Mathematics
Colloquium 2004-5
30.5.2005 Igor
Pak, MIT
Title: Convex polytopes, rigidity, and classical
geometry
Abstract
Cauchy Theorem states that
simplicial polytopes in three dimensions are rigid, so in
principle one should be able to "construct" a polytope from its graph
and edge lengths. Actually doing this is more complicated even in the
most simple special cases... In this talk we review
several approaches to the problem. We start with the
Alexandrov "existence theorem", then switch to a remarkable Kapovich-Millson
"universality theorem" on planar linkages, and then outline
Sabitov's polynomials for nonconvex polyhedra. The latter work
is related to Sabitov's proof of the "bellows conjecture" that flexible polyhedrons
must keep its volume constant under continuous deformation. We conclude
with joint results of Fedorchuk and myself on the degrees of Sabitov
polynomials and sketch our proof of the Robbins conjecture on the area
of cyclic polygons.
The talk should be accessible to anyone who have seen icosahedron, or at least heard about it...
The talk should be accessible to anyone who have seen icosahedron, or at least heard about it...
23.5.2005 Wolf
Prize Laureate Lecture(partially supported by the Lehrer fund)
Serguei
Novikov, University of Maryland
Title: Quasiperiodic Functions and Dynamical
Systems Describing the Motion of Electrons.
16.5.2005 Peter Lax, Sackler Distinguished Lecturer - 2005, Courant
Institute - New York University
Title:
The Zero Dispersion Limit
Abstract
There are many dispersive systems,
such as the KdV equation where the zero dispersion limit is of
interest,It turns out that this limit,with fixed initial values, exists
in the weak but not strong sense. This will be illustrated with
numerical examples;a rigorous proof uses the machinary of complete
integrability.The limiting flow is non-Markovian.
9.5.2005 Adi Ditkowski, Tel-Aviv University
Title: On Properties of Hyperbolic Equations
and Nonreflecting Boundary Conditions
Abstract
Many hyperbolic systems, such as
the Maxwell's and Acoustics equation, have plane wave solutions. If we
define a domain D in R^m, m >= 2, with
smooth boundary, then at a boundary point, we can classify the waves as
incoming, outgoing and stationary or tangential waves. In this work we
show that for systems which couple the outgoing and incoming waves, as
the examples above, the incoming and outgoing waves are not separate
subspaces. This imply that an incoming wave may be presented as a linear
combination of outgoing waves. We use this phenomenon to prove, that for
such systems, an absolutely absorbing, local and linear boundary
condition, does not exist. Joined work with Michael Sever.
2.5.2005 Joseph Avron, Technion.Title: Swimming at low Reynolds numbers
Abstract
Micro-organisms and microbots swim
at low Reynolds numbers. The theory associate to this can be viewed as a
gauge theory. The notion of parallel transport (connection)
is dictated by solutions of the Navier-Stokes equation in
the limit of low Reynolds numbers. The gauge theory of swimming
will be briefly reviewed and some applications to the swimming of
microbots discussed. I shall also describe an application to the
peculiar mode of natural swimming known as metaboly and used
by Euglena. I will show movies of Euglena swimming and simulations
of competitions between microbots..
11.4.2005 Distinguished Lectures
in Topology
(Supported through the
Michael Bruno Memorial Awards)
Octav
Cornea, Universite de Montreal, Canada
Title: Morse theory: alive and well !
Abstract
In this talk destined to a general
audience I will describe the main tools and constructions in Morse
theory. While much of this is classical I will also describe some more
recent results concerning some deeper relations between Morse theory and
algebraic topology as well as the role of Morse theoretical ideas in
symplectic topology.
28.3.2005 Barry Simon, CaltechTitle: The lost proof of Loewner's Theorem
Abstract
A real-valued function, F, on an
interval (a,b) is called matrix monotone if F(A) < F(B) whenever A
and B are finite matrices of the same order with eigenvalues in (a,b)
and A < B. In 1934, Loewner proved the remarkable theorem that F is
matrix monotone if and only if F is real analytic with continuations to
the upper and lower half planes so that Im F > 0 in the upper half
plane.
This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been ``lost'' in that the proof is not mentioned in various books and review article presentations of the subject, and I have found no references to the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.
This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been ``lost'' in that the proof is not mentioned in various books and review article presentations of the subject, and I have found no references to the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.
21.3.2005 Distinguished Lectures
in Partial Differential Equations (Supported through the
Michael Bruno Memorial Awards)
Leonid
Ryzhik, University of Chicago USA
Title: Fronts in flows: speed-up and extinction
Abstract
It is well known that an ergodic
wind is quite effective in extinguishing a small fire (lighting a
match is but one careless example), or spreading a large fire.
Mathematically, this should be reflected in such objects as the decay of
the heat kernels, the size of the first Dirichlet eigenvalue, or
the spreading vs. extinction of the solutions of the
advection-reaction-diffusion PDE's in terms of the geometric properties
of the flow, such as the presence of invariant cells as opposed to
open streamlines. I will describe some results in this direction
as well as some coupled problems when the fluid convection is
generated by temperature variations. In particular, I hope to describe
some lower bounds on the propagation rates that reflect the
non-trivial effect of convection.
14.3.2005 Evgenii Shustin, Tel-Aviv UniversityTitle: The tropical algebraic geometry
Abstract
The tropical algebraic geometry is
a part of the tropical mathematics, where the objects are defined over
the so-called tropical semi-field, the real numbers equipped with the
operations of maximum and ordinary addition. The tropical geometry
is tightly linked with the classical one via non-Archimedean valuation
fields, large complex limits and other procedures, which represent
tropical object as limits of the classical ones. It turns out that
the tropical limits inherit unexpectedly many properties of the
classical stuff, which sometimes allows one to reduce hard
algebraic-geometric problems to tropical geometry problems, i.e.
geometry of polyhedral complexes, convex (lattice) polytopes etc., and
are often easier to solve. We illustrate this relation by examples,
including recent applications to computation of Gromov-Witten and
Welschinger invariants.
28.2.2005 Blumenthal
Lectures in GeometryDmitri
Burago, Pennsylvania State
University
Title: Choosing "good" coordinates: from
asymptotic geometry of tori to boundary rigidity
Abstract
The conclusion of many theorems
and conjectures in geometry is the assertion that two manifolds are
isometric. There is a good reason why such problems are often
extremely difficult. Even if we are explicitly given two metrics on the
plane, it is virtually impossible to decide whether they are isometric.
The metrics can be given simply by their metric coefficients E(x,y),
F(x,y), G(x,y) (using the notations that became rather common in
geometry of surfaces since Gauss), and still this is essentially an
undoable task. The difficulty is that in the background of the problem,
there always is a change of coordinates that sends metric
coefficients of one metric to that of the other one. It is
virtually impossible to prove that two manifolds are isometric without
finding (i.e. constructing) this change of coordinates. By a coordinate
system (in a region in an n-dimensional manifold) we usually mean a
collection of n real valued functions, which can also be viewed as an
identification of the region with a region in a model space.
We will discuss a more sophisticated understanding of coordinate systems, with many more (or even "much", that is with a continuum of) coordinate functions, and with a certain type of structures on the set of these functions. The ultimate goal is always replacing the statement that "there exists a diffeomorphism such that..." by proving that two geometrically well-defined sets coincide.
At first glance, the structures that arise here may seem rather unnatural and not so well connected to the original problems . I hope to be able to advocate that their use is actually well-justified. For good or bad, there is essentially the same "perverted" scheme of thinking (of course, augmented by concrete tricks in each specific case) that allowed to resolve a number of questions whose formulations are extremely natural and require almost no math background.
We will discuss a more sophisticated understanding of coordinate systems, with many more (or even "much", that is with a continuum of) coordinate functions, and with a certain type of structures on the set of these functions. The ultimate goal is always replacing the statement that "there exists a diffeomorphism such that..." by proving that two geometrically well-defined sets coincide.
At first glance, the structures that arise here may seem rather unnatural and not so well connected to the original problems . I hope to be able to advocate that their use is actually well-justified. For good or bad, there is essentially the same "perverted" scheme of thinking (of course, augmented by concrete tricks in each specific case) that allowed to resolve a number of questions whose formulations are extremely natural and require almost no math background.
17.1.2005 Yair Glasner, University of
Illinois at Chicago
Title:On infinite primitive permutation groups.
Abstract
This is a joint work with Tsachik
Gelander.
We investigate finitely generated groups by studying their actions on sets. Our focus is on groups that have some geometric flavor to them such as: linear groups, hyperbolic groups, mapping class groups, and more.
A group action on a set is called primitive if there is no non-trivial invariant equivalence relation on the set. One can think of primitive actions as the basic building blocks in the study of permutation groups (in much the same way that irreducible representations are the basic building blocks for linear actions). A group is called primitive if it admits a faithful primitive action on a set.
In all of the geometric settings cited above we obtain a complete classification of finitely generated primitive groups. Here is an example of a theorem that is simple to state: "A finitely generated linear group with a simple Zarisky closure is primitive." A slightly more technical statement gives a complete characterization of finitely generated primitive linear groups.
As a corollary we prove a conjecture of Neumann and Higman on the Frattini subgroup of an amalgamated free product.
The talk will be non-technical and suitable for a wide mathematical audience.
We investigate finitely generated groups by studying their actions on sets. Our focus is on groups that have some geometric flavor to them such as: linear groups, hyperbolic groups, mapping class groups, and more.
A group action on a set is called primitive if there is no non-trivial invariant equivalence relation on the set. One can think of primitive actions as the basic building blocks in the study of permutation groups (in much the same way that irreducible representations are the basic building blocks for linear actions). A group is called primitive if it admits a faithful primitive action on a set.
In all of the geometric settings cited above we obtain a complete classification of finitely generated primitive groups. Here is an example of a theorem that is simple to state: "A finitely generated linear group with a simple Zarisky closure is primitive." A slightly more technical statement gives a complete characterization of finitely generated primitive linear groups.
As a corollary we prove a conjecture of Neumann and Higman on the Frattini subgroup of an amalgamated free product.
The talk will be non-technical and suitable for a wide mathematical audience.
10.1.2005 Roman Bezrukavnikov, Northwestern University
Title: Symplectic resolutions and representation
theory.
Abstract
To make progress in representation
theory one often tries to link algebraic study of representation to
geometry of an appropriate algebraic variety (usually related to a
generalized flag variety). Our joint work with Mirkovic and
Rumynin provides such a link in the setting of representations in
positive characteristic. This allows to settle some numerical questions
of representation theory, but also yields an interesting (in my view)
geometric structure on the variety; the latter also appears in other
context, and conjecturally can be generalized to any resolution of
singularities carrying an algebraic symplectic form. I will discuss
examples and properties of this geometric structure.
3.1.2005 Kostya Khanin, Heriot-Watt University, Edinburgh, UK
Title: Renormalizations, multi-dimensional
continued fractions and KAM theory.
27.12.2004 Vadim Kaloshin, Caltech, USA
Title:Periodic and Quasiperiodic Motions
for Planar Circular Restricted 3-Body Problem with applications to
planar Sun-Jupiter-Pluto system
20.12.2004 Benny Sudakov, Princeton
University
Title:Probabilistic Reasoning and Ramsey Theory
Abstract
"Ramsey Theory" refers to a large
body of deep results in mathematics concerning the partition
of large collections. Its underlying philosophy is captured
succinctly by the statement that "In a large system complete
disorder is impossible". Since the publication of the seminal paper of
Ramsey in 1930, this subject has grown with increasing
vitality, and is currently among the most active areas in
Combinatorics. One of the most important factors in the
development of Ramsey Theory was the successful application of
the so-called "Probabilistic Method". This method was initiated more than
fifty years ago by Paul Erdos, and became one of the most powerful and widely used methods in Discrete Mathematics.
In this talk I will describe some classical results of Ramsey Theory together with recent progress on some old questions of Erdos which was made using elegant probabilistic arguments. We will also discuss the problem of converting existence arguments into deterministic constructions, in particular, the recent explicit constructions of Bipartite Ramsey graphs.
fifty years ago by Paul Erdos, and became one of the most powerful and widely used methods in Discrete Mathematics.
In this talk I will describe some classical results of Ramsey Theory together with recent progress on some old questions of Erdos which was made using elegant probabilistic arguments. We will also discuss the problem of converting existence arguments into deterministic constructions, in particular, the recent explicit constructions of Bipartite Ramsey graphs.
29.11.2004 Stephen Miller, Rutgers
UniversityTitle:Expander graphs, the Riemann Hypothesis,
and the difficulty of solving discrete logarithms
Abstract
Many cryptographic applications
rely on the presumed difficulty of solving discrete logarithms (DLOGs)
in cyclic groups, especially exotic examples such as the group of points
of an elliptic curve over a finite field. However, it is not clear
whether or not such DLOGs have equivalent difficulty for different
groups.
I will discuss a recent result in this direction which, in many circumstances, shows that different DLOG problems have equivalent difficulty. This can be practically used to verify that certain curves proposed as international standards are indeed "typical", and not chosen so as to give standards bodies an unfair advantage over the general public. In another direction, it is used to estimate the security of an upcoming, potentially widely-used anti-piracy system.
Though the proofs involve notions from graph theory and analytic number theory, an effort will be made to make this talk self-contained and suitable for a general audience. (Joint work with David Jao and Ramarathnam Venkatesan, Microsoft Research Cryptography and Anti-piracy group.)
I will discuss a recent result in this direction which, in many circumstances, shows that different DLOG problems have equivalent difficulty. This can be practically used to verify that certain curves proposed as international standards are indeed "typical", and not chosen so as to give standards bodies an unfair advantage over the general public. In another direction, it is used to estimate the security of an upcoming, potentially widely-used anti-piracy system.
Though the proofs involve notions from graph theory and analytic number theory, an effort will be made to make this talk self-contained and suitable for a general audience. (Joint work with David Jao and Ramarathnam Venkatesan, Microsoft Research Cryptography and Anti-piracy group.)
8.11.2004 Ron Aharoni, TechnionTitle:A solution of the Erdos-Menger
conjecture
valid also for infinite graphs, in the following strong form:
given sets A and B of vertices in a graph, there exists a family P of disjoint
A-B paths, and an A-B separating set S, such that S consists of a choice of
precisely one vertex from each path in P.
joint work with Eli Berger.
Abstract
An old conjecture of Erdos is proved, saying that Menger's theorem
isvalid also for infinite graphs, in the following strong form:
given sets A and B of vertices in a graph, there exists a family P of disjoint
A-B paths, and an A-B separating set S, such that S consists of a choice of
precisely one vertex from each path in P.
joint work with Eli Berger.
1.11.2004 Ze'ev Rudnick, Tel-Aviv UniversityTitle:
Eigenvalues statistics and lattice points
Organizer: Paul Biran (biran@math.tau.ac.il)