Stanford University Geometric Analysis Seminar 1999-2000
Unless otherwise noted, all seminars begin at 4:00 pm in room 380-381T,
and last approximately one hour. There is tea at 3:30 in the
lounge area on the third floor of the Math Department Building (Building 380).
Spring 2000 Schedule
March 29 Tobias Ekholm, Stanford University
Title: Total curvatures of holonomic links
Abstract: A closed holonomic space curve is curve
of the form $(f,f',f'')$, where
$f$ is a function on the circle. Vassiliev introduced holonomic curves
in 1997 and proved that any link type has a holonomic representative.
Borrowing an idea from classical works of Milnor, we study the infima of
total curvature and of the sum of total curvature and total absoulte
torsion over all holonomic representatives of a given (topological)
link type. The later infimum equals $2\pi$ times the braid index
(minimal number of strands in a closed braid representative) of
the link type and thus gives a differential geometric characterization of
this combinatorially defined invariant. The former
infimum can be estimated in terms of the difference between the
braid-index and the bridge-index (the infimum of total
curvature over all representatives) of the link.
April 5 Jingyi Chen, University of British Columbia, Vancouver,
Canada
Title: Holomorphic connections and lagrangian cycles
Dinner: We will be taking the speaker out to dinner at a
nearby restaurant after the seminar.
April 12 Dan Ostrov, Santa Clara University
Title: Determining shapes from discontinuous shading data and
solving related first order PDEs where the flux function is discontinuous
Abstract: Shape from shading is the study of how to determine a
3-D surface from a 2-D picture of the surface (plus as minimal an amount
of additional information as possible.) When the picture has
discontinuities (i.e., a bright part of the picture borders a darker
part), difficulties arise in determining existence, uniqueness, and a
method of computation for the solution of the underlying PDE describing
the surface. We will explain a method of resolving these questions
involving a control theory representation for the PDE, which will also
allow us to answer larger questions about much more general first order
PDEs with discontinuous flux/Hamiltonian functions.
Dinner: We will be taking the speaker out to dinner at a
nearby restaurant after the seminar.
April 19 Michael Hutchings, Stanford University
Title: The Double Bubble Conjecture
Abstract: The double bubble conjecture states that the
least-area way
to enclose and separate 2 given volumes in R^3 is the "standard double
bubble", consisting of three spherical caps meeting at 120 degree angles,
and easily produced using actual soap. The proof of this conjecture has
recently been completed, combining the work of a number of people. We
will explain the basic tools used in the proof. Many more general
questions remain open.
More information is available at
Williams
College and in
Science magazine.
April 26 Ben Andrews, Australian National University, Canberra
Title: Evolution of surfaces in space forms
Abstract: I will discuss some new work aimed at developing
geometric heat equations as tools which can be applied to prove theorems
in geometry. I will concentrate on problems involving hypersurfaces.
There are many evolution equations available, including the motion
by mean curvature and motion by other functions of curvature.
While the mean curvature flow seems at first sight the most natural,
I will show that for specific purposes it is better (even necesssary!)
to look at other flows. I will discuss examples relating to surfaces
in 3-manifolds, particularly in spheres and hyperbolic spaces.
May 3 Sergiu Klainerman, Princeton University
Title: Vectorfield Methods and Regularity Estimates for
Quasilinear Wave Equations
Location and Time Change: This talk will be 12:15-1:15 pm
in room 380-383N, on the third floor of the Math Building.
May 10 Gigliola Staffilani, Stanford University
Title: Global well-posedness for KdV below L^2 norm
Abstract:A result of Kenig-Pone-Vega states that the KdV
equations on the real
line is locally well-posed for any initial data in the Sobolev space
H^s, s>-3/4 and on the circle for s>-1/2. Bourgain also shows that in
a certain sense this result is sharp. In this talk I will present a
recent result obtained in collaboration with Colliander, Keel,
Takaoka and Tao, in which we prove that in both cases, for the range
of Sobolev exponents mentioned above, the solutions indeed exist for
all times.
May 17 Michael Taylor, University of North Carolina at
Chapel Hill
Title: Fluid flows on rough domains
Special Lecture, Monday, May 22 at 2:30 pm Gang Tian, Massachusetts
Institute of Technology
Location: Room 380-383N
Title: The Ricci Flow in Complex Geometry
May 24 Rafe Mazzeo, Stanford University
Title: Resolvents, Riemannian products and Martin boundaries
Abstract: I will describe some recent work with Andras Vasy
which involves the
beginnings of a program to carry out many of the goals of scattering
theory on symmetric spaces of arbitrary rank and their perturbations.
I will review what is known about the (asymptotically) rank one situation,
and then discuss the particular example of the product of asymptotically
hyperbolic spaces in some detail.
Winter 2000 Schedule
January 12 Jeff Brock, Stanford University
Title: The Weil-Petersson metric and the geometry of hyperbolic
3-manifolds
January 19 Helmut Friedrich, Max Planck Institut, Potsdam, Germany
Title: Asymptotically Simple Spacetimes
January 26 Kai Cieliebak, Stanford University
Title: Symplectic properties of hypersurfaces in Euclidean space
February 2 Mu-Tao Wang, Stanford University
Title: Calibrated geometry and mean curvature flow
February 9 Hyam Rubinstein, University of Melbourne, Australia
and AIM
Title: The shape of 3-manifolds
Dinner: We will have our quarterly Geometric Analysis
Seminar dinner
after this talk. Seminar attendees (this includes faculty,
students, and visitors!) and their companions
are invited to a barbecue at Leon Simon's house 6:30 - 8:30 pm. Leon's
address is 14 Ryan Court, off of Stanford Avenue and near the Nixon
School on the university campus. Phone number: 858-3639. Please spread
the word, and try to attend this festive event! RSVP to Helen Moore at
moore@math.stanford.edu so
we'll
know how much barbecue, salad, and drinks to provide. Also please advise
us of any dietary restrictions.
February 16 Kenji Nakanisi, Kobe University, Japan
Title: Scattering theory for nonlinear Klein-Gordon equation with
Sobolev critical power
February 23 Giulia Furioli, Universite Orsay, France
Title: Besov's regularity persistence for the Koch-Tataru
solutions to Navier-Stokes in R^3
March 1 Robert Bartnik, University of Canberra, Australia
Title: Local existence for the Einstein equations
Abstract: I will review the details of the Choquet-Bruhat
proof of local existence for the Einstein equations.
March 8 Boris Vainberg, University of North Carolina at
Charlotte
Title: Spectrum of Schrodinger operators with sparse or slow
decaying potentials
Abstract: We study wide classes of Schrodinger operators with
reach a.c. spectrum
and, sometimes, dense singular spectrum. For 1-D operators, we prove the
existence of the a.c. spectrum (on the semiaxis) when the potential
satisfies some conditions related to higher order first integrals of the
KdV equation (generalization of results by Deift and Killip on square
summable potentials), and the existence of pure a.c. spectrum when a high
order derivative of the potential is summable (the potential may decay
very slow). We give a description of the a.c. and singular (possibly,
dense) spectrum for multidimensional operators with sparse potentials. One
of the possible applications is multiscattering by a system of distant
obstacles.
March 15 Daniel Weinholtz, Stanford University
Title: Some remarks on closed curves in space
Abstract: Any closed curve in an at least three-dimensional
Euclidean space can be bounded by two parallel hyperplanes, such that
it touches each of them at least twice in an alternating manner.
We will prove this and thus positively decide a conjecture of Kusner and
Sullivan on the minimal diameter of the projection of a closed curve in
space into a hyperplane.
March 22 Spring break
Fall 1999 Schedule
October 27 Martin Chuaqui, Universidad Catolica de Chile
Title: John Domains, Quasidisks and the Nehari Class
November 3 DINNER: We will have a special Geometric Analysis
Seminar dinner
after the talk on November 3. Seminar attendees (this includes faculty,
students, and visitors!) and their companions
are invited to a barbecue at Leon Simon's house 6:00 - 8:00 pm. Leon's
address is 14 Ryan Court, off of Stanford Avenue and near the Nixon
School on the university campus. Phone number: 858-3639. Please spread
the word, and try to attend this festive event! RSVP to Leon at
lms@math.stanford.edu so
he'll
know how much barbecue, salad, and drinks to provide. Also please advise
him of any dietary restrictions.
November 3 Julie Levandosky, Stanford University
Title: Smoothing Properties of Nonlinear Dispersive Equations
Abstract: We will consider the regularity of solutions to certain
nonlinear dispersive equations. In particular, we will discuss the
Korteweg-de Vries (KdV) and Kadomtsev-Petviashvilli (KP) equations,
models for water-wave propagation in shallow water, but we will also
discuss results for a very general class of dispersive equations.
We will show sufficient conditions on equations of this type for which
singularities in the initial data instantly disappear. Specifically,
we will relate the smoothness of the solution with the amount of decay
at spatial infinity of the initial data, showing that for initial data
decaying sufficiently at spatial infinity, the solution will in fact
be C^\infty. Please note the November 3 DINNER announcement above.
November 10 Daniel Weinholtz, Stanford University
Title: A method to exclude boundary branch points of minimal
surfaces
Abstract: By looking at the third derivatives of Dirichlet's
energy into suitable directions one finds criteria on the asymptotic
behaviour of a minimal surface spanned by a sufficiently smooth
boundary curve which prohibit the surface to be a local minimizer of
Dirichlet's energy (and hence area).
November 17 Marc Soret, Univ. de Tours, France/Johns Hopkins
University
Title: Recurrence of surfaces in Euclidean space
Abstract: Consider a surface M of $R^3$ that is topogically
the punctured sphere. If a tubular neighbourhood of M, with fixed
positive radius, is embedded, then M is conformally the punctured
sphere. More generally we give extrinsic criteria for surfaces in
R^3 to be recurrent, and give one application to minimal surfaces.
Dinner: We will be taking the speaker out to dinner at a nearby
restaurant after the seminar.
November 24 No seminar, due to Nov. 25 holiday.
December 1 No speaker this week.
December 8 Frank Pacard, Universite Paris XII, France
Title: Linear and nonlinear aspects of Ginzburg-Landau vortices
Abstract: We present a joint work with
Tristan Riviere concerning existence and
uniqueness questions for Ginzburg-Landau
vortices.
More precisely, we describe precisely
some branches of critical points of the
Ginzburg-Landau functional
\[
E(u) = \int |\nabla u|^2 + \frac{1}{2 \e^2}
\int (1 - |u|^2)^2,
\]
as the parameter $\e$ tends to $0$, here
$u$ is a complex valued function defined
in some bounded domain of ${\mathbb R}^2$.
In particular we prove that, provided $\e$
is small enough, all solutions of
\[
\Delta u + \frac{u}{\e^2} (1- |u|^2) =0,
\]
which are defined in the unit ball and have
boundary data given by $u = e^{i \theta}$
are " radialy symmetric", which means that
they are of the form $u = S (r) \, e^{i \theta}$.
Applications to the gauge invariant
Ginzburg-Landau functional are also given.
Dinner: We will be taking the speaker out to dinner at a nearby
restaurant after the seminar.
Go to:
The current schedule for the Geometric Analysis Seminar
Mathematics Department
Stanford University
For more information, send email to
moore@math.stanford.edu
Last Modified:
May 24
, 2000