Seminars Sorted by Series
Members’ Seminar
We introduce and construct the "AC geometry" from the Gaussian
free field and use it to prove various facts about Schramm-Loewner
evolutions.
Random Geometry and SLE II
We introduce and construct the "AC geometry" from the Gaussian
free field and use it to prove various facts about Schramm-Loewner
evolutions.
Regularity and Analyticity for the dissipative Quasi-Geostrophic Equations
The Generalized de Rham-Witt Complex Over a Field is a Complex of Zero-Cycles
The Renormalisation Group I
A very long random walk, seen from so far away that individual
steps cannot be resolved, is the continuous random path called
Brownian motion. This is a rough statement of Donsker's theorem and
it is an example of how models in statistical mechanics...
On the Geometric Langlands Functoriality for the Dual Pair Sp_{2n}, SO_{2m}
I will report on a the following work in progress. Let X be a
smooth connected curve over an algebraically closed field. Consider
the dual pair H=SO_{2m}, G=Sp_{2n} over X with H split. Let Bun_G
and Bun_H be the stacks of G-torsors and H-torsors on...
Color Coding, Balanced Hashing and Approximate Counting
Color Coding is an algorithmic technique for deciding
efficiently if a given input graph contains a path of a given
length (or another small subgraph of a certain type). It
illustrates well the phenomenon that probabilistic reasoning can be
helpful...
Members Seminar, cancelled today
The Topography of Random Waves
Random waves have been investigated since the 1940's in
connection with modeling telephone signals (Rice), to model sea
waves (Longuet-Higgins), and since the 1970's by Berry and others
to model quantum wave-functions of classically chaotic
systems...
Gap Theorem and Finite Diffeomorphism Theorem in Conformal Geometry
New Entire Solutions for Semilinear Elliptic Equations
Motivated by some recent progress in the study of concentration
phenomena for singularly perturbed elliptic nonlinear equations and
the analogies with some problems in geometric analysis, we prove
existence of new entire solutions of the focusing...
LERF, the Lubotzky-Sarnak Conjecture and the Topology of Hyperbolic 3-Manifolds
The Lubotzky-Sarnak Conjecture asserts that the fundamental
group of a finite volume hyperbolic manifold does not have Property
\tau. Put in a geometric context, this conjecture predicts a tower
of finite sheeted covers for which the Cheeger...
Spherical Cubes and Rounding in High Dimensions
What is the least surface area of a shape that tiles Rd under
translations by Zd? Any such shape must have volume 1 and hence
surface area at least that of the volume-1 ball, namely (–d). Our
main result is a construction with surface area O(–d)...
Mathematical Questions Arising from Bose-Einstein Condensation
Israel Michael Sigal
Bose-Einstein condensation was predicted by Einstein in 1925 and
was experimentally discovered 70 years later. This discovery was
followed by a flurry of activity in the physics community with
hundreds of papers published every year and with...
Trace Formulae and Locally Symmetric Spaces
Trace formulae and relative trace formulae can be used to study
the rich geometry of locally symmetric spaces. I will explain some
illustrative examples coming from unitary groups in this talk.
The Sum of Squares of the Wavelengths of a Surface
This talk is intended for a general audience. We define and
discuss a spectral invariant of closed Riemannian surfaces, namely
the zeta regularized trace of inverse of the Laplacian. Physically
this corresponds to the sum of squares of the...
Expansion in Linear Groups and Applications
Nonlinear Problems for Nonlocal Diffusions
Hidden Structures in the Family of Convex Functions in R^n and the New Duality Transform
(Joint work with Shiri Artstein-Avidan). We discuss in the talk
an unexpected observation that very minimal basic properties
essentially uniquely define some classical transforms which
traditionally are defined in a concrete and quite involved
form...
Pseudo-Hermitian Geometry in 3-D
A pseudo-hermitian structure in 3-D is a contact form and an
almost complex structure on the kernel of the contact form. There
is a natural notion of area and mean curvature for a surface in
such a geometry. I will discuss some work on the structure...
Isoperimetric and Concentration Inequalities, and Their Applications
The classical isoperimetric inequality in Euclidean space
asserts that among all sets of given Lebesgue measure; the
Euclidean ball minimizes surface area. Using a suitable
generalization of surface area, isoperimetric inequalities may
be...
Categorical Probability Theory
The Noether Lefschetz Locus
Ania Otwinowska
Integral Conformal Invariants
Alice Chang
We will survey the role played by some classes of higher order
integral conformal invariants in conformal geometry which include
the integral of Q-curvature and those related to the Gauss-Bonnet
integrand. We will also discuss a class of integral...
Finite Approximation of Group Actions and Graph Metrics
Sofic groups are the groups whose Cayley graph that can be
approximated by finite graphs. This class of groups contains many
known classes, e.g. profinite and amenable groups. Quite a few
conjectures are known to hold for sofic groups. The most...
The Regularized Determinant of a Four-Manifold
Matt Gursky
I will give an overview of a variational problem in conformal
geometry/spectral theory. Beginning with a formula for the
determinant of the laplacian for a surface and the work of
Osgood-Phillips-Sarnak, I will explain some attempts to
generalize...
Local Entropy and Projections of Dynamically Defined Fractals
If a closed subset X of the plane is projected orthogonally onto
a line, then the Hausdorff dimension of the image is no larger than
the dimension of X (since the projection is Lipschitz), and also no
larger than 1 (since it is a subset of a line)...
Recent Progress on QUE (Quantum Unique Ergodicity)
Enrico Bombieri and the Prime Number Theorem
We survey some of the points of contact between the two subjects
of the title. The talk is intended for non-specialists.
The Decomposition Theorem and Abelian Fibrations
The Algebra and Combinatorics of Box Splines
What do the following seven things have in common: *
zero-dimensional homogeneous polynomial ideals, * multivariate
polynomial interpolation, * box splines, * hyperplane arrangements,
* parking functions on graphs, * f-vectors of matroids, *
integer...
Smectic Topology, Tomography, and Topography
Randy Kamien
Smectic liquid crystals correspond to certain foliations of
Euclidean space. I will discuss the topology of defects in the
smectics. Typically the index of these topological defects is
characterized via the fundamental group of the manifold of...
Deformation Spaces of Geometric Structures
The Riemann Hypothesis--150 Years and Counting
Brian Conrey
In this talk we will relate some of the colorful history of one
of the world's great math problems.
Number Theory Related to Quantum Chaos
Quantum chaos is concerned with properties of eigenvalues and
eigenfunctions of "quantized Hamiltonians". For instance, can
classical chaos be detected by looking at the spacings between
eigenvalues? Another problem is if classical ergodicity
forces...
Moduli Spaces of Bundles -- With Some Twists
Semisimple Lie groups seem to be very rigid objects. In
arithmetic situations the fact that a semisimple group may
degenerate into a non-semisimple one in a family is well known. In
geometric situations this phenomenon has not been applied that
much...
Finding the Symmetry Group and the Three-Dimensional Shape of Symmetric Molecules that we Don't Know How to Crystallize
I will report on a joint work with Ronny Hadani (UT Austin) and
Amit Singer (Princeton). We give an algorithmic solution to the two
problems that appear in the title. The input is the numerical data
(roughly, random plane sections of the molecule)...
Function Theory on Symplectic Manifolds
Leonid Polterovich
It has been recently observed that function spaces associated to
a symplectic manifold exhibit unexpected properties and surprising
structures, giving rise to new tools and intuition in symplectic
topology. In the talk I shall discuss these...
The Members Seminar talks will resume at the beginning of the Second Term (January 11, 2010).
Pseudoholomorphic Curves and Dynamics in Dimension Three
The talk describes how pseudoholomorphic curve based techniques
can be used to understand better the dynamics of autonomous
Hamiltonian systems of two degrees of freedom restricted to a
compact non-degenerate energy surface.
Gelfand Pairs and Invariant Distributions
First I will introduce the notion of Gelfand pair and its
connection to invariant functions and distributions on the group,
and give some examples. We will start from finite groups and
compact groups and then go on to reductive groups over local...
Pretentiousness in the Analytic Theory of Numbers
Following the brilliant insight of Riemann, that a good
understanding of the distribution of prime numbers is equivalent to
a good understanding of the location of zeros of pertinent
L-functions, analytic number theory has traditionally centered
on...
An Extension Criterion for Lattice Actions on the Circle
Marc Burger
We establish a necessary and sufficient condition for an action
of a lattice by homeomorphisms of the circle to extend continuously
to the ambient locally compact group. This condition is expressed
in terms of the real bounded Euler class of this...
Heegaard Floer Homology, Khovanov Homology and Contact Geometry
John Baldwin
I'll give a brief survey of Heegaard Floer homology, one of the
latest and most powerful descendants of the Gauge theories of the
80's and 90's, and I'll discuss established and conjectured
connections between Heegaard Floer homology and Khovanov...