Mathematics Colloquium, 2003-4
Fall 2003
12.1.2004
Alexandre
Chorin, Berkeley UniversityTitle: Statistical Projection and Prediction
5.1.2004
Alexander Givental, Berkeley UniversityTitle:Symmetries of Gromov-Witten theory
Abstract
Gromov-Witten
invariants of a
symplectic manifold are defined via intersection theory in moduli
spaces
of pseudo-holomorphic curves in these manifolds. The invariants satisfy
numerous identities universal in the sense that their form is
independent on the choice of the target symplectic manifold. The
peculiar structure formed by the invariants and the universal
identities
has been subject to extensive study and has lead to the theory of
Frobenius manifolds and other concepts of axiomatic Gromov-Witten
theory. Some recent work shows that the axiomatic structure posesses a
certain loop group of hidden symmetries. In the first lecture we plan
to
outline the approach to axiomatic Gromov-Witten theory which emphasizes
the role of this symmetry group. In the second lecture we intend to
discuss the place of the so-called Virasoro constraints - some
conjectural indentities which play a key role in Gromov-Witten theory
at
its current state.
29.12.2003 George
Zaslavsky, New-York UniversityTitle:
Pseudo-chaotic Dynamics
22.12.2003 Shlomo
Sternberg, Harvard UniversityTitle:The Euler Maclaurin formula in higher
dimensions
Abstract
We provide an
elementary proof of
the Euler Maclaurin formula with remainder for a simple convex polytope
in several dimensions. This formula applies to smooth functions which
are symbols in the sense of partial differential equations. I will
begin
by reviewing the classical Euler Maclaurin formula.
Joint work with Y.Karshon and J. Weitsman
Joint work with Y.Karshon and J. Weitsman
15.12.2003 Alexander
Goncharov, Brown University and Max Plank Institute (Bonn)Title: Positivity and Higher Teichmuller Theory
11.12.2003 (special colloquium) Shmuel Friedland, University of Illinois at
Chicago.
Title: Multi-dimensional entropy and the monomer-dimer problem
Abstract
In the first
part of the talk we
will define and discuss the concept of multi-dimensional entropy,
sometimes called the Shannon capacity, and bring examples from statistical mechanics and information theory.
In the second part of the talk we will discuss some theoretical and computational aspects of the classical monomer-dimer problem.
sometimes called the Shannon capacity, and bring examples from statistical mechanics and information theory.
In the second part of the talk we will discuss some theoretical and computational aspects of the classical monomer-dimer problem.
8.12.2003 Nira
Dyn, Tel-Aviv University 24.11.2003 Boris
Kunyavski, Bar-Ilan Title: Characterizing Finite Solvable Groups and Finite-Dimensional Solvable Lie Algebras
Abstract
Our main result
characterizes the
finite solvable groups in terms of identities in two variables
(as
the
finite nilpotent groups are characterized by Engel's identities). The proof is a little surprising: we use
not only methods of the theory of finite groups but also those of arithmetic geometry and computer algebra.
A parallel result gives a similar characterization of finite dimensional solvable Lie algebras defined over
a field of characteristic different from 2, 3, 5.
This is a joint project with T.Bandman, G.-M.Greuel, F.Grunewald, D.Nikolova, G.Pfister, and E.Plotkin.
finite nilpotent groups are characterized by Engel's identities). The proof is a little surprising: we use
not only methods of the theory of finite groups but also those of arithmetic geometry and computer algebra.
A parallel result gives a similar characterization of finite dimensional solvable Lie algebras defined over
a field of characteristic different from 2, 3, 5.
This is a joint project with T.Bandman, G.-M.Greuel, F.Grunewald, D.Nikolova, G.Pfister, and E.Plotkin.
17.11.2003 Arie
Levant, Tel-Aviv University Title: Mathematical "Black Box" Control
Abstract
The task of the
mathematical
control is to make the solutions of a dynamic system (here - ordinary
differential equation) satisfy some prescribed requirements. The goal
is
attained controlling an input variable parameter called control. The
chosen control strategy has also to satisfy some non-mathematically
formulated practical-feasibility conditions.
Historically the first and still one of the main control problems is to make an output variable of the controlled system to follow some signal (the output regulation problem). The problem is aggravated when the exact mathematical model of the controlled process is unknown and only the output is available.
Smooth Single-Input-Single-Output (SISO) dynamic systems are naturally classified by the order of the first total output derivative which explicitly depends on the control (the relative degree). In practice and especially in mechanical systems, the relative degree is often constant and known.
A family is proposed of universal finite-time convergent controllers assigned to each relative degree. Each controller provides for the real-time exact tracking of smooth unknown-in-advance signals by the output of a general SISO dynamic system considered as a "black box". Only two controller parameters are to be tuned. The controller is robust with respect to measurement noises, and the control signal can be designed arbitrarily smooth. The controller contains a high-order asymptotically-optimal robust exact differentiator as its integral part.
The application field is very wide and especially includes tracking and targeting problems of any kind, robotics, avionics, real-time or numeric differentiation, image and signal processing. Experimental results are demonstrated.
Historically the first and still one of the main control problems is to make an output variable of the controlled system to follow some signal (the output regulation problem). The problem is aggravated when the exact mathematical model of the controlled process is unknown and only the output is available.
Smooth Single-Input-Single-Output (SISO) dynamic systems are naturally classified by the order of the first total output derivative which explicitly depends on the control (the relative degree). In practice and especially in mechanical systems, the relative degree is often constant and known.
A family is proposed of universal finite-time convergent controllers assigned to each relative degree. Each controller provides for the real-time exact tracking of smooth unknown-in-advance signals by the output of a general SISO dynamic system considered as a "black box". Only two controller parameters are to be tuned. The controller is robust with respect to measurement noises, and the control signal can be designed arbitrarily smooth. The controller contains a high-order asymptotically-optimal robust exact differentiator as its integral part.
The application field is very wide and especially includes tracking and targeting problems of any kind, robotics, avionics, real-time or numeric differentiation, image and signal processing. Experimental results are demonstrated.
10.11.2003
Jean-Louis Colliot-Thélène, Université de
Paris-Sud, Orsay Title:
The Local-Global Principle for Rational Points and Zero-Cycles
If a quadratic form over a number
field has a nontrivial zero over each completion of the field, then it
has a nontrivial solution over the number field itself. In other words,
the local-global principle holds for rational points on quadrics
(Minkowski, Hasse). Ths celebrated result was extended to
projective varieties which are homogeneous spaces of connected linear
algebraic groups (Kneser, Harder).
However this ``principle" fails for many classes of varieties, among which we find varieties birational to (nonprojective) homogeneous spaces of linear algebraic groups as well as (projective) homogeneous spaces under abelian varieties (e.g. curves defined by a cubic form in three variables). In 1970, a common mould was found for many of the available examples : Manin showed that the theory of the Brauer group yields further necessary conditions for the existence of a rational point over a number field. There are classes of varieties for which one hopes that these necessary conditions are sufficient.
The lecture will explain the Brauer-Manin condition, it will survey classes of varieties for which this condition is sufficient, will mention that there are further conditions (having to do with the cohomology of noncommutative groups).
A less demanding condition on a variety is the existence of a zero-cycle of degree one : namely, one requires the existence of rational points in finite extensions of the ground field, the degrees of these extensions having no common divisor. It is an open question whether the analogue of the Brauer-Manin condition is sufficient for the existence of a zero-cycle of degree one on an arbitrary nonsingular variety over a number field. The lecture will also survey positive results in this direction.
However this ``principle" fails for many classes of varieties, among which we find varieties birational to (nonprojective) homogeneous spaces of linear algebraic groups as well as (projective) homogeneous spaces under abelian varieties (e.g. curves defined by a cubic form in three variables). In 1970, a common mould was found for many of the available examples : Manin showed that the theory of the Brauer group yields further necessary conditions for the existence of a rational point over a number field. There are classes of varieties for which one hopes that these necessary conditions are sufficient.
The lecture will explain the Brauer-Manin condition, it will survey classes of varieties for which this condition is sufficient, will mention that there are further conditions (having to do with the cohomology of noncommutative groups).
A less demanding condition on a variety is the existence of a zero-cycle of degree one : namely, one requires the existence of rational points in finite extensions of the ground field, the degrees of these extensions having no common divisor. It is an open question whether the analogue of the Brauer-Manin condition is sufficient for the existence of a zero-cycle of degree one on an arbitrary nonsingular variety over a number field. The lecture will also survey positive results in this direction.
Spring 2004
7.6.2004 Uri
Yechiali, Tel-Aviv University. 24.5.2004
Mikhail Verbitsky, University of Glasgow and Institute of Theoretical
and Experimental Physics Moscow
Title: Hyperkaehler Geometry
Abstract
Compact
hyperkaehler manifolds are
equally relevant in algebraic geometry and differential geometry. From
algebraic-geometric point of view, a hyperkaehler manifold is a compact
Kaehler manifold equipped with a holomorphic symplectic structure. From
differential-geometric point of view, a hyperkaehler
manifold is a Riemannian manifold equipped with a triple of Kaehler structures $I, J, K$ which satisfy the quaternionic relation $IJ = - JI = K$. The interplay between quaternionic and holomorphic symplectic geometry makes it possible to define and study the geometric objects intrinsic to hyperkaehler geometry (vector bundles, subvarieties, Hodge structures). In this aspect, hyperkaehler geometry becomes just as rich and interesting as the complex algebraic geometry.
manifold is a Riemannian manifold equipped with a triple of Kaehler structures $I, J, K$ which satisfy the quaternionic relation $IJ = - JI = K$. The interplay between quaternionic and holomorphic symplectic geometry makes it possible to define and study the geometric objects intrinsic to hyperkaehler geometry (vector bundles, subvarieties, Hodge structures). In this aspect, hyperkaehler geometry becomes just as rich and interesting as the complex algebraic geometry.
17.5.2004 Yosef
Yomdin, Weizmann InstituteTitle: Closed trajectories of
plane systems of ODE's, Moments, Compositions, and Algebraic Geometry
Abstract
A system of
ordinary differential
equations on the plane is said to have a center at one of its singular
points if all its trajectories around this point are closed. It is a
classical problem to give explicit necessary and sufficient conditions
for a system to have a center (Center-Focus problem). Another closely
related question is to count isolated closed trajectories of a plane
system (second part of Hilbert's 16-th problem). Recently in both
these problems new connections have been found with some questions in
classical analysis and algebra. In particular, this concerns the
vanishing problem of certain moment-like expressions and of iterated
integrals on one side, and the structure of the composition
factorization of analytic functions on the other. The resulting
information allows one to better understand the algebraic geometry of
the "center equations". We present an overview of some recent
developments in this direction.
10.5.2004 Noam
Elkies, Harvard UniversityTitle: Belyi Functions in
Arithmetic Geometry.
Abstract
Abstract
A Belyi function
on a compact
Riemann surface S is a rational function f: S -> P^1
whose ramification locus is contained in the preimage of
{0,1,infinity}. Thus S admits such a function f if and only if S
is the closure of a finite unramified cover of C-{0,1}. These are
named after Belyi due to his remarkable theorem (1979) that S admits
such a function if and only if S is isomorphic to an algebraic curve
defined over a number field. Belyi functions occur surprisingly
often in parts of modern number theory; we give examples ranging from
Fermat's Last Theorem and the ABC conjecture for polynomials, to
modular curves, to several open research questions.
3.5.2004 Daniel Yekutieli,
Tel-Aviv
UniversityTitle: False Discovery Rate
Controlling Methods for Modeling Complex Data.
Abstract
Abstract
FDR controlling
procedures have
proven to be essential tools in dealing with large multiplicity
problems. In this talk I will present three examples of complex
statistical analysis of large data sets. These will be used to review
the Benjamini and Hochberg FDR approach, explain its popularity, but
also to show how the need to analyze larger and more complex data sets
brings about new FDR controlling methodologies.
29.3.2004 Saharon
Rosset, IBM Research Laboratories, New-York.Title: The Role of Regularization in Modern Data
Analysis
15.3.2004 Semyon
Alesker, Tel-Aviv University Title: Theory of valuations on convex sets
8.3.2004 Iddo
Eliazar, Bar-Ilan University Title: A Growth-Collapse Model: Levy Inflow,
Geometric Crashes, and Generalized Ornstein-Uhlenbeck Dynamics
Abstract
We introduce and study a
stochastic growth-collapse model. The growth process is a steady random
inflow with stationary, independent, and non-negative increments.
Crashes occur according to an arbitrary renewal process, they are
geometric, and their magnitudes are random and are governed by an
arbitrary distribution on the unit interval. If the system's pre-crash
level is $X>0$, and the crash magnitude is $0<C<1$, then an
`avalanche' of size $CX$ takes place - after which the system collapses
to the post-crash level $(1-C)X$. This results in a stochastic
$growth->collapse->growth->collapse...$ system evolution,
governed by generalized, non-Markovian, Ornstein-Uhlenbeck dynamics. We
explore the behavior of this growth-collapse model and compute various
system statistics including: mean; variance; auto-correlation; Laplace
transform; and - in the case of heavy-tailed inflows - probability
tails. Special emphasis is devoted to `scale-free' systems where the
crash magnitudes are governed by a power-law distribution. We show
that:
(i) there is a hidden regular Ornstein-Uhlenbeck structure underlying
all scale-free systems; (ii) when crash-rates tend to infinity these
systems yield regular Ornstein-Uhlenbeck limits; and, (iii) when the
inflow is selfsimilar these systems yield Linnik-distributed equilibria.
Organizer: Paul Biran
(biran@math.tau.ac.il)