Archive
2007-2008
| September 27, 2007 | |
| Louis Nirenberg
Courant Institute of Mathematical Sciences |
A Geometric Problem and the Hopf Lemma |
|
A.D. Alexandrov proved that a compact hypersurface embedded in Rn,
having constant mean curvature is a sphere. A generalization is presented. Some
open problems will be described. |
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| September 28, 2007 | |
| Louis Nirenberg
Courant Institute of Mathematical Sciences |
Degree Theory and Some Inequalities |
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Degree theory is described for some maps which are not continuous, and estimates
for degree are given in terms of Sobolev and other norms. |
|
| October 11, 2007 | |
| Peter Trapa
University of Utah |
Kazhdan-Lusztig-Vogan Polynomials and Applications |
|
Given any reductive Lie group G (like the group of n-by-n
invertible complex matrices), one may consider a remarkable set of polynomials,
the KLV polynomials of the title, which controls a great deal of the
representation theory and geometry associated to G. For a large class of
groups, an effective algorithm for computing these polynomials has been known
for over 25 years from the work of Kazhdan-Lusztig, Beilinon-Bernstein,
Brylinski-Kashiwara, and Lusztig-Vogan. The algorithm has been implemented in
many special cases over the years. Recently a particular case, the split real
form of the exceptional group E8, received a great deal of attention. (The
calculation itself was carried out on a UW math department computer maintained
by William Stein who nobly endured the many crashes which the computation
caused.) The purpose of this talk is to give an overview of the mathematics
surrounding the theory of KLV polynomials, with a view toward applications of
their computation. |
|
| October 16, 2007 | |
| Chi-Kwong Li
College of William and Mary |
Quantum Computing, Higher Rank Numerical Ranges, Totally Isotropic Subspaces and Matrix Equations |
|
An introduction will be given to quantum error correction codes, which leads to
the study of the higher rank numerical range Λk(A)
of an n × n matrix A.
The solutions of several conjectures and open problems concerning the convexity
and nonemptyness of the higher rank numerical range will be discussed. In
addition, the results are used to derive a formula for the maximum dimension of
a totally isotropic subspace of a square matrix, and verify the solvability of
certain matrix equations. If time permits, some dilation theorems, and
extensions of the results to infinite dimensional operators will also be
discussed. This is based on joint work with Yiu-Tung Poon (Iowa State University) and Nung-Sing Sze (University of Connecticut). |
|
| November 13, 2007 | |
| Douglas Arnold
University of Minnesota |
The Geometrical Basis of Numerical Stability |
|
The accuracy of a numerical solution to a partial differential equation
depends on the consistency and stability of the discretization
method. While consistency is usually elementary to establish,
stability of numerical methods can be subtle, and for some key PDE
problems the development of stable methods is extremely challenging.
After illustrating the situation through simple but surprising examples,
we will describe a powerful new approach--the finite element exterior
calculus--to the design and understanding of discretizations for a
variety of elliptic PDE problems. This approach achieves stability by
developing discretizations which are compatible with the geometrical
and topological structures, such as de Rham cohomology and Hodge
decompositions, which underlie well-posedness of the PDE problem being
solved. |
|
| November 27, 2007 | |
| Mihalis Dafermos
Cambridge University and M.I.T. |
The Stability Problem for Black Hole Spacetimes |
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The notion of black hole plays a central role in general relativity.
Nonetheless, the most basic mathematical questions about black holes remain
unanswered, in particular, the question of their stability with
respect to perturbation of initial data. In this talk, I will discuss how this
problem is mathematically formulated, emphasizing its relation to decay
properties for solutions of wave equations. I will then discuss recent progress
on various related problems. |
|
| November 29, 2007 | |
| Matti Lassas
Helsinki University of Technology |
Inverse Problems, Invisibility, and Artificial Wormholes |
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There has recently been considerable interest in the possibility, both
theoretical and practical, of invisibility of objects to different types of
waves. We construct several examples of cloaking enclosures covered with
anisotropic materials. These examples have a close connection to earlier works
carried out for the case of conductivity equation [1,2], a case that is
important in electrical impedance tomography (EIT). These results have also a
close connection to counterexamples for inverse problem. For instance, consider
the Calderon problem, that is, whether the Dirichlet-to-Neumann map determines
uniquely the conductivity. The problem has a positive answer in all dimensions
n ≥ 2 if the conductivity is isotropic (under suitable regularity
hypothesis). In two dimensions, it is also known that an anisotropic
conductivity can be found from the boundary measurements up to a change of
coordinates. However, in all of these results it is assumed that the
conductivity is bounded both below and above by strictly positive constants. If
this condition is violated, one can cover any object with a properly chosen
anisotropic material so that the covered object appears in all boundary
measurement similar to a domain with constant conductivity. Clearly, this kind
of counterexample gives us theoretical instructions how to cover an object so
that it appears "invisible" in zero frequency measurements. In this talk we
consider similar kind of result for all frequencies. We note that on practical
level, the engineered materials needed for invisibly cloaking are inherently
prone to dispersion, so that realistic cloaking must currently be considered as occurring at a single wavelength, or very narrow
range of wavelengths. We review the results on the counterexamples based on singular transformations that push isotropic electromagnetic parameters forward into singular, anisotropic ones. We will consider in detail the existence of the finite energy solutions when cloaking a ball or an infinite cylinder. The analogous technique can be used to construct artificial wormholes, that is, devices that act like invisible tunnels guiding electromagnetic radiation having a given frequency. The results have been done in collaboration with A. Greenleaf, Y. Kurylev and G. Uhlmann. References: |
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| February 19, 2008 | |
| John Voight
University of Vermont |
Number Field Enumeration |
|
How quickly can one enumerate number fields of fixed degree with bounded
absolute discriminant? We discuss some mathematically and computationally
interesting aspects of this question. For totally real number fields, a
particular case of interest, we exhibit an algorithm which improves upon known
methods by the use of elementary calculus (Rolle's theorem and Lagrange
multipliers). |
|
| February 21, 2008 | |
| Anatoly Vershik
Steklov Institute of Mathematics & St. Petersburg State University |
New Ideas in the Theory of Metric Spaces |
|
Recently a remarkable and completely forgotten for many years paper by P.S.
Urysohn about a universal metric space finally attracted the attention of
several mathematicians. This is a Polish space which is universal (=each Polish
space can be embedded into it) and super homogeneous (=any two isometric compact
subsets are globally isometric). Many new unexpected properties of this space
were discovered. For example, it is an abelian group with invariant metric, it
has property of rigidity, etc. |
|
| February 26, 2008 | |
| Mihai Putinar
University of California, Santa Barbara |
Positive Polynomials |
|
A historical approach to some positivity problems of real algebraic geometry
will be given from the perspective of early functional analysis. The spectral
theorem for commuting self-adjoint operators will be the key to a variety of
Positivstellensatze, in conjunction with a powerful decision theorem of Tarski.
Recent applications to non-linear, non-convex optimization for polynomials
functions will also be discussed. |
|
| March 11, 2008 | |
| Simon Brendle
Stanford University |
Ricci Flow in Higher Dimension and the Sphere Theorem |
|
We describe recent joint work with Richard Schoen on the Ricci flow in higher
dimension. We discuss various algebraic conditions on the Riemann curvature
tensor. These conditions are closely related to the notion of positive isotropic
curvature (PIC) and can be shown to be preserved under Ricci flow. Using these
ideas, we prove a new convergence result for a class of initial data that
includes all manifolds with 1/4-pinched sectional curvatures. As a corollary, we
give an affirmative answer to a question posed by H. Rauch in 1951. |
|
| May 8, 2008 | |
| Richard Melrose
Massachusetts Institute of Technology |
Semiclassical Limits and the Index Theorem |
|
I will start by describing the semiclassical limit for smoothing operators and
its connection to K-theory. This will be used to define the index map of Atiyah
and Singer. As time permits I will indicate the proof, which is now not very
hard, and some applications to smooth K-theory. |
|
| May 20, 2008 | |
| Jingyi Chen
University of British Columbia |
Special Lagrangian Submanifolds in Cn |
|
In this talk, I will first discuss some recent results on graphical special
Lagrangian submanifolds (joint work with Warren and Yuan) and then discuss
minimal cones related to special Lagrangian geometry (joint work with Yuan). |
|
| May 27, 2008 | |
| Plamen Stefanov
Purdue University |
Travel Time Tomography and Tensor Tomography |
|
Let (M,g) be a compact Riemannian manifold with boundary. We study
the following inverse problem: can we recover the metric g, up to an isometry,
from knowing the distance function between boundary points or from knowing the
outgoing point and direction of any incoming ray (the scattering relation).
Linearizing, we get the following integral geometry problem: recover a symmetric
2-tensor, up to a potential one, from integrals along all maximal geodesics. We
will discuss the recent progress on those problem, obtained in collaboration
with Gunther Uhlmann. We emphasize on a microlocal
approach to this geometry problem. We will discuss results for simple manifolds
(convex boundary, no conjugate points), some non-simple ones, and those two
problems with partial data. We will also discuss related non-linear problems and linear integral geometry problems for more general geometries. Those two problems arise naturally in seismology, medical imaging, and it is of independent interest in geometry. Physically, the boundary distance function measures travel times of waves. It is encoded, together with the scattering relation, in boundary measurements related to hyperbolic PDEs. |
|
| June 5, 2008 | |
| Olga Holtz
UC Berkeley and TU Berlin |
Zonotopal Algebra, Analysis and Combinatorics |
|
A wealth of geometric and combinatorial properties of a given linear
endomorphism X of RN is captured in the study of
its associated zonotope Z(X), and, by duality, its associated
hyperplane arrangement H(X). This well-known line of study is
particularly interesting in case n := rank X << N. We
enhance this study to an algebraic level, and associate X with three
algebraic structures, referred herein as external, central, and internal. Each
algebraic structure is given in terms of a pair of homogeneous polynomial ideals
in n variables that are dual to each other: one encodes properties of the
arrangement H(X), while the other encodes by duality properties of
the zonotope Z(X). The algebraic structures are defined purely in
terms of the combinatorial structure of X, but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to either
of Z(X) or H(X). The theory is universal in the
sense that it requires no assumptions on the map X (the only exception
being that the algebro-analytic operations on Z(X) yield
sought-for results only in case X is unimodular), and provides new tools
that can be used in enumerative combinatorics, graph theory, representation
theory, polytope geometry, and approximation theory. Special attention in this
talk will be paid to the case when X is the incidence matrix of a graph
(and therefore unimodular), when the general theory provides interesting
combinatorial information about the graph, refining the statistics recorded by
its Tutte polynomial and related generating functions. |
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