Concordia-McGill Analysis Seminar
Starting June 1, 2006, the seminar web page will move back to
this location.
WINTER 2006
Vladimir Peller (Michigan State University)
Analytic approximation of rational matrix functions
Wednesday, May 31, 2:30pm (NOTE DIFFERENT DAY)
McGill University, BH 920
Abstract: If $f$ is a rational function without poles on the unit
circle that has at least one pole in the unit disc and $g$ is its
best uniform (on the unit circle) approximation by functions analytic
in the unit disk, then $g$ is also rational and its degree is less
than the degree of $f$.
This is a classical well-known result.
In the case of matrix functions, the corresponding problem is
considerably more complicated. First of all, a best approximant
is almost never unique. It is natural to impose additional
requirements to obtain the "very best" (or superoptimal) approximant.
The next step is to give the right definition of the degree
of a rational matrix function. Finally, the problem is how to
estimate the degree of the superoptimal approximant in terms of
the degree of the initial matrix function.
It turns out that for ``almost all'' rational matrix functions a
result similar to the scalar one holds. However, it can happen that
the degree of the superoptimal approximant can jump. In the case of
$2\times 2$ matrix functions, if the degree of the initial rational
matrix function is $k$, then the degree of the superoptimal
approximant is at most $2k-3$. This is the best possible estimate.
Andrew McIntyre (Concordia, CRM-ISM)
Determinant of Laplacians on Riemann surfaces
Friday, April 7, 1:00pm (NOTE EARLIER TIME)
Concordia University, LB 921-4
Abstract: The Laplacian acting on a line bundle over a Riemann
surface has a regularized determinant which arises in
geometry and in physics. In the late '80s it was shown
to have a product expansion, which is essentially
equivalent to the Selberg trace formula, i.e. the
functional equation for the Selberg zeta function.
I will discuss a new generalization of this product
expansion which reflects the holomorphic structure of
the relevant moduli spaces, and describe ongoing work
relating it to geometric invariants.
Roman Dwilewicz (University of Missouri at Rolla)
Cauchy-Riemann Theory. A short introduction.
Thursday, March 30, 2:00pm
Concordia University, LB 921-4
Abstract:Cauchy-Riemann (CR) theory nicely combines Complex Analysis,
Partial Differential Equations, Geometric Analysis,
Geometry (Algebraic and Differential) and other areas.
In the talk I present basic CR problems, their connections
to the above mentioned areas, and state some approximation
and extension theorems.
Zbigniew Slodkowski (University of Illinois at Chicago)
Invariant Extensions of Holomorphic Motions
Friday, March 24, 2:00pm
Concordia University, LB 921-4
Abstract:
Holomorphic motions are isotopies of injections of a subset of the Riemann
sphere into the Riemann sphere that depend holomorphically on the parameter.
(The injections themselves are usually not holomorphic.) This notion arose
naturally in the context of dynamics of rational maps, where it was introduced
by Mane, Sad and Sullivan. Holomorphic motions found applications in other
areas as well, most notably in the Teichmuller theory. For most of these
applications, questions concerning extensibility of holomorphic motions,
in particular existence of invariant equations are crucial. A series of
problems in this direction was posed by Sullivan and Thurston.
C. Zhu (University of Montreal)
Approximation by rational functions in weighted Hardy spaces
Friday, March 17, 2:30pm
McGill University, Burnside Hall 920
Abstract:
In weighted Hardy spaces over the unit disc with weight functions
satisfying Muckenhoupt's (Ap) condition, we consider a complete
system of rational functions and study the order of the approximation
by linear combinations of the elements in the system.
Michael Levitin (Heriot-Watt University)
Symmetry tricks in spectral geometry
Friday, March 10, 2:00pm
McGill University, Burnside Hall 920
Abstract:
We show the use of symmetry tricks in
solving two problems of spectral geometry. One is a
partial case $g=2$ of the question ``How large can
the first eigenvalue be on a surface of genus $g$?''.
Another is a modification of the famous Mark Kac's
question, ``Can one hear the shape of a drum?''. The
questions (or, rather, our answers) are surprisingly
related.
Alexander Strohmaier (Universitaet Bonn)
Analytic Continuation of Resolvent Kernels on noncompact
Symmetric Spaces
Monday, March 6, 2:30pm
McGill University, Burnside Hall 920
Abstract:
We investigate special situations when the resolvent kernel
of the Laplace operator on a manifold X admits a meromorphic
continuation across the spectrum.
It is well known that for euclidean space the resolvent kernel extends
to a holomorphic function on a double cover of the complex plane in case the
dimension is odd, and a logarithmic cover otherwise. We show that a similar
statement holds if X=G/K is a symmetric space of noncompact type. In this
case the answer depends on the rank of X rather than the dimension.
In case G has only one conjugacy class of Cartan subalgebras the resolvent
kernel extends to a holomorphic function on a branched cover of the complex
plane with the only branching point being the bottom of the spectrum.
Sidney Trudeau (McGill University)
On permuted sums of lacunary sequences
Friday, March 3, 2:30pm
McGill University, Burnside Hall 920
Abstract:
Many results that hold for lacunary sequences with large ratio
extend to sequences with smaller ratio by realizing the latter as finite
unions of the former. We show that while a set of permuted sums is a
Littlewood-Paley p-set for large enough ratio, the result does not
immediately extend to smaller ratios.
Mohammad Sababheh (McGill University)
Contructive proofs of Hardy-type inequalities
Monday, February 27, 2:30pm
McGill University, Burnside Hall 920
Abstract:
We discuss the general method of proving Hardy inequalities.
In particular, we discuss some constructions related to the proof of the
Littlewood conjecture in some detail, and then we use this to prove some
Hardy-type inequalities.
Yuri Gliklikh (Voronezh State University)
Necessary and sufficient conditions for global
in time existence of solutions of ordinary,
stochastic and parabolic differential equations
Friday, February 17, 2:00pm
Concordia University, LB 921-4
Abstract:
We derive necessary and sufficient conditions for global in time
existence of solutions of ordinary differential, stochastic differential
and parabolic equations. The conditions are formulated in terms of
complete Riemannian metrics on extended phase spaces (conditions
with two-sided estimates) or in terms of derivatives of proper functions
on extended phase spaces (conditions with one-sided estimates).
Ivo Klemes (McGill University)
Alexandrov's inequality and some conjectures on Toeplitz Matrices
Friday, February 10, 2:30pm
McGill University, Burnside Hall 920
Abstract:
We prove some sharp determinant inequalities
for certain "Toeplitz-like" matrices over C. The method
involves Alexandrov's inequality for mixed discriminants.
The motivation of the problem is Littlewood's conjecture
on the L1 norm of exponential sums. In certain cases, we prove
analogous determinant results over Zp using polarized
Bazin-Reisz-Picquet identities.
Javad Mashreghi (Universite Laval)
On the derivative of Blaschke products
Friday, February 3, 2:30pm
McGill University, Burnside Hall 920
Abstract (pdf file)
Vitali Bergelson (Ohio State University)
Ergodic Theorems Along Polynomials
Friday, January 20, 2:15pm (Note earlier time)
McGill University, Burnside Hall 920
Abstract:
Various recurrence and convergence results obtained in recent
years indicate that dynamical systems exhibit regular behavior along
polynomial times. While these results were mainly motivated by
applications to number theory and combinatorics, this phenomenon deserves
attention from the point of view of potential applications to physics.
For example, Poincare recurrence theorem as well as convergence theorems of
von Neumann and Birkhoff type hold along any sequence of the form p(n),
n=1,2,... where p(n) is a polynomial with integer coefficients satisfying
p(0) = 0, and it would be of interest to give physical interpretation of
these facts. After reviewing some known results we shall discuss the
ntriguing dichotomy between theorems related to polynomial and
exponential behavior. The last part of the talk will be devoted to open
problems and conjectures.
Paul Gauthier (Universite de Montreal)
Overconvergence of Taylor series
Friday, January 13, 2:30pm (To be confirmed)
McGill University, Burnside Hall 920
FALL 2005
Dimiter Dryanov (Concordia)
Canonical Sets in Approximation Theory
Friday, December 9, Time 2:00pm
Concordia University, LB 540 (Library Buildling, downtown)
Abstract:
A canonical set permits construction of the solution of
a given problem by using data on this set. According
to a result due to Markoff and Bernstein, the best
polynomial L1-approximant to a function, under some
conditions, is in fact its Lagrange interpolant with
nodes the set of the zeros of a Chebyshev polynomial of
second kind which is a canonical set for the problem in
the univariate case. I shall present results on best
multivariate L1-approximation by blending functions which
are kernels of differential operators. The best approximants
are characterized in terms of canonical sets and they are
constructed as new transfinite Hermite interpolants on
the canonical sets. I shall discuss also a multivariate
extension of a well known Duffin and Schaeffer polynomial
inequality in terms of canonical sets. Canonical set
constructions can be used to obtain new cubature formulae for
numerical integration and for optimal recovery of multivariate
functions.
Maria Roginskaya (Chalmers University of Technology)
Singularity of pluriharmonic measures on the unit sphere
Monday, November 14, Time 2:30pm
Concordia University, LB 540 (Library Buildling, downtown)
Abstract:
A measure on the sphere in C^n is called pluriharmonic if its
harmonic extension inside the ball is pluriharmonic, i.e. is harmonic on
any complex line. The existance of a (positive) singular pluriharmonic
measure implies existance of a non-trivial inner function and has been
proven by A.B.Aleksandrov in 1984. On the other hand a pluriharmonic
measure can not have the lower Hausdorff dimension less than 2n-2. In this
talk I want to discuss different aproaches to the lower estimate on the
Hausdorff dimesnion of a pluriharmonic measure (both positive and signed
cases). In particular I show that the aforementioned construction of a
singular measure results in the measure of the lower Hausdorff dimension
2n-1 (largest possible).
Dimiter Vassilev (UC Riverside)
Asymptotic behavior and Lp space regularity of solutions to non-linear
elliptic equations related to sharp Hardy-Sobolev inequalities and
applications
Friday, November 11, Time 2:30pm
McGill University, BH 920
Abstract:
We consider the problem of finding the extremals of some Hardy-Sobolev
inequalities. The solution of this problem requires certain results
on the Lp regularity and asymptotic behavior of the extremals, which
are of independent interest. We present the latter in a more general
setting, which will allow applications to the Yamabe equation on a
non-compact manifold.
Andrei Comech (Texas A&M University)
Solitary waves in nonlinear Hamiltonian systems
Friday, October 28, Time 2:00pm
Concordia University, LB 540 (Library Buildling, downtown)
Abstract:
We are interested in properties of solitary wave
solutions in nonlinear Hamiltonian systems, such as
nonlinear Schrodinger, Klein-Gordon, and (generalized)
Korteweg -- de Vries equations.
In particular systems, we consider stability of solitary
waves and properties of global attractors.
We will start with reviewing the history of the subject.
Our ultimate goal:
To prove that the attractor of nonlinear dispersive equations
is finite-dimensional and consists of the set of all solitary
waves. That is, that each finite energy solution approaches
the set of ``nonlinear eigenfunctions''.
Emmanuel Fricain (Lyon & McGill)
Bases de noyaux reproduisant dans les espaces de De-Branges
Friday, October 21, Time 2:30pm
McGill University, BH 920
Georgi Karadzhov (Bulgarian Academy of Sciences, Concordia)
Carleson type theorems for certain convolution operators
Friday, October 7, Time 2:00-3:00pm
Concordia University, LB 540 (Library Buildling, downtown)
Maciej Zworski (UC Berkeley)
Fractal Weyl laws for semiclassical resonances
Friday, September 23, Time 2:00-3:00pm
McGill, Burnside Hall 920
Klaus Schmidt (Vienna)
Invariant Sets and Measures of Toral Automorphisms
Friday, September 2, Time 2:30-3:30pm
McGill, Burnside Hall 920
Abstract: Hyperbolic toral automorphisms have an abundance of closed
invariant subsets and invariant probability measures, and the
construction of these objects (e.g. by using Markov partitions) is well
understood. In contrast, invariant sets and measures of nonhyperbolic
toral automorphisms have very surprising properties and their
construction presents many open questions.
SUMMER 2005
Scott Rodney (McMaster)
Properties of Weak Solutions to Sub-Elliptic Boundary Value and
Neumann Problems and the Necessity of Sobolev and Poincare
Inequalities.
Friday, July 29, Time 2:30-3:30
Concordia (Library building, 5th floor) Room LB 540
Abstract:
In a recent paper of Eric Sawyer and Richard Wheeden it is
proved that an operator is L^q subelliptic (weak solutions can be
shown to be Holder continuous of order \alpha) provided one assumes a
doubling and containment condition on the metric of Fefferman and
Phong, the e existence of a family of smooth cut off functions relative
to Fefferman&Phong balls, and Sobolev and Poincare inequalities. I
will be concerned with the necessity of the Sobolev and Poincare
inequalities as well as the existence of weak solutions to certain
boundary value problems.
Past Years' Seminars:
2004/2005 Seminars
2003/2004 Seminars
2002/2003 Seminars
2001/2002 Seminars
2000/2001 Seminars
1999/2000 Seminars
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