Department of Mathematics

Calendar


Yesterday

Junior Analysis Seminar, Qinfeng Li, Purdue University, MATH 731

Tuesday, March 31, 2015, 3:30 - 4:30 PM EDT

Introduction to the Regularity Theory of Perimeter Minimizer

Abstract: The theory of sets of finite perimeter provides, in the broader framework of GMT(Geometric Measure Theory), a particularly well suited framework for studying the existence, regularity, and structure of singularities of minimizers in those geometric variational problems. Also, a lot of deeper results in the theories of currents and varifolds, which play a primary role in GMT, can be fully appreciated in this simplified framework. In this talk, I'll mention some basic facts of sets of finite perimeter, and then quickly move to the regularity part of the theory. Although regularity results can be proved under weaker minimality assumptions, for example, $(\Lambda, r)$-minimizer, I will be mainly focusing on the idea in the case of perimeter minimizer in $\mathbb{R}^n$. I'll derive the formula of the first variation of perimeter and prove the monotonicity formula for stationary sets. Then I'll quickly outline heuristically the idea of the proof of the analyticity of the reduced boundary of perimeter minimizer. Then I'll give a less-detailed proof of Simon's Theorem, which says there are no non-trivial (n-1)-dimensional stable minimal cones in $\mathbb{R}^n$ for $n \le 6$. Simon's theorem, together with the consequence of the monotonicity formula and Federer's dimension reduction argument, proves that for $2 \le n \le 7$, there are no singular points for perimeter minimizer. If time permits, I will explain briefly how the ideas can be modified to prove the $C^{1,\alpha}$ regularity theorem of integral k-varifolds with bounded mean curvature.

Department of Mathematics Colloquium, Prof. Robert Hardt, Rice University, MATH 175

Tuesday, March 31, 2015, 4:30 - 5:30 PM EDT

Some Variational Homology and Cohomology Theories for a Metric Space

ABSTRACT. Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, H. Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space and used this to define flat chains (as well as dual flat cochains). H.Federer and W.Fleming(1960) also used these to obtain mass-minimizing representatives for ordinary singular homology of Euclidean Lipschitz neighborhood retracts. Such sets include all smooth submanifolds and polyhedra, but not all algebraic varieties or even subspaces of some Banach spaces. In 2013 work with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate variational topologies for chains and cochains in general metric spaces. With these, we homologically characterize Lipschitz path connectedness. We also obtain several facts about spaces that satisfy local linear isoperimetric inequalities including a duality between these metric homology and cohomology theories. Refreshments will be served in the Math Library Lounge at 4:00 p.m.


Today

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

Wednesday, April 1, 2015, 2:30 - 3:30 PM EDT

Introduction of Navier-Stokes Equations, III

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. Uli Walther, Purdue University, MATH 731

Wednesday, April 1, 2015, 3:30 - 4:30 PM EDT

The logarithmic Complex

Abstract: Let f be a polynomial (or a holomorphic function). Ultimately, the goal is to understand the ideal ann($f^s$) in the Weyl algebra $D_n[s$] of n variables $x_1,...,x_n$ with a parameter s (or the sheaf of differential operators with parameter s). We consider a certain "logarithmic" complex that combines sheaves of logarithmic differentials with the Liouville form from synplectic geometry. If f has good homological properties and is locally strongly Euler homogeneous, this logarithmic complex resolves the symmetric algebra of the Jacobian of f. This implies that this symmetric algebra is a Cohen--Macaulay domain and that implies that ann($f^s$) is generated by derivations. In particular, this proves for a majority of hyperplane arrangements a conjecture (+a generalization) of Terao on ann($f^s$). I will show by example that the generalization does not always hold. Given time, I might talk about what all this has to do with the monodromy conjecture.

Commutative Algebra Seminar, Professor Chris Francisco, Oklahoma State University, MATH 215

Wednesday, April 1, 2015, 4:30 - 5:30 PM EDT

Generalizing the Borel Property

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

Wednesday, April 1, 2015, 4:30 - 6:30 PM EDT

A New Approach to Arakelov Geometry

Abstract: We will introduce descent techniques and relate these notions to classical Arakelov constructions.


Tomorrow

Ph.D. Thesis Defense, Luis Acuna Valverde, BRNG 1254

Thursday, April 2, 2015, 1:30 - 3:00 PM EDT

HEAT KERNELS, SCHRODINGER OPERATORS AND SMALL TIME ASYMPTOTICS Committee: Banuelos (Chair), Sa Barreto, B.Davis, Baudoin

Automorphic Forms and Representation Theory Seminar, Professor Martin Luu, University of Illinois at Urbana-Champaign, UNIV 217

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

Numerical Local Langlands Duality and Weil's Rosetta Stone

ABSTRACT: The analogy between number fields, curves over finite fields, and Riemann surfaces has a long and fruitful history, in particular with respect to the various Langlands dualities. More recently, through the work of Kapustin and Witten, quantum physics has been added as a fourth pillar to the story. In this talk I will describe some local aspects of these analogies. In particular, I will discuss the answer to the question of what the analogue of the numerical local Langlands duality should be in the geometric and quantum setting.

Operator Algebras Seminar, Professor Bhishan Jacelon, Purdue University, MATH 731

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

Nuclear Dimension of Amenable Group C*-algebras

PDE Seminar, Prof. Yuanzhen Shao, Vanderbilt University, BRNG B261

Thursday, April 2, 2015, 3:30 - 4:30 PM EDT

Degenerate and Singular Elliptic Operators on Manifolds with Singularities

Abstract: In this talk, we will introduce the concept of manifolds with singularities and study a class of elliptic differential operators that exhibit degenerate or singular behavior near the singularities. Based on this theory, we investigate several linear and nonlinear parabolic equations arising from geometric analysis and PDE. Emphasis will be given to geometric flows with “bad” initial metrics.

Probability Seminar, Shuwen Lou, University of Illinois at Chicago, REC 308

Thursday, April 2, 2015, 3:30 - 4:30 PM EDT

Fractal properties of rough differential equations driven by fractional Brownian

Abstract: We will introduce fractal properties of rough differential equations driven by frational Brownian motion with Hurst parameter H>1/4. We will first survey some known results on density and tail estimates of such processes. Then we will show the Hausdorff dimension of the sample paths is equal to min(d,1/H), where d is the dimension of the process. Also we will show that with positive probability, the level sets in the form of ${t:Xt=x}$ has Hausdorff dimension 1−dH when dH<1, and are almost surely empty otherwise.


Friday

Secret Seminar, Ryan Spitler, Purdue University, MATH 731

Friday, April 3, 2015, 11:30 - 1:30 PM EDT

Expander Graphs and Property T

Abstract: I will discuss various beautiful connections between spectral graph theory, classical spectral theory, and representation theory.


Next Week

Computational & Applied Mathematics Seminar, Professor Zhiliang Xu, University of Notre Dame, REC 108

Monday, April 6, 2015, 3:30 - 4:30 PM EDT

A RKDG Method with Conservation Constraint to Improve CFL Condition for Solving Conservation Laws and an Energetic Variational Approach to Model Biofilm

Abstract: In this talk, two different works will be discussed. The first work is to develop a new formulation of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving conservation laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. From both numerical experiments and the analytic estimate of CFL number, we find that: 1)this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2)the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method. The second work is to develop a biofilm model which is based on first and second laws of thermodynamics and is derived using energetic variational approach and phase-field coupling. Newly developed unconditionally energy-stable numerical splitting scheme is implemented for computing numerical solution of the model efficiently. Model simulations predict biofilm cohesive failure and show that higher EPS elasticity yields formation of streamers with complex geometries that are more prone to detachment. These results are consistent with experiments.

Department of Mathematics Colloquium, Dr. Andrew Pollington, National Science Foundation, MATH 175

Tuesday, April 7, 2015, 4:30 - 5:30 PM EDT

Diophantine Approximation, A Conjecture of Wolfgang Schmidt, Schmidt Games and Irregularities of Distribution

Abstract: We describe Littlewood's conjecture in Diophantine approximation and some related questions and also present some connections to work on irregularities of distribution. Some of the work I will describe is Joint with Dmitry Bhadzian, Sanju Velani and some with William Moran. Refreshments will be served in the Math Library Lounge at 4:00 p.m.

Ph.D. Thesis Defense, Byeongho Lee, UNIV 019

Thursday, April 9, 2015, 3:00 - 4:30 PM EDT

$G$-Frobenius Manifolds Committee: R. Kauffman (Chair), McClure, Yeung, Albers

Probability Seminar, Rohini Kumar, Wayne State University, REC 308

Thursday, April 9, 2015, 3:30 - 4:30 PM EDT

TBA

Ph.D. Thesis Defense, Andrew Homan, REC 309

Friday, April 10, 2015, 3:00 - 4:30 PM EDT

Applications of Microlocal Analysis to Some Hyperbolic Inverse Problems Committee: Stefanov (Chair), Sa Barreto, Datchev, and Li


Two Weeks

Ph.D. Thesis Defense, Christina Alvey, BRNG B206

Monday, April 13, 2015, 9:30 - 11:00 AM EDT

Investigating Synergy: Mathematical Models for the Coupled Dynamics of HIV and HSV-2 and Other Endemic Diseases Committee: Feng (Chair), Buzzard, Yip, and John Glasser.

Computational & Applied Mathematics Seminar, Dr. John Glasser, CDC, REC 108

Monday, April 13, 2015, 3:30 - 4:30 PM EDT

TBA

Modeling the Impact of Vaccination on the Epidemiology of Pertussis to Deduce the Optimal Number and Timing of Booster Doses in Sweden

Bio sketch: John Glasser studied biology, population biology, and epidemiology and biostatistics at Princeton, Duke and Harvard Universities, served as an EIS officer in the Division of Reproductive Health, and returned to Harvard to study mathematical biology with Richard Levins. Afterwards, he joined the Immunization Division, which became the National Immunization Program (NIP) and ultimately part of the National Center for Immunization and Respiratory Diseases. The project about which he’ll speak today began before the NIP was assimilated into the NCIRD with its separate divisions of bacterial and viral diseases. Abstract: In Sweden, pertussis vaccination began in 1953, but was discontinued 1979-95. This 17-year hiatus, together with a longstanding tradition of reporting to national registries and recently enhanced surveillance, enabled us to evaluate one explanation for the changing epidemiology of pertussis throughout the developed world. We modeled the hypothesis that immunity wanes, but can be boosted by exposure to infectious individuals. As vaccination controls disease among children, however, opportunities for boosting decrease. Because residual immunity moderates disease, symptoms increase with time since disease or vaccination. With some parameters estimated independently and others by fitting to laboratory-confirmed case reports from 1987-96, we compared predictions to observations from 1997-2013. Then we simulated transmission of B. pertussis until 2029 with infant vaccinations alone and together with preschool or also adolescent boosters. Allowing for considerable age-specific under-reporting, especially of immunity-modified disease, model predictions in Stockholm, Skåne, and Västra Götaland are concordant with observations. Vaccination decreases typical disease among young children and increases immunity-modified disease among older ones. Revaccination should damp multi-annual cycles and may reduce susceptibility among children and young adults. Our modeling supports Swedish health authorities’ decision to introduce a preschool booster in 2007 and a school-leaving dose in 2016. While vaccination has substantially reduced morbidity among children, pertussis in infants may still be life-threatening. As parents with immunity-modified disease are their main source of infection, we recommend continued serological surveillance. Meanwhile, vaccination of pregnant women, neonates and adults caring for young infants warrants consideration.

Ph.D. Thesis Defense, Jieun Kim, EE 236

Tuesday, April 14, 2015, 2:00 - 3:30 PM EDT

Mathematical approaches to food nutrient content estimation with a focus on phenylalanine Committee: Boutin(Chair), Buzzard, Yip, and C. Boushey

Department of Mathematics Colloquium, Feng Luo, Rutgers, MATH 175

Tuesday, April 14, 2015, 4:30 - 5:30 PM EDT

TBA

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

PhD. Thesis Defense, Peter Weigel, MATH 731

Thursday, April 16, 2015, 10:30 - 12:00 PM EDT

Orderability and rigidity in contact geometry Committee: Lempert (Co-Chair), P. Albers (Co-Chair), Bell, and R. Kaufmann


Three Weeks

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.

Department of Mathematics Colloquium, 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.

Ph.D. Thesis Defense, Britian Cox, BRNG B212

Wednesday, April 22, 2015, 10:30 - 12:00 PM EDT

Supercuspidal Representations arising from stable vectors. Committee: Shahidi and J.K. Yu (co-chairs), Goldberg, and Liu

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.

Ph.D. 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.

Computational & Applied Mathematics Seminar, Dr. Andrew Hill, Emory University, CDC, REC 108

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

TBA


April

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.


May

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

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

TBA


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

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