical Biology Seminar, Dr. Andrew Hill, Centers for Disease Control and Prevention (CDC), REC 108

Friday, April 24, 2015, 1:30 - 2:30 PM EDT

The Role of Latent Infection in Models of Tuberculosis Transmission

Abstract: Mathematical modeling of tuberculosis (TB) transmission is complicated by the fact that generally a small proportion of people who are infected will develop TB disease quickly, whereas most will harbor the bacteria for a long latent period before progressing to disease. Variants of SEIR models are typically employed to model TB, but the slow dynamics present challenges in developing credible models, especially in low-burden settings where activation of chronic latent infection can be the main driver of disease. In this talk, I discuss various approaches to modeling TB, describe recent work with CDC colleagues to estimate the prevalence of TB infection in the United States, and outline work in development to model outbreaks in homeless shelters. Biosketch: Andrew Hill is a statistician with the Division of Tuberculosis Elimination at the U.S. Centers for Disease Control and Prevention. He received his Ph.D. in mathematics from the University of Canterbury, New Zealand. After teaching at the University of Auckland, Agnes Scott College, and Emory University where he was a NIH postdoctoral fellow in the Biostatistics Department, he joined the CDC in 2007.


Two Weeks

PhD Thesis Defense, Yiran Wang, BRNG B212

Monday, April 27, 2015, 3:00 - 5:00 PM EDT

The resolvent of the Laplacian on non-trapping asymptotically hyperbolic manifolds. Committee: Sa Barreto (Chair), Bauman, Stefanov, and Datchev

Computational & Applied Mathematics Seminar, Professor Jianfeng Lu, Duke University, REC 108

Monday, April 27, 2015, 3:30 - 4:30 PM EDT

Transition Path Processes and Coarse-Graining of Stochastic System

Abstract: Understanding rare events like transitions of chemical system from reactant to product states is a challenging problem due to the time scale separation. In this talk, we will discuss some recent progress in mathematical theory of transition paths. In particular, we identify and characterize the stochastic process corresponds to transition paths. The study of transition path process helps to understand the transition mechanism and provides a framework to design and analyze numerical approaches for rare event sampling and simulation.

Automorphic Forms and Representation Theory Seminar, Professor Terrence Blackman, University of Denver, UNIV 217

Thursday, April 30, 2015, 1:30 - 2:30 PM EDT

Spectral Correspondences for Maass Waveforms on Quaternion Groups

ABSTRACT: I will discuss aspects of the arithmetic and geometry of quaternion algebras. In particular, I will address a special case of the Jacquet-Langlands correspondence. I.e., the spectral correspondence between spaces of Maass forms on the multiplicative group of a division quaternion algebra and spaces of Maass forms on a related congruence subgroup of the modular group.

PhD Thesis Defense, Nick Stull, BRNG 1254

Thursday, April 30, 2015, 2:00 - 3:30 PM EDT

Unique Continuation From Infinity for Perturbations of the Complex Hyperbolic Space. Committee: Sa Barreto (Chair), Banuelos, Danielli, Stefanov.

Probability Seminar, Elnur Emrah, University of Wisconsin, REC 308

Thursday, April 30, 2015, 3:30 - 4:30 PM EDT

TBA


Three Weeks

Computational & Applied Mathematics Seminar, Professor Bingyu Zhang, University of Cincinnati, REC 108

Monday, May 4, 2015, 3:30 - 4:30 PM EDT

TBA


June

PhD Thesis Defense, Heejun Choi, REC 302

Monday, June 15, 2015, 10:00 - 12:00 PM EDT

On several efficient algorithms for some partial differential equations. Committee: Shen (Chair), Cai, Chen, and Dong.


July

PhD Thesis Defense, Jonathan Montano, BRNG 1206

Friday, July 17, 2015, 2:00 - 4:00 PM EDT

Generalized multiplicities and depth of blowup algebras. Committee: Ulrich (Chair), Caviglia, Goins, and Heinzer.


August

Computational & Applied Mathematics Seminar, Professor Krzysztof J. Fidkowski, University of Michigan, REC 108

Monday, August 31, 2015, 3:30 - 4:30 PM EDT


September

Computational & Applied Mathematics Seminar, Professor Chun Liu, Penn State University , REC 108

Monday, September 14, 2015, 3:30 - 4:30 PM EDT

Graduate Student Invited Colloquium, Bernd Sturmfels, University of California, MATH 175

Tuesday, September 22, 2015, 4:30 - 5:30 PM EDT


November

Computational & Applied Mathematics Seminar, Professor David Kopriva, Florida State University, REC 108

Monday, November 16, 2015, 3:30 - 4:30 PM EST

TBA


December

Computational & Applied Mathematics Seminar, Professor Lin Lin, UC Berkeley, REC 108

Monday, December 7, 2015, 3:30 - 4:30 PM EST