Current Seminars
updated 10/03/ 2001
As of October 3-5
Statistical Mechanics Seminar
Topic: Stability of Matter in a non-perturbative, Relativistic Model of Quantum Electrodynamics
Presenter: Elliott Lieb, Princeton University
Date: Wednesday, October 3, 2001, Time: 2 p.m., Location: Jadwin 343
Abstract: The relativistic "no pair" model of quantum electrodynamics uses the Dirac operator, D(A) for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A. There are arbitrarily many electrons and fixed nuclei with charges Z. This many-body system is shown to have finite ground state energy for suitable alpha and Z if and only if one uses the positive spectral subspace of the full Dirac operator D(A) to define an electron and not the free Dirac operator D(0). This formulation of QED is somewhat unusual because it means that the electron Hilbert space is inextricably linked to the photon Fock space. But such a linkage appears to better describe the real world of photons and electrons.
Department Colloquium
Topic: Geometric and Topological Rigidity for Hyperbolic 3-Manifolds
Presenter: David Gabai, Princeton University
Date: Wednesday, October 3, 2001, Time: 4:30 p.m., Location: Fine Hall 314
Discrete Mathematics
Topic: List coloring of graphs with high chromatic number
Presenter: Benny Sudakov, Princeton University and the Institute for Advanced Study
Date: Thursday, October 4, 2001, Time: 4:00 p.m., Location: Fine Hall 224
Topology Seminar
Topic: Some three-dimensional applications of HF^+
Presenter: Peter Ozsvath, Princeton University
Date: Thursday, October 4, 2001, Time: 4:00 p.m., Location: Fine Hall 314
Princeton/IAS Number Theory Seminar ***Note room change
Topic: Hecke operators and equi-distribution of integer points on a family of homogeneous varieties.
Presenter: Hee Oh, Princeton University
Date: Thursday, October 4, 2001, Time: 4:30 p.m., Location: Fine Hall 401
Abstract: Let f be a homogeneous polynomial with integer coefficients, and let V_m be the variety defined by f=m. In the early sixties, Linnik raised the problem of understanding the distribution of the integer points V_m(Z) as m tends to infinity. In complete generality it seems hopeless to attack this question, except when the number of variables of f is much bigger than the degree of f in which case the Hardy-Littlewood method can be applied. In this talk we discuss Linnik's problem when f arises from invariant theory, explaining how the Hecke operators then play a role here. This is a joint work with W. T. Gan.
Geometry Analysis Seminar
Topic: Manifolds with positive curbature almost everywhere
Presenter: B. Wilking, University of Pennsylvania
Date: Friday, October 5, 2001, Time: 3:00 p.m., Location: Fine Hall 314
Week of October 8-12
Analysis Seminar
Topic: TBA
Presenter: Kenji Nakanishi, Nagoya University and Princeton University
Date: Monday, October 8, 2001, Time: 4:00 p.m.., Location: Fine Hall 314
PACM Colloquium
Topic: Disclinated states in nematic elastomers
Presenter: Eliot Fried, University of Illinois at Urbana-Champaign
Date: Monday, October 8, 2001, Time: 4:00 p.m., Location: Fine Hall 214
Abstract: We present a theory for uniaxial nematic-elastomers with variable asphericity. As an application of the theory, we consider the time-independent, isochoric extension of a right circular cylinder. Numerical solutions to the resulting differential equation are obtained for a range of extensions. For sufficiently large extensions, there exists an isotropic core of material surrounding the cylinder axis where the asphericity vanishes and in which the polymeric molecules are shaped as spherical coils. This region, corresponding to a disclination of strength $+1$ manifesting itself along the axis, is bounded by a narrow transition layer across which the asphericity drops rapidly and attains a non-trivial negative value. The material thereby becomes anisotropic away from the disclination so that the polymeric molecules are shaped as ellipsoidal coils of revolution oblate about the cylinder radius. In accordance with the area of steeply changing asphericity between isotropic and anisotropic regimes, a marked drop in the energy density is observed. The boundary of the disclination core is associated with the location of this energy drop. For realistic choices of material parameters, this criterion yields a core on the order of $10^{-2}$ microns, which is consistent with observations in conventional liquid-crystal melts. Also occurring at the core boundary, and further confirming its location, are sharp transitions in the behavior of the constitutively determined contribution to the deformational stress and a minimum in the pressure. Furthermore, the constitutively determined contribution to the orientational stress is completely concentrated at the core boundary.
Algebraic Geometry Seminar
Topic: Rational connectedness and rational points over function fields
Presenter: T. Graber, Harvard University
Date: Tuesday, October 9, 2001, Time: 4:30 p.m., Location: Fine Hall 314
Abstract: I will discuss two results. The first states that a variety over the function field of a curve which is geometrically rationally connected always has a rational point. The second result is a sort of converse to the first. It gives a necessary and sufficient condition, in terms of rationally connected subvarieties, for a family of varieties parametrized by a base B to have a section when restricted to an arbitrary curve in B. This is joint work with J. Harris, B. Mazur, and J. Starr.
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.
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
Geometry 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
Geometry Analysis Seminar
Topic: Minimizing Oseen-Frank energy: a geometric approach
Presenter: M. Giaquinta
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
Geometry 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 October 29-November 2
PACM Colloquium
Topic: TBA
Presenter: Yves Meyer
Date: Monday, October 29, 2001, Time: 4:00 p.m., Location: Fine Hall 214
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
Geometry 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
Geometry 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
Geometry Analysis Seminar
Topic: TBA
Presenter: D. Panagiota, Columbia University
Date: Friday, November 30, 2001, Time: 3:00 p.m., Location: Fine Hall 314
Week of December 3-7
Algebraic Geometry
Topic: TBA
Presenter: T. Pantev, University of Pennsylvania
Date: Tuesday, December 4, 2001, Time: 4:30 p.m., Location: Fine Hall 314