Current Seminars
updated 10/10/ 2001

 

As of October 10-12

 

Statistical Mechanics Seminar

Topic:        Stability of the Relativistic Electron-Positron Field in Hartree-Fock Approximation in the Presence of a Strong Nuclear

                  Charge

Presenter:   Heinz Siedentop, University of Munich

Date:          Wednesday, October 10, 2001, Time: 2 p.m., Location: Jadwin 343

Abstract:    A nucleus can create electron-positron pairs. It is important to know that this process does not cause the energy of the electron-positron field to become unbounded from below. In the case of Hartree-Fock states boundedness holds and, mathematically, is based on a Sobolev type inequality, namely that the modulus of the free Dirac operator is an upper bound on the modulus of the atomic Dirac operator multiplied by some positive constant less than one. In the talk we will discuss the electron-positron field in Hartree-Fock approximation, we will relate this stability to the above inequality, and prove it.

 

Department Colloquium
Topic:         Algorithms for quantum computers

Presenter:   Peter Shor, AT & T
Date:          Wednesday, October 10, 2001, Time: 4:30 p.m., Location: Fine Hall 314

Abstract:    Quantum computers are hypothetical devices which use the principles of quantum mechanics to perform computations. For some difficult computational problems, including the cryptographically important problems of prime factorization and finding discrete logarithms, the best algorithms known for classical computers are exponentially slower than the algorithms known for quantum computers. Although they have not yet been built, quantum computers do not appear to violate any fundamental principles of physics. I will give a mathematical model of quantum computation, explain how quantum mechanics provides this extra computational power, and briefly describe several fundamental algorithms in quantum computation, including the algorithm for efficient prime factorization. 

 

Special PACM Colloquium  ***Note special date

Topic:        Distinguished Limits, Multiple Scales, and Asymptotically Adaptive Numerics for Atmospheric Flows 

Presenter:   Rupert Klein, Konrad-Zuse-Zentrum fur informationstechnik Berlin

Date:          Wednesday, October 10, 2001, Time: 4:30 p.m., Location: Fine Hall 224

Abstract:    Atmospheric flows are characterized by a considerable variety of singular asymptotic limit regimes. This is evident from the large collection of simplified model equations to be found in contemporary textbooks on theoretical meteorology.  The presence of such singular flow regimes induces multiple length and time scale solutions which call for adaptive grids and time steps.  However, these regimes also involve dominant balances of selected terms within the governing equations.  The associated numerical dynamic range problems lead to numerical stiffnesses and to amplifications of numerical truncation errors.  This presentation will describe a generalized asymptotic framework for analysing these effects and will discuss first examples of numerical methods that overcome the numerical difficulties through "asymptotic adaptivity".

 

Discrete Mathematics Seminar

Topic:        Explicit Unique-Neighbor Expanders 

Presenter:   Michael Capalbo, Institute for Advanced Study

Date:          Thursday, October 11, 2001, Time: 4:00 p.m., Location: Fine Hall 224

 

Topology  Seminar

Topic:        Braids and symplectic four-manifolds with abelian fundamental group

Presenter:   Paul Seidel, Institute for Advanced Study

Date:          Thursday, October 11, 2001, Time: 4:00 p.m., Location: Fine Hall 314

 

Geometric Analysis Seminar

Topic:        Boundary regularity and structure for Poincare-Einstein metrics

Presenter:   Rafe Mazzeo, Stanford University

Date:          Friday, October 12, 2001, Time; 3:00 p.m., Location: Fine Hall 314

 

Week of October 15-19

 

PACM Colloquium

Topic:         Mathematical and computational modeling of the martensitic phase transformation 

Presenter:    Mitchell Luskin, University of Minnesota

Date:           Monday, October 15, 2001, Time: 4:00 p.m., Location: Fine Hall 214

Abstract:    We present a mathematical model and computational results for the martensitic phase transformation of a thin film as the film is cyclically heated and cooled. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a Monte Carlo method to nucleate the phase transformation since the film would otherwise remain in metastable local minima of the energy.

 

Algebraic Geometry

Topic:        Higgs bundles and mirror symmetry

Presenter:   M. Thaddeus, Columbia University and IAS

Date:          Tuesday, October 16, 2001, Time: 4:30 p.m., Location: Fine Hall 314

Abstract:    We conjecture that the moduli spaces of Higgs bundles studied by Hitchin and Simpson satisfy the requirements to be mirror partners in the sense of Strominger-Yau-Zaslow. More precisely, the moduli space with structure group G is foliated by special Lagrangian tori, and carries a flat gerbe B whose equivalence classes of trivializations on the tori can be identified with the moduli space having the dual structure group. We have verified this for G = SL(n).  Moreover, the mirror relationship leads us to suspect a relationship between the Hodge numbers of these spaces, which we have verified for G = SL(2) and SL(3).  This is joint work with Tamas Hausel. 

 

Statistical Mechanics Seminar

Topic:        Nonconvergent pertubative expansions for unstable invariant tori in Hamiltonian mechanics

Presenter:   Giovanni Gallavotti, University of Rome

Date:          Wednesday, October 17, 2001, Time: 2 p.m., Location: Jadwin 343

Abstract:    We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter.  We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential.  They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order $N$ is bounded by the $(N+1)$-st power of the argument times a power of $N!$. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory.

 

Department Colloquium

Topic:         Hypersurfaces of prescribed curvature and energy inequalities in General Relativity

Presenter:   Gerhard Huisken, Universität Tübingen

Date:          Wednesday, October 17, 2001, Time: 4:30 p.m., Location: Fine Hall 314

 

Ergodic Theory and Statistical Mechanics Seminar

Topic:        Sample path properties of the stochastic flow and the dynamics of an oil spill (joint work with Dima Dolgopyat and

                 Vadim Kaloshin)

Presenter:   Leonid Koralov, Princeton University

Date:          Thursday, October 18, 2001, Time: 2:00 p.m., Location: Fine 224

Abstract:    We consider a stochastic flow driven by a finite dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multi point motion. An application of these results allows us to describe the limiting shape of a domain in $\R^2$ (say, an oil spill on the surface of the ocean) carried by the stochastic flow.

 

Topology  Seminar

Topic:        An invariant of homology cobordism

Presenter:   Nikolai Saveliev, University of Miami and Princeton University

Date:          Thursday, October 18, 2001, Time: 4:00 p.m., Location: Fine Hall 314

 

Geometric Analysis Seminar

Topic:        Minimizing Oseen-Frank energy: a geometric approach

Presenter:   M. Giaquinta, Scuola Normale Superiore

Date:          Friday, October 19, 2001, Time: 3:00 p.m., Location: Fine Hall 314

 

Week of October 22-26

 

PACM Colloquium

Topic:         Wavelet Methods for Medical Tomography  

Presenter:    Bradley Lucier, Purdue University 

Date:           Monday, October 22, 2001, Time: 4:00 p.m., Location: Fine Hall 214

Abstract:    The mathematics of Computed Tomography (CT) and Positron Emission Tomography (PET) medical imaging is based on inverting the Radon transform. The Radon transform is a linear, smoothing operator, so its inverse, while linear, is unbounded, and the presence of noise (especially for PET imaging) makes applying this inverse problematic. David Donoho introduced Wavelet Shrinkage applied to Wavelet-Vaguelette decompositions to solve this problem. This talk describes how Donoho's method can be cast in a variational framework, how to choose the scaling of shrinkage parameters, and gives experimental results that compare our method with the so-far standard method, Filtered Back Projection.

 

Algebraic Geometry

Topic:         Quantum cohomology on flag manifolds, finite difference Toda lattices, and quantum groups 

Presenter:   Y.P. Lee, UCLA

Date:          Tuesday, October 23, 2001, Time: 4:30 p.m., Location: Fine Hall 314

Abstract:    A "completely integrable" system, the finite difference Toda lattices, is constructed from quantum groups $U_q(g)$ for any complex simple Lie algebras $g$ by defining a homomorphism from the center of $U_q(g)$ to finte difference operators. The image consists of the commuting hamiltonians of the finite difference Todalattices. (This part will only be briefly sketched.).  We will prove that a generating function of one-point quantum $K$-invariants, the $J$-function, on (complete) flag manifolds of type $A_r$ is the common eigenfunction of the commutating hamiltonians. We also conjecture that this statement holds for arbitrary simple Lie algebra.  This is based on the joint work with A. Givental in math.AG/0108105.  If time permits, I will also discuss some problems in quantum $K$-theory in general, and how this result could be related to the quantum $K$-ring by a "Floer $K$-theory"

 

Department Colloquium

Topic:        TBA

Presenter:  Tobias Colding, Princeton University
Date:         Wednesday, October 24, 2001, Time: 4:30 p.m., Location: Fine Hall 314

 

Topology  Seminar

Topic:        TBA

Presenter:   Thomas Parker, Institute for Advanced Study

Date:          Thursday, October 25, 2001, Time: 4:00 p.m., Location: Fine Hall 314

 

Geometric Analysis Seminar

Topic:        TBA

Presenter:   Tobias Colding, Princeton University

Date:          Friday, October 26, 2001, Time: 3:00 p.m., Location: Fine Hall 314

 

Week of November 5-9 

 

PACM Colloquium

Topic:         Complex fluids: liquid crystals, mixtures and polymeric materials  

Presenter:    Chun Liu, Penn State University 

Date:           Monday, November 5, 2001, Time: 4:00 p.m., Location: Fine Hall 214

Abstract:    In this talk, several dynamical systems modeling specific types of complex fluids are introduced. The relation between these and other existing models will be discussed. We will also study the relations between the variational procedure; the basic energy law; stability; and the higher order energy estimates. The different non-Newtonian properties such systems exhibit is of particular interest. Finally we will study a differential-integral equation system that allows us to consider couplings and interactions of different spatial length scales.

 

Algebraic Geometry

Topic:         p-adic representations and differential equations 

Presenter:   L. Berger, Brandeis University

Date:          Tuesday, November 6, 2001, Time: 4:30 p.m., Location: Fine Hall 314

Abstract:    I will explain how to associate objects of a differential nature to a p-adic representation. Using recent results of Andr\'e on the structure of p-adic differential equations, these constructions allow us to give a proof of Fontaine's monodromy conjecture: every "de Rham" p-adic representation is potentially semi-stable. 

 

Department Colloquium

Topic:        TBA

Presenter:   Bjorn Engquist, Princeton University

Date:         Wednesday, November 7, 2001, Time: 4:30 p.m., Location: Fine Hall 314

 

Geometric Analysis Seminar

Topic:        Singular Yamabe metrics, explosion for superprocess, and thinness

Presenter:   D. Latutin, ETH

Topic:        TBA

Presenter:   Tobias Colding, Princeton University

Date:          Friday, November 9, 2001, Time: 3:00 p.m., Location: Fine Hall 314

 

Week of November 12-16

 

Algebraic Geometry

Topic:        TBA 

Presenter:  V. Vatsal, University of British Columbia

Date:         Tuesday, November 13, 2001, Time: 4:30 p.m., Location: Fine Hall 314

 

Geometric Analysis Seminar

Topic:         TBA

Presenter:    C. L. Terng, Northeastern University

Date:           Friday, November 16, 2001, Time: 3:00 p.m., Location: Fine Hall 314

 

Week of November 19-23

 

Algebraic Geometry

Topic:        TBA 

Presenter:  B. de Oliveira, University of Pennsylvania

Date:         Tuesday, November 20, 2001, Time: 4:30 p.m., Location: Fine Hall 314

 

Week of November 26-30

 

Geometric Analysis Seminar

Topic:        TBA

Presenter:   Panagiota Daskalopoulos, Columbia University

Date:          Friday, November 30, 2001, Time: 3:00 p.m., Location: Fine Hall 314

 

Week of December  3-7

 

PACM Colloquium

Topic:         Lattice Boltzmann Method for fluid flows

Presenter:    Shiyi Chen, Johns Hopkins University 

Date:           Monday, December 3, 2001, Time: 4:00 p.m., Location: Fine Hall 214

 

Algebraic Geometry

Topic:        TBA 

Presenter:  T. Pantev, University of Pennsylvania

Date:         Tuesday, December 4, 2001, Time: 4:30 p.m., Location: Fine Hall 314