#
Math Colloquium

**
Math colloquium meets
on Mondays at 12:15 in Schreiber 006,
Tel Aviv University.
**

#
Fall 2005

###
07.11.2005, 12:15
Alexander Olevskii
, Tel Aviv University
Uniqueness in Fourier analysis: a new phenomenon.

###
14.11.2005, 13:00 (Note the non-standard time!)
Jon Keating, University of Bristol
Random matrices and the Riemann Zeta function.

###
21.11.2005, 12:15
Yakov Eliashberg, Stanford, USA
Applications of symplectic geometry to low-dimensional topology.
(This is the first lecture in the framework of
DISTINGUISHED LECTURES IN TOPOLOGY
supported through the Michael Bruno Memorial Awards.)

ABSTRACT: Symplectic geometry enters low-dimensional topology in several
ways. First of all, there is a number of canonical constructions
which associate with smooth manifolds and their submanifolds some
symplectic or contact manifolds and their Lagrangian and Legendrian
submanifolds. Then symplectic and contact invariants of the
constructed objects become smooth invariants of the original
manifolds. This approach was successfully applied by L. Ng for his
study classical knots in the 3-space. I will briefly review his
results and also discuss some ongoing attempts to use this approach
for defining new invariants of 4-manifolds. Gromov's theory of
holomorphic curves is another door through which symplectic
geometry enters
low-dimensional topology, and I will discuss some of the
constructions and results in this direction.
###
28.11.2005, 12:15
Pierre Lochak, Universite Paris 6, France
Thurston with Grothendieck?
From diffeomorphisms of surfaces to Grothendieck-Teichmueller theory.

ABSTRACT. Using concrete examples I will explain how Thurston's viewpoint
on the diffeomorphisms of surfaces and its subsequent ramifications should
be quite relevant for exploring the landscape delineated in Grothendieck's
`Esquisse d'un programme'. This includes introducing `origamis' which
are a priori of a purely topological nature and can also be viewed as
a higher dimensional analog of `dessins d'enfants'; in particular
the Galois group of the rational numbers acts faithfully on them.
###
05.12.2005, 12:15
Michael Polyak, Technion, Haifa
Counting lines and other geometric shapes (or
where rigid algebraic geometry meets smooth topology).

ABSTRACT.
In complex enumerative geometry one counts
algebraic-geometric objects with certain properties,
e.g. a number of rational curves of degree d passing
through 3d-1 points. I will discuss a real counterpart
of such problems, starting from some simple examples and
relating them to the theory of finite type invariants.
I will also discuss a general setting to produce such
invariants using maps of configuration spaces and
homology intersections.
###
12.12.2005, 12:15
Victor Palamodov, Tel Aviv University
Rigidity of Riemannian manifolds and the Inverse Kinematic
problem

ABSTRACT:
Whether a Riemannian manifold $M$ is uniquely
determined by lengths of its closed geodesics? The (affirmative)
answer is known in few special cases: Michel, Gromov, Croke,
Uhlmann,... If $M$ is conformal to a bounded domain in Euclidean
space, the problem is to recover the conformal coefficient as
function of Euclidean coordinates from knowledge of lengths of all
geodesics connecting boundary points. The particular case is known
as the inverse kinematic problem in geophysics: to reconstruct the
velocity of elastic waves in the globe from day surface
measurements of travel times. This problem has a long history:
Herglotz, V. Markushevich, M. Gerver, V. Romanov, R. Muhometov, G.
Beylkin, J. Bernstein,...
The newest progress in the problems will be described in the talk. The basic
notions will be explained as well as the main arguments, which are quite
elementary.
###
19.12.2005, 12:15
Yuval Peres , University of California, Berkeley
Point processes, the stable marriage algorithm, and Gaussian
power series.

ABSTRACT: We consider invariant point processes, i.e., random
collections of points with distribution invariant under isometries:
the simplest example is the Poisson point process. Given a point
process M in the plane, the Voronoi tesselation assigns a polygon (of
different area) to each point of M. The geometry of "fair"
allocations is much richer: There is a unique "fair" allocation that
is "stable" in the sense of the Gale-Shapley stable marriage
problem. Zeros of power series with Gaussian coefficients are a
different source of point processes, where the isometry invariance is
connected to classical complex analysis. In the case of independent
coefficients with equal variance, the zeros form a determinantal
process in the hyperbolic plane, with conformally invariant
dynamics. Surprisingly, in this case the number of zeros in a disk
has a coin-tossing interpretation. (Talk based on joint works with
C. Hoffman, A. Holroyd and B. Virag).
###
26.12.2005, 12:15
Albert Fathi , Ecole Normale Superieure de Lyon
Existence of critical C^1 subsolutions of the Hamilton-Jacobi
equation.

###
09.01.2006, 12:15
Dan Romik , University of California, Berkeley
Random Young tableaux, Young diagrams, permutations and sorting networks.

ABSTRACT. Plancherel measure is an important probability model that encodes the
probabilistic behavior of lengths of increasing subsequences in random
permutations. It has been shown in recent years to be in many ways a
discrete analogue of the GUE (Gaussian Unitary Ensemble) random matrix
model.
I will introduce a new model, uniform random square Young tableaux, which
turns out to be a natural deformation of Plancherel measure that can be
analyzed using similar techniques. I will survey some new results on this
model and some applications to the combinatorics of random permutations and
random "sorting networks", which are ways to sort a list of N distinct
numbers from increasing to decreasing order by applying a minimal-length
sequence of adjacent transpositions. The talk is based on joint works with
Boris Pittel, Omer Angel, Alexander Holroyd, Balint Virag, Scott Sheffield
and Rick Kenyon.
###
16.01.2006, 12:15
Leonid Polterovich , Tel Aviv University
Title: Symplectic maps: algebra, geometry, dynamics.

ABSTRACT: Symplectic maps appear as a natural generalization of
area-preserving diffeomorphisms of surfaces. They play a central role in the
mathematical model of classical mechanics.
We will focus on the following topics:
(a) growth rate of symplectic maps and the trichotomy
hyperbolic/parabolic/elliptic in the context of diffeomorphisms;
(b) obstructions to symplectic actions of finitely generated groups,
including a symplectic version of the Zimmer program on actions of lattices.
We discuss results in these directions based on modern methods of symplectic
topology
###
30.01.2006, 12:15
Steven Schochet , Tel Aviv University
The Incompressible Limit

ABSTRACT: The incompressible limit is a singular limit of a system of partial
differential
equations that describes fluid flow. Because there are several variants of
the
equations for fluid flow and a variety of classes of initial data and
boundary
conditions that may be considered, there are in fact a large number of
incompressible limits. After its introduction almost a hundred years ago,
interest in the incompressible limit was renewed about thirty years ago, and
it
has become one of the prototypical problems in the theory of singular limits
for partial differential equations.
In this talk, which is aimed at a general mathematical audience, I will
describe
the relevant equations and their physical background, the nature of singular limits including in particular the relation to the theory of averaging for
ODEs, some basic theory and examples of partial differential equations, a
variety of results that have been obtained over the years concerning the
incompressible limit, key ideas from some of their proofs, and some recent
results and open problems.

#
Spring 2006

###
06.03.2006, 12:15
Sergey Fomin
, University of Michigan, USA
Cluster algebras.

ABSTRACT: The talk will survey the basic definitions and results of the emerging
theory of cluster algebras, viewed from a combinatorial perspective. Joint
work with Andrei Zelevinsky.
###
13.03.2006, 12:15
Emmanuel Giroux
, ENS Lyon, France
Open books in contact geometry.

ABSTRACT: we will first describe the notion of an open book and how it
appears under distinct names in different areas of mathematics---dynamical
systems, complex algebraic geometry, algebraic and geometric topology. Then
we will see that these various aspects of open books are facets of their
general relations with contact geometry. Finally, we will discuss a number
of questions raised by these relations.
###
27.03.2006, 12:15
Alfred Inselberg , Tel Aviv University, Israel & San Diego SuperComputing Center, USA
Visualizing R^N and some applications.

ABSTRACT. The desire to understand the underlying geometry of multidimensional problems
motivated several visualization methodologies to augment our limited 3-dimensional perception. Parallel Coordinates are rigorously developed
obtaining a 1-1 mapping between subsets of N-space and subsets of 2-space.
It leads to representations for lines, flats, curves, hypersurfaces
and constructions algorithms in N-space involving intersections,
proximity, interior point construction and topologies of flats useful in
approximations. This is a VISUAL Multidimensional Coordinate System. It
is illustrated on some applications to Air Traffic Control, Data Mining on
multivariate datasets and Decision Support systems (based on hypersurfaces)
capable of doing Feasibility, Trade-Off and Sensitivity Analyses for complex
multivariate processes. There will be several interactive demonstrations.
###
03.04.2006, 12:15
Shlomo Sternberg , Harvard University, USA
The Ehrhart formula for symbols and a
generalization of Euler's constant.

###
10.04.2006
Passover break

###
17.04.2006
Passover break

###
24.04.2006, 12:15
Viatcheslav Kharlamov
, Universite Louis Pasteur, Strasbourg
On Dif=Def problem in real algebraic geometry.

ABSTRACT. Deformation equivalent real algebraic manifolds are equivariantly
(respecting the complex conjugation involution) diffeomorphic. The Dif=Def
problem asks for what classes of algebraic manifolds diffeomorphic real
structure are deformation equivalent. Our goal is to present two recent
results. One extends the class of manifolds for which Dif=Def holds to cubic
four-folds. The other gives first examples of Dif\ne Def manifolds: it shows
that Dif=Def does not hold for Campedelli surfaces.
###
08.05.2006, 12:15
Mikhail Sodin
, Tel Aviv University, Israel
Random complex zeroes.

ABSTRACT: In the 90-ies, physicists introduced a unique class
of Gaussian analytic functions with remarkable unitary
invariance of zero points. In the talk, I am going
to discuss some recent progress toward understanding
the zeroes of these functions.
The talk is based on joint works with Fedor Nazarov,
Boris Tsirelson, and Alexander Volberg.
###
15.05.2006, 12:15
Alexander Elashvili
, Institute of Mathematics of Georgien Academy of Science, Tbilisi, Georgia
Lie algebras and singularity theory.

ABSTRACT:
Let O_n:=C{x_1,...,x_n} be the algebra of convergent power series in n
variable. For f in O_n define the ideal I(f)=(f,d_1f,...,d_nf),where
d_if is partial derivation by variable x_i, and consider factor -algebra
A(f):=O_n/I(f). We said that f has isolated singularity in 0 if A(f) is
finite dimensional vector space over C. Let L(f):=Der(A(f)) be the Lie
algebra of all derivations of algebra A(f). In this talk we describe
structural properties and some numerical invariants of algebras L(f).
###
22.05.2006, 12:15
Steve Zelditch , Johns Hopkins University, USA
(Raymond and Beverly Sackler distinguished lectures in Mathematics)

###
Counting possible universes in string/M theory.

ABSTRACT:
According to string, the vacuum state of our universe is 10
dimensional: the product of the usual 4 dimensional (Minkowski)
spactime and a small 3 complex dimensional Calabi-Yau manifold.
The problem is to determine how many candidates there are for this
CY 3-fold. Physicists often quote the number as 10^{500}.
The first purpose of my talk is to explain the problem in mathematical
terms. Although the setting is sophisticated, the problem boils down
to the combination of a lattice point problem
and a problem on critical points of Gaussian random
holomorphic functions. I will give a rigorous asymptotic formula for
the number of candidate universes and discuss how close it comes to pinning
down a specific number in certain string models. The counting formula is
reminiscent
of counting metastable states of glassy systems as the dimension of the
system increases, with the third betti number of the CY 3-fold playing the role of
dimension.
No prior acquaintance with string theory is assumed.
###
05.06.2006, 12:15
Pavel Bleher
, Indiana University-Purdue University Indianapolis, USA

###
Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions.

ABSTRACT:
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC)
has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on
the Yang-Baxter equations, and it represents the free energy in terms of an $N\times N$ Hankel
determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed
in terms of the partition function of a random matrix model with a nonpolynomial interaction.
We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC
in the disordered phase.
The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest
descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign
matrices (the ASM problem) is a special case of the the six-vertex model. We compare the
obtained exact solution of the six-vertex model with known exact results for the
1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free
fermion line. We prove the conjecture of Zinn-Justin that the partition function of
the six-vertex model with DWBC has the asymptotics,
$Z_N\sim CN^\kappa e^{N^2f}$, as $N\to\infty$,
and we find the exact value of the exponent $\kappa$.
###
12.06.2006, 12:15
Shmuel Weinberger
, University of Chicago, USA
(Blumenthal Geometry Lecture)

###
Using the telescope as a microscope: Large scale
determination of small scale structure.

ABSTRACT: Just as physicists occasionally use cosmological evidence to
reflect
on the subatomic world, in geometry, there are a number of situations
where one can link the structure at the largest and smallest of
scales. The mind does this naturally: watching television we
actually see a large number of small pixels, but since we are
naturally considering a larger scale, we clump them together, and
hypothesize the structure that should be there, were the objects
"homogenous" and not pixilated.
In this lecture, I will start by giving simple minded examples where
large scale structure constrains the small scale and will continue
and describe situations where one can actually determine the small
scale structure from the large.
#####
Organizer: Semyon Alesker
, e-mail: semyon AT post DOT tau.ac.il