Mathematics Department - Colloquium - Spring 2012

Colloquium - Spring 2012



Mathematics Department Colloquia take place on Friday afternoons from 4:00-5:00PM in the Hill Center, Room 705, on Busch Campus. Also, due to recent construction on Route 18, most on-line maps and driving instructions are out of date. Here are updated driving directions. If you need information on public transportation , you may want to check the New Jersey Transit page for information on fares and schedules for the Northeast Corridor Line. Taxis are available at the New Brunswick train station (fare about $7) and can take you to and from the Hill Center (Victory Cabs, (732) 545-6666). The Rutgers Campus Bus System provides free inter-campus transportation, with the A and H buses taking passengers between Busch Campus and College Avenue, with the A providing a faster ride from College Avenue and the H providing a faster ride from the Hill Center : please visit their website for bus schedules and maps, including real-time tracking of campus buses.

Unfortunately, colloquium cancellations do occur from time to time. Please feel free to call our department (732)-445-3921 before embarking on your journey.

Colloquium participants and hosts may wish to also consult the Rutgers University academic calendar, as well as its calendars of religious holidays and of weather emergencies and university closings.

Organizer(s)

Lisa Carbone, Konstantin Mischaikow, Anders Buch

Archive




Past Talks


Friday, May 4th

Richard Rimanyi, University of North Carolina at Chapel Hill (NOTE NEW ROOM!!)

"Global singularity theory"

Time: 4:00 PM
Location: DIMACS #431
Abstract: The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3-space must have double points. A map from the projective plane to the plane must have an odd number of cusp points. To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In this lecture, we will explore the theory of Thom polynomials and their applications to enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the theory of diagrams of linear maps (quivers).


Friday, April 27th

Lizhen Ji, University of Michigan

"Geometry and analysis of moduli spaces of Riemann surfaces"

Time: 4:00 PM
Location: Hill 705
Abstract: Moduli spaces of Riemann surfaces are fundamental objects of mathematics and have been intensively and extensively studied since Riemann. A lot of work has been done on the algebraic geometry and algebraic topology aspects of the moduli space. In this talk, I will describe some problems and results on the geometry and analysis of the moduli spaces.


Friday, April 20th

Luis Silvestre, University of Chicago

"Partial regularity for fully nonlinear elliptic PDE"

Time: 4:00 PM
Location: Hill 705
Abstract: We will discuss regularity issues for fully nonlinear elliptic equations of second order. We prove that solutions to a fully nonlinear elliptic equation F(D^2u)=0 are classical outside a set of dimension at most n-epsilon, where n is the dimension and epsilon is a small constant depending on the ellipticity bounds of F and dimension. We do not make any convexity assumption on the equation, but we assume that F is differentiable. This is a joint work with Scott Armstrong and Charles Smart.


Friday, April 6th

John McCarthy, Washington University in St Louis

"Operator Monotone Functions of Several Variables "

Time: 4:00 PM
Location: Hill 705
Abstract: Self-adjoint n-by-n matrices have a natural partial ordering, namely A is less than or equal to B if the matrix B - A is positive semi-definite. In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called n-matrix monotone. The condition depends on the dimension n, but if a function is n-matrix monotone for all n, then it must extend analytically to a function that maps the upper half-plane to itself. I will describe Loewner's results, and then discuss what happens if one wants to characterize functions f of two (or more) variables that are matrix monotone in the following sense:

If A = (A_1, A_2) and B = (B_1,B_2) are pairs of commuting self-adjoint n-by-n matrices, with A_1 <= B_1 and A_2 <= B_2, then f(A) <= f (B).


Friday, March 30th

Gang Tian, Princeton University

"Structures of almost Einstein manifolds"

Time: 4:00 PM
Location: Hill 705
Abstract: Almost Einstein manifolds are generalizations of Einstein manifolds. They appear naturally in the regularity theory of the elliptic Einstein equation. Roughly speaking, they satisfy the Einstein equation in a suitable $L^1$-sense. I will show some recent results on the structure of such manifolds. I will also show some applications to the Kahler geometry. This is a joint work with B. Wang.


Friday, March 23rd

William Minicozzi, Johns Hopkins University

"Singularities and dynamics of mean curvature flow"

Time: 4:00 PM
Location: Hill 705
Abstract: I will give a brief introduction to mean curvature flow (MCF) of hypersurfaces and survey recent progress with Toby Colding on the dynamics of mean curvature flow near a singularity. MCF is a nonlinear heat equation where the hypersurface evolves to minimize its surface area and the major problem is to understand the possible singularities of the flow and the behavior of the flow near a singularity.


Friday, March 9th

Laszlo Lovasz , Eotvos Lorand University and IAS

"Graph limits and their applications"

Time: 4:00 PM
Location: Hill 705
Abstract: We introduce and motivate the notions of convergent graph sequences and graph limits. The most important applications of these constructions are extremal graph theory and the theory of graph property testing. We are going to show how analytic techniques allow us to pose and in some cases answer general questions about graphs: which inequalities between subgraph densities are valid, what is the possible structure of extremal graphs, which graph properties are testable, and which of them are testable in a nondeterministic sense.


Friday, March 2nd

Fang-Hua Lin, Courant Institute

"Elliptic Equations with Periodic Coefficients and Homogenization"

Time: 4:00 PM
Location: Hill 705
Abstract: First I shall review some earlier results concerning elliptic equations with periodic coefficients. Most of them were motivated by the theory of homogenization though many results are of independent interest. Then I shall discuss various uniform estimates for the Dirichlet, Neumann problems. These estimates can be applied to solve convergence rates problems in homogenization and many other problems. All the recent results are obtained in joint works with C. Kenig and Z.W.Shen.


Friday, February 24th

Jinchao Xu, Center for Computational Mathematics and Applications, Penn State University

"Optimal and Practical Algebraic Solvers for Discretized PDEs"

Time: 4:00 PM
Location: Hill 705
Abstract: An overview of fast solution techniques (such as multi-grid, two-grid, one-grid and nil-grid methods) will be given in this talk on solving large scale systems of equations that arise from the discretization of partial differential equations (such as Poisson, elasticity, Stokes, Navier-Stokes, Maxwell, MHD, and black-oil models). Mathematical optimality, practical applicability and parallel (CPU/GPU) scalability will be addressed for these algorithms and applications.


Friday, February 17th

Sun-Yung Alice Chang, Princeton University

"Conformal invariants: perspectives from geometric PDE (D'atri Lecture)"

Time: 4:00 PM
Location: Hill 705
Abstract: We will survey properties of a class of integral conformal invariants in conformal geometry and their connection to geometric quantities on conformally compact Einstein manifolds in ADS/CFT setting. Special emphasis will be on the role played by non-linear elliptic PDE.


Friday, February 10th

Olga Kharlampovich, Hunter College

"First order properties and algebraic geometry in groups in the presence of negative curvature "

Time: 4:00 PM
Location: Hill 705
Abstract: I will do a survey of the subject.


Friday, February 3rd

Mina Teicher, Bar Ilan University

"Braid group techniques in Algebraic geometry"

Time: 4:00 PM
Location: Hill 705
Abstract: We will give an over view of the techniques related to the braid group in topology of algebraic varieties and the connections to the open questions on the braid group.


Friday, January 27th

Michael L. Overton, Courant Institute of Mathematical Sciences, NYU

"Optimization of Polynomial Roots, Eigenvalues and Pseudospectra"

Time: 4:00 PM
Location: Hill 705
Abstract: The root radius and root abscissa of a monic polynomial are respectively the maximum modulus and the maximum real part of its roots; both these functions are nonconvex and are non-Lipschitz near polynomials with multiple roots. We begin the talk by giving constructive methods for efficiently minimizing these nonconvex functions in the case that there is just one affine constraint on the polynomial's coefficients.

We then turn to the spectral radius and spectral abscissa functions of a matrix, which are analogously defined in terms of eigenvalues. We explain how to use nonsmooth optimization methods to find local minimizers and how to use nonsmooth analysis to study local optimality conditions for these nonconvex, non-Lipschitz functions.

Finally, the pseudospectral radius and abscissa of a matrix $A$ are respectively the maximum modulus or maximum real part of elements of its pseudospectrum (the union of eigenvalues of all matrices within a specified distance of $A$). These functions are also nonconvex but, it turns out, locally Lipschitz, although the pseudospectrum itself is not a Lipschitz set-valued map.

We discuss applications from control and from Markov chain Monte Carlo as examples throughout the talk. Coauthors of relevant papers include Vincent Blondel, Jim Burke, Kranthi Gade, Mert Gurbuzbalaban, Adrian Lewis and Alexandre Megretski.


Friday, January 20th

Manish Patnaik, Yale University

"Automorphic Forms on Loop Groups"

Time: 4:00 PM
Location: Hill 705
Abstract: I will survey some recent advances in the theory of automorphic forms on certain infinite dimensional loop groups (also known as affine Kac-Moody groups). First, I will describe some local constructions of convolution Hecke algebras on these groups, and explain their connection with the Double Affine Hecke Algebras. Then I will describe the global theory of Eisenstein series for these groups, and explain how they can be used to study certain questions arising from the usual finite-dimensional theory of automorphic forms. Finally, I will describe a geometric analogue of the above constructions involving the certain moduli spaces of bundles on an algebraic surface.

The work is joint in parts with A. Braverman, H. Garland, and D. Kazhdan.


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