Location:
Time:
Tuesday 17:00-18:00
Seminar web page: http://www.math.mcgill.ca/student_seminars
Seminar email:student_seminars@math.mcgill.ca
Organizers:
Eugene Kritchevski,
office: 1029 Burnside Hall, McGill, email: ekritc@math.mcgill.ca
Ivo Panayotov, office: 1029 Burnside Hall, McGill, email: ipanay@math.mcgill.ca
The seminar is sponsored by
Centre de recherches mathématiques (CRM)
Date: May 22, 2007
Speaker: Isidore Fleischer,
CRM
Title: Limits `Continuity Space` -- an expose
Abstract:
"Continuity space" is an invention of Ralph Kopperman:
it results from
weakening the metric axioms by dropping symmetry
and passing to a poset in place
of the reals. The
resulting structure has been applied to domain theory, structure spaces,
hull-kernel topology, etc. This paper will explore the
extent to which these result are new.
Important: there is a location change for
this talk at Université
de Montréal, Pavillon André Eisenstadt, room 4336
This
seminar has three main purposes:
(1) To learn
about topics in Probability and Analysis which are not necessarily covered in
standard classes
(2) To explore
applications of Probability and Analysis in mathematical physics and other
areas in mathematics
(3) To bring
together students who use Analysis or Probability theory in their work
We
plan to have two types of talks:
(1) expository
talks explaining known (but not too well-known) material
(2)
presentations on original research
The
seminar is mainly intended for students, but postdocs
and professors are welcome to participate.
We challenge the
speakers to maintain their presentation accessible to anyone with basic
knowledge in probability
and measure
theory. Please feel free to contact the organizers if you would like to give a
talk. Also, if you would
like to hear a talk
on a particular subject, we will do our best to find a speaker.
Schedule for the Winter 2007
term
Date: February 6, 2007
Speaker: Claude Gravel, UdeM
Title: Branching Processes: the fundamentals
Abstract:
First, we define branching processes and we derive the basic results
without
using Markov chains and martingales. Only some ideas from probability
are
required like generating Functions. Second, we make a brief comment on age
dependant
branching processes and size-dependant branching processes.
Date: February 13, 2007
Speaker: Eugene Kritchevski, McGill
Title: The Magic of Cauchy Distributions
Abstract:
A random variable has a Cauchy distribution C(x,y),
y>0, if its probability
density
is given by f(t)=y/((x-t)^2+y^2)/pi. In this talk I will try to defend the
point of
view
that the Cauchy distribution is at least as fundamental as the Gaussian. First,
I
will derive the main properties of Cauchy random variables concerning
fractional
linear
transformations and linear combinations of independent Cauchy random
variables, and I will explain the meaning of
these properties in terms of the Dirichlet
problem in the upper half plane. Second, I
will attempt to give an intuitive explanation
of the central limit theorem for the Cauchy law:
for an i.i.d. sequence of random
variables
(Xn), their average(X1+X2+…+Xn)/n
is close to Cauchy, for large n. Third,
if time permits, I will discuss some
spectacular properties of Cauchy distributions in the
spectral theory of random Schrödinger operators.
Date: February 20, 2007
STUDY BREAK – THE SEMINAR IS CANCELLED
Date: February 27, 2007
Speaker: Eugene Kritchevski, McGill
Title: The Magic of Cauchy Distributions: PART 2
Date: March 6, 2007
Speaker: Igor Wigman, CRM
Title: On the distribution of the zeroes of random trigonometric polynomials.
Abstract: The fundamental subject of the current talk is to study the number
of
solutions of a "random" equations, or, equivalently, the number of
roots of
a
"random" functions lying in some ensemble. The most studied example
of
this
kind is the ensemble of random polynomials where we put some distribution
measure (say, Gaussian) on the coefficients.
Another important example is the
ensemble of trigonometric polynomials, which
could be either stationary or not.
This
study of this problem was initiated by Littlewood and
Offord, Erdos and Offord.
However,
it was not before Mark Kac, that this problem got a
real breakthrough.
In
this talk I will present the results by the above researchers, as well as those
by
Qualls,
Dunnage, Das, Ibragimov and
Maslova, Maslova and Farahmand.
Date: March 13, 2007
Speaker: Hugues Lapointe, UdeM
Title: Classical and quantum ergodicity
Abstract:
The
discrete
spectrum of eigenvalues. An interesting problem is to
study the asymptotic
properties
of the eigenfunctions. In this talk, I will be
interested in the case where
the geodesic flow on M is ergodic. This property has a deep impact on the behavior
of the eigenfunctions,
called quantum ergodicity. I will explain this
result, due to
Schnirelman, Zelditch and Colin-de-Verdière,
and give some ideas of the proof.
Date: March 20, 2007
Speaker: Marco Bertola,
Title: Random Matrices: a short introduction with emphasis on connections with
orthogonal
polynomials and related topics.
Abstract:
In this talk I will introduce the prototype of many random matrix models;
this is simply a probability measure of
particular type on the space of Hermitean
matrices
of size N. I will touch upon some of the far reaching applications of the
theory -besides the questions of probabilistic
nature- such as enumerative geometry,
number theory, string theory.
I
will explain how the theory of orthogonal polynomials on the real line plays a
role
(and
a prominent one) in the study of the topic and why their asymptotic analysis
(for
large degrees) is crucial in deriving some “universality theorems” which play
the role of “central limit theorems” in this
context.
Both
the orthogonal polynomials and their asymptotic analysis are connected to the
study
of certain Riemann surfaces (algebraic curves in some cases) referred to as
“spectral
curves” of the model; I will explain how this connection arises and how
it
is useful in addressing certain generalizations of orthogonal polynomials to
pseudo-orthogonal
polynomials and their asymptotics.
I
will not assume any prior knowledge of the subject of random matrices, but I
will
liberally use (elementary) measure theory and
L^2 spaces, linear algebra
(eigenvalues etc.) and elementary theory of Riemann
surfaces.
Date: March 27, 2007
Speaker: Alexandre Girouard, UdeM
Title: Spectrum of Neumann and Dirichlet Laplacian on Bounded Domains
Abstract: We
will review some of the most basic (and fundamental) properties of
the spectrum of
the
bounded domain.
These include discreteness of the spectrum, regularity of the
corresponding eigenfunctions and a bit more.
Date: April 3, 2007
Speaker: Dmitry Jakobson, McGill
Title: Limits of eigenfunctions on arithmetic flat tor. (PART I)
Abstract: We
discuss the classification of possible limits of
on arithmetic
flat tori R^n/Z^n. We
explain how a result from number theory about
solving systems
of Pell’s equations helps to show that limits in dimension n are “no
worse” than eigenfunctions in dimension (n-2), and how to construct
limits in
dimension n that
are “as bad” as eigenfunctions in dimension (n-3). We
shall also
discuss L^p bounds for eigenfunctions and
some related problems including a
generalization
of 2-dimensional results of Cooke and Zygmund to dimentions 3 and 4.
Date: April 10, 2007
Speaker: Eugene Kritchevski,
Title: Hierarchical