Analysis Seminar
Seminars are usually held on Mondays or Fridays at Concordia or at McGill
For suggestions, questions etc. please contact Dmitry Jakobson
(jakobson@math.mcgill.ca),
Alina Stancu (stancu@mathstat.concordia.ca),
or Alexey Kokotov (alexey@mathstat.concordia.ca)
SUMMER 2013
Friday, June 21, 13:30-14:30, Burnside 920
Guofang Wang (Freiburg)
Mass and geometric inequalities
Absract: In this talk, we start from a new mass for
asymptotically flat manifolds and prove a positive mass
theorem and a Penrose type inequality for asymptotically
flat graphs, by using classical Alexandrov-Fenchel
inequalities in the euclidean space. Then we define a new
mass for asymptotically hyperbolic manifolds and prove
a positive mass theorem and a Penrose type inequality
for asymptotically hyperbolic graphs, by establishing
new weighted Alexendrov-Fenchel inequalities in hyperbolic space
WINTER 2013
Monday, January 14, 13:30-14:30,
Burnside 920
Phan Thanh Nam (Universite de Cergy-Pontoise)
Bogoliubov spectrum of interacting Bose gases
Abstract:In 1947 Bogoliubov predicted that the excitation
spectrum of some certain N-body Bose gas can be approximated by that
of the so-called Bogoliubov Hamiltonian in Fock space. This prediction
was proved recently for systems with short range potentials by Seiringer
(2011) and Grech-Seiringer (2012). In this talk, we shall give abstract
conditions on which Bogoliubov's theory is valid. Then we apply the method
to some Coulomb systems. This is joint work with Mathieu Lewin, Sylvia
Serfaty and Jan Philip Solovej.
Wednesday, February 6, 13:00-14:00, Burnside 1234
Liz Vivas (Purdue/IMPA)
Geodesics in the space of Kahler metrics
Abstract:
Let (X,\omega) be a compact Kahler manifold. It is known that the set
H of Kahler forms cohomologous to \omega has the natural structure of
an infinite dimensional Riemannian manifold. We address the question
whether any two points in H can be connected by a smooth geodesic, and
show that the answer, in general, is no. This is joint work with
Laszlo Lempert.
Friday, February 8, 13:30-14:30,
Burnside 920
Sergei Tabachnikov (Penn State)
Tire tracks geometry, hatchet planimeter, Menzin's conjecture,
and complete integrability
Abstract:
This talk concerns a simple model of bicycle motion: a bicycle is a
segment of fixed length that can move in the plane so that the velocity
of the rear end is always aligned with the segment. The trajectory of the
front wheel and the initial position of the bicycle uniquely determine
its motion and its terminal position; the monodromy map sending the initial
position to the terminal one arises. This circle mapping is a Moebius
transformation, a remarkable fact that has various geometrical and dynamical
consequences. Moebius transformations belong to one of the three types:
elliptic, parabolic and hyperbolic. I shall outline a proof of a 100 years
old conjecture: if the front wheel track is an oval with area at least Pi
then the respective monodromy is hyperbolic. I shall also discuss the
related Backlund-Darboux transformation on curves, in the continuous and
discrete settings, its complete integrability, and its unexpected relation
with the binormal (smoke ring, filament) equation, a much studied
completely integrable PDE.
Joint Analysis and Probability seminar
Monday, February 11, 13:30-14:30,
Burnside 920
Linan Chen (McGill)
Gaussian free field, random measure and KPZ on R^4.
Abstract:
A highlight in the study of quantum physics was the work of Knizhnik,
Polyakov and Zamolodchikov (1988), in which they proposed a relation
(KPZ relation) between the scaling dimension of a statistical physics
model in Euclidean geometry and its counterpart in the random geometry.
Recently, Duplantier and Sheffield used the 2D Gaussian free field to
construct the Liouville quantum gravity measure on a planar domain, and
gave the first mathematically rigorous formulation and proof of the KPZ
relation in that setting. We have applied a similar approach to generalize
part of their results to R^4 (as well as to R^(2n) for n>=2). To be
specific, we construct a random Borel measure on R^4 which formally
has the density (with respect to the Lebesgue measure) given by the
exponential of an instance of the 4D Gaussian free field. We also
establish the KPZ relation corresponding to this random measure.
This is joint work with Dmitry Jakobson.
Joint Analysis and Mathematical Physics
Thursday, February 14, 15:00-17:00,
Burnside 1120
Michal Wrochna (University of Gottingen)
Hadamard states: from solutions of the Klein-Gordon equation to
renormalised quantum fields
Abstract: In Quantum Field Theory on curved space-time or in external
potentials, a central problem is to find solutions of the underlying
Klein-Gordon equation with prescribed singularity structure. Such
solutions are used to construct quantum states which generalize the
Minkowski vacuum state. In the first part of the talk, I will explain the
original motivation and modern formulations of the problem in terms of
microlocal analysis. In the second part, I will present two recent
constructions. This will include the static case, in which spectral
methods are available, and the case of asymptotically homogeneous
space-times, where pseudo-differential methods will be used instead (based
on a joint work with C. Gerard). As an outlook, I will comment on
perspectives in systems with Coulomb potential.
Joint seminar with Probability
Monday, February 18, 14:00-15:00,
Burnside 920
Dan Stroock (MIT)
A Probabilist's Thoughts about a Theorem of L. Hormander
Abstract:
pdf
Joint Analysis and Mathematical Physics
Thursday, February 21, 15:00-17:00,
Burnside 1120
Falk Linder (Hamburg University)
Perturbative Approach to AQFT and KMS states
Abstract: I will introduce the notions and important concepts
of the functional approach to QFT, established in the last decade by
Brunetti, Duetsch and Fredenhagen. The explicit construction of the
free theory will be done and the relations to the canonical approach
are shown. The inductive construction of the interacting theory will
be presented, following the ideas of Epstein and Glaser. The problem
of the existence of Vacuum and KMS states for the perturbatively
constructed, interacting theories is discussed.
Friday, February 22, 13:30-14:30,
Burnside 920
Eric Schippers (Manitoba)
A correspondence between conformal field theory and Teichmuller
theory
Abstract:
The rigorous construction of two-dimensional conformal field theory,
according to a program initiated by Segal and Kontsevich, requires results
in geometry, topology, algebra and analysis. One of the analytic problems
is the construction of a complex structure on the moduli space of
Riemann surfaces with boundary parametrizations, and the holomorphicity
in this moduli space of the operation of sewing. David Radnell and I
discovered that this moduli space is the quotient of quasiconformal
Teichmuller space by a discrete group action, which led to the solution
of these problems and others.
In this talk, I will give a non-technical introduction to
quasiconformal Teichmuller theory, sketch the correspondence between
the moduli spaces, and indicate some of the consequences for conformal
field theory and Teichmuller theory. Joint work with David Radnell
(American University of Sharjah) and Wolfgang Staubach (Uppsala University).
Joint Analysis and Mathematical Physics
Thursday, February 28, 15:00-17:00,
Burnside 1120
Laurent Bruneau (University of Cergy-Pontoise)
Applications of repeated interaction systems
Abstract: After describing the general philosophy of repeated
interaction systems we will present two concrete models:
1. return to equilibrium in a QED cavity (one-atom maser experiment)
2. a toy model describing the establishment of a dc current in a
tight-binding band.
Friday, March 1, 13:30-14:30, Burnside 920
Alexei Penskoi (Moscow University)
Constructing explicitly parametrized minimal
tori in spheres via Takahashi's theorem
Abstract: It is well-known that a surface
isometrically immersed in a Euclidean
space by harmonic functions is minimal.
Takahashi generalized this result
to the case of an isometric immersion of a surface
by Laplace-Beltrami eigenfunctions
with the same eigenvalue. It turns out that
in this case the image is minimal in
a standard sphere. Such surfaces carry
metrics that are extremal for the normalized
eigenvalues. This motivates the following question:
can one use Takahashi's theorem to construct
explicitly minimal surfaces in spheres in order
to find new extremal metrics?
Mini-course in Analysis and Mathematical Physics
Jan Derezinski (Warsaw)
March 13, 14, 20, 21, 27; 15:00-17:00, Burnside 1120
Operators and Perturbations
Abstract:
The main purpose of the course is to develop general theory
of perturbations of linear operators on Hilbert spaces, with the
emphasis on Schrodinger operators. Many concrete examples will be
described in detail. These examples illustrate a number of interesting
points relevant for quantum mechanics and probability theory.
List of subjects that will be (partially) covered:
1) Reminder of basic spectral theory
- Unbounded operators
- Closed operators
- Spectrum
- Pseudoresolvents
- Unbounded operators on Hilbert spaces
- (Essential) self-adjointness
- Relative boundedness
- Scale of Hilbert spaces
- Closed and closable positive forms
- Relative form boundedness
- Friedrichs extensions
2) Reminder of basic harmonic analysis and its applications
- Young inequality
- Sobolev inequalities
- Application: self-adjointness of Schrodinger operators
3) Momentum and Laplacian in 1 dimension
- Momentum on half-line
- Momentum on an interval
- Laplacian on half-line
- Laplacian on an interval
4) Orthogonal polynomials
- Orthogonal polynomials in weighted L2 spaces
- Self-adjointness of Sturm-Liouville operators
- Classical orthogonal polynomials as eigenvectors of certain
Sturm-Liouville operators
- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
5) Finite rank perturbations and their renormalization
- Aronszajn-Donoghue Hamiltonians
- Delta potentials
- Friedrichs Hamiltonians
- Bound states and resonances of Friedrichs Hamiltonians
- Exponential decay from a unitary dynamics
6) Potential 1/|x|2
- Hardy inequality
- Modified Bessel equation
- Bessel equation
- Operator -d2+(m2-1/4)/|x|2.
Friday, March 15, 13:30-14:30, Burnside 920
Alberto Enciso (Madrid)
Knotted vortex tubes in steady Euler flows
Abstract: In this talk we will review recent results on the
existence of knotted and linked thin vortex tubes for steady solutions
to the incompressible Euler equation in R3. More precisely,
given a
finite collection of (possibly linked and knotted) disjoint thin tubes
in R3, we will see that they can be transformed with a
Cm-small
diffeomorphism into a set of vortex tubes of a steady solution to the
Euler equation that tends to zero at infinity. The interest in the
existence of steady knotted thin vortex tubes can be traced back to
Lord Kelvin, and in fact these structures have been recently realized
experimentally. The talk is based on joint work with D. Peralta-Salas.
Monday, March 18, 13:30-14:30, Burnside 920
Junfang Li (University of Alabama, Birmingham)
Hardy inequalities on mean convex domains
Abstract: I will report a recent joint work with Roger Lewis and
Yanyan Li. In this work, we prove that Hardy inequalities with a sharp
constant hold on weakly mean convex domains. Moreover, we show that the
weakly mean convexity condition cannot be weakened. We also prove
various improved Hardy inequalities on mean convex domains along the
line of Brezis-Marcus. I will also outline several related open questions.
Wednesday, March 20, 13:30-14:30, Burnside 708
Nam Le (Columbia)
The linearized Monge-Ampere equation and its geometric applications
Abstract:
In this talk, we will introduce the linearized Monge-Ampere equation
and discuss its boundary regularity in joint works with Ovidiu Savin
and Truyen Nguyen.
Linearized Monge-Ampere equation is an interesting combination of the
linear elliptic equation and the Monge-Ampere equation.
Though highly degenerate, linearized Monge-Ampere equation has the
same regularity results as those of
the Poisson equation. Though linear, it has the same challenging aspects
of the fully nonlinear Monge-Ampere equation.
We will also describe applications of our regularity results to fourth
order, fully nonlinear geometric partial differential equations
such as affine maximal surface and Abreu equations in affine and
complex geometry.
Friday, March 22, 13:30-14:30, Burnside 920
Ivana Alexandrova (Albany)
Resonances in Scattering by Two magnetic Fields at Large Separation
and a Complex Scaling Method
Abstract:
We study the quantum resonances in magnetic scattering in two
dimensions. The scattering system consists of two obstacles by which
the magnetic fields are completely shielded. The trajectories trapped
between the two obstacles are shown to generate the resonances near the
positive real axis when the distance between the obstacles goes to
infinity. The location of the resonances is described in terms of
the backward apmlitues for scattering by each obstacle. A difficulty
arises from the fact that even if the supoorts of the magnetic fields
are largely separated from each other, the corresponding vector
potentials are not expected to be well seperated. To overcome this, we
make use of a gauge transformation and develop a new type of complex
scaling method. The obtained result heavily depends on the magnetic
fluxes of the solenoids. This indicates that the Aharonov-Bohm effect
influences the location of the resonances. This is joint work with
Hideo Tamura.
Monday, March 25, 13:30-14:30, Burnside 920
Renjie Feng (McGill)
The supremum of L^2 normalized random holomorphic fields
Abstract:
We prove that the expected value and median of the supremum
of L^2 normalized random holomorphic fields of degree n on
m-dimensional Kahler manifolds are asymptotically of order
\sqrt{m log(n)}. The estimates are based on the entropy
methods of Dudley and Sudakov combined with a precise analysis
of the relevant distance functions and covering numbers using off-diagonal
asymptotics of Bergman kernels.
Wednesday, April 10, 13:00-14:00, Burnside 708
Philippe Poulin (United Arab Emirates University)
Weighted Paley-Wiener Spaces and MC-Spaces
Abstract: In their study of weighted Paley-Wiener spaces,
Lyubarskii and Seip exhibited structural properties shared by a larger
class of de Branges spaces, which we will call the MC-spaces. In this talk
we will re-state their results in their full generality. If time permits,
we will show how their method can be used for getting concrete
realizations of the MC-spaces.
Friday, April 12, 13:30-14:30, Burnside 920
Leonid Parnovski (University College, London)
Spectral theory of multidimensional periodic and almost-periodic
operators: Bethe-Sommerfeld conjecture and the integrated density
of states.
Abstract:
I will make a survey of recent results on the spectrum of periodic and,
to a smaller extent, almost-periodic operators. I will consider two
types of results:
1. Bethe-Sommerfeld Conjecture. For a large class of multidimensional
periodic operators the numbers of spectral gaps is finite.
2. Asymptotic behaviour of the integrated density of states of periodic
and almost-periodic operators for large energies.
Friday, April 26, 13:30-14:30, Burnside 920
Andrew McIntyre (Bennington College)
Chern-Simons invariants, determinant of Laplacian, and tau functions
Abstract:
This is joint work with Jinsung Park, Korea Institute for Advanced Study.
Suppose X is a compact 2-manifold, of fixed genus 2 or more, with
hyperbolic metric. It is known (Belavin-Knizhnik, Bost, Takhtajan-Zograf)
that the determinant of the Laplacian on X is the modulus squared of a
holomorphic function F on the Teichmuller space of such X, times a
"conformal anomaly". It has been gradually understood (Polyakov, Krasnov,
Takhtajan-Teo, Schlenker) that the conformal anomaly is the exponential
of a regularized volume of a certain infinite-volume hyperbolic 3-manifold
M whose conformal boundary is X. It is a result of Zograf that the function
F may be written as a Selberg zeta-like product for the 3-manifold M.
(These results are a baby case of physicists' conjectured "holography".)
This raises the question of the meaning of the phase of F. Park realized
that the phase of F may be interpreted in terms of a regularized
Atiyah-Patodi-Singer eta invariant of M. In our work, we define a
regularized Chern-Simons invariant for M, which forms a natural
complexification of the regularized volume. We relate it to the
regularized eta invariant. The Bergman tau function, introduced
and studied by Kokotov-Korotkin, makes a surprise appearance.
Monday, May 6, 11:00-12:00, Burnside 920 (moved from May 7!)
Tom Lagatta (NYU)
Geodesics of Random Riemannian Metrics
Abstract:
In Riemannian geometry, geodesics are curves which locally minimize
lengths. In general, it is a difficult and interesting question to determine
which geodesics of a manifold are in fact globally minimizing. In settings
of non-positive curvature (e.g., hyperbolic space), the Cartan-Hadamard
theorem says that all geodesics are minimizing, so the presence of positive
curvature (e.g., sphere) is necessary to destabilize this minimization
property. With Janek Wehr, we have used the point-of-view of the particle
technique to show that for random perturbations of 2-dimensional
Euclidean space, enough positive curvature arises to destabilize
"generic" geodesics. I will present this work, as well as discuss the
extension to the more general setting of symmetric random geometries.
No background in geometry or probability will be required for this
talk, and it will be accessible to graduate students.
Monday, May 6, 13:30-14:30, Burnside 920
Thomas Hoffmann-Ostenhof (University of Vienna)
Spectral Minimal Partitions
Abstract:
Spectral minimal partitions are related to nodal domains. They have many
interesting properties. There are close relations to Courant's nodal
theorem. In terms of spectral minimal partitions the case of
equality for this theorem can be characterized.
In this talk joint work with Bernard Helffer, Susanna Terracini and
Virginie Bonnaillie-Noel will be described.
Friday, May 10, 13:30-14:30, Burnside 920
Leonardo Marazzi (University of Kentucky)
Generic properties of surfaces with cusps
Abstract:
I will talk about scattering theory for compactly supported metric
perturbations of the hyperbolic metric on non-compact finite area surfaces.
The main result I want to discuss is that generically, for these type of
perturbations, there are no embedded eigenvalues and infinitely many
resonances. I will take a closer look at this phenomenon using Fermi's
Golden Rule. This is joint work with P. Hislop and P. Perry.
Monday, May 13, 13:30-14:30 (to be confirmed)
Semyon Klevtsov (Koln)
The talk is CANCELLED
Friday, May 17, 11:00-12:00, Univ. de Montreal, Room 5183
(time and room changed!)
Egor Shelukhin (CRM)
Braids and L^p-norms of area-preserving diffeomorphisms
Abstract:
We survey results on the large-scale metric properties of groups
of volume-preserving diffeomorphisms of surfaces endowed with the
hydrodynamic L^2-metric (or more generally the L^p-metric). The simplest
such property is the unboundedness of the metric, which we establish
for the last unknown case among surfaces - the two-sphere.
Our methods involve quasimorphisms on spherical braid groups and
differential forms on configuration spaces.
This talk is based on a joint work with Michael Brandenbursky.
Friday, May 24, 13:30-14:30, Burnside 920
Joel Spruck (Johns Hopkins)
The half space property and entire minimal graphs in MxR
Abstract:
An important question is to understand when two natural objects,
for example two complete minimal hypersurfaces S1, S2 in a
Riemannian manifold N, must intersect.
In this talk I consider this question when N=MxR where M is a complete
n dimensional
Riemannian manifold, S1=Mx{0} and S2 is a properly immersed
minimal hypersurface in N.
We want to find conditions on M so that if S1 and S2 do not intersect,
then S2 is a slice
Mx{c} for some constant c. The celebrated theorem of
Bomberi-De Giorgi-Miranda,
which says that an entire positive minimal graph over
R^n must be a totally geodesic slice,
is perhaps the first such result. Another foundational result
is the Hoffman-Meeks half space
theorem which states that if S is a properly immersed minimal surface
in R^3=R^2xR+,
then S=R^2x{c} for a nonnegative constant c. Since there are
rotationally invariant minimal
hypersurfaces (catenoids) that are bounded above and below, the
Hoffman-Meeks theorem is false for M=R^n.
ANALYSIS-REALTED TALKS ELSEWHERE, WINTER 2013
CRM-ISM colloquium
Friday, February 1, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O.,
salle SH-3420, 16:00-17:00
Elliott Lieb
(Princeton)
Proof of a 35 Year Old Conjecture for the Entropy of SU(2) Coherent
States, and its Generalization
Abstract:
35 years ago Wehrl defined a classical entropy of a quantum density matrix
using Gaussian (Schrodinger, Bargmann, ...) coherent states.
This entropy, unlike other classical approximations, has the virtue
of being positive. He conjectured that the minimum entropy occurs for
a density matrix that is itself a projector onto a coherent state and
this was proved soon after. It was then conjectured that the same thing
would occur for SU(2) coherent states (maximal weight vectors in a
representation of SU(2)). This conjecture, and a generalization of it,
have now been proved with J.P. Solovej. (arxiv: 1208.3632).
After a review of coherent states in general, a summary of the proof
will be given. Obviously, one would like to prove similar conjectures
for SU(n) and other Lie groups. This is open and the audience is invited
to join the fun. Another question the audience is invited to think
about is the meaning of all this for group representation theory. If
this conjecture is correct, it must have some general significance.
CRM-ISM colloquium
Friday, February 15,
Universite de Montreal, Pav. Andre-Aisenstadt, 2920, chemin de la Tour,
SALLE 5340, 16:00-17:00
Nilima Nigam
(Simon Fraser University)
Eigenproblems, numerical approximation and proof
Abstract: In this talk, we investigate the role of numerical
analysis and scientific computing in the construction of rigorous proofs
of conjectures. We focus on eigenproblems, and present recent progress on
three unusual, conceptually simple, eigenvalue problems. We explore how
validated numerics and provable convergence and error estimates are
helpful in proving theorems about the eigenvalue problems. The first of
these problems concerns sharp bounds on the eigenvalue of the
Laplace-Beltrami operator of closed Riemannian surfaces of genus higher
than one. One may ask: for a fixed genus, and a given fixed surface area,
which surface maximizes the first Laplace eigenvalue? The second of these
concerns eigenvalue problems for the Laplacian, with mixed
Dirichlet-Neumann data. If the Neumann and Dirichlet curves meet at an
angle which is Pi or larger, reflection strategies will not work. The
third problem is about the famous Hot Spot conjecture: the extrema of the
2nd Neumann eigenfunction of the Laplacian in an acute triangle will be
at the vertices.
CRM-ISM colloquium
Thursday, March 28,
Universite de Montreal, Pav. Andre-Aisenstadt, 2920, chemin de la Tour,
SALLE 5340, 16:00-17:00
Victor Guillemin (MIT)
Moser averaging
Abstract:
Moser averaging is a method for detecting periodic trajectories in
classical mechanical systems which are small perturbations of periodic
systems. (The Kepler system: the earth rotating about the sun, is
probably the most familiar example of a system of this type.)
In this talk I'll describe how, in the late nineteen seventies,
Weinstein and Colin de Verdiere adapted Moser's techniques to the
quantum mechanical setting and describe some recent applications of
their results to inverse problems.
Montreal Probability seminar
Thursday, March 28,
Concordia University, Room TBA, 16:30-17:30
Tai Melcher (University of Virginia)
Smoothness properties for some infinite-dimensional heat kernel measures
Abstract:
Smoothness is a fundamental principle in the study of measures on
infinite-dimensional spaces, where an obvious obstruction to overcome is
the lack of an infinite-dimensional Lebesgue or volume measure. Canonical
examples of smooth measures include those induced by a Brownian motion,
both its end point distribution and as a real-valued path. More generally,
any Gaussian measure on a Banach space is smooth. Heat kernel measure
is the law of a Brownian motion on a curved space, and as such is the
natural analogue of Gaussian measure there. We will discuss some recent
smoothness results for these measures on certain classes of
infinite-dimensional groups, including in some degenerate settings.
Some parts of this talk are joint work with Fabrice Baudoin,
Daniel Dobbs, and Masha Gordina.
CRM-ISM colloquium
Friday, April 12, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O.,
salle SH-3420, 16:00-17:00
Narutaka Ozawa (RIMS, Kyoto university)
Quantum correlations and Tsirelson's problem
Abstract:
The EPR paradox tells us quantum theory is incompatible with classic
realistic theory. Indeed, Bell has shown that quantum correlations of
independent bipartite systems have more possibility than the classical
correlations. To study what the possibilities are, Tsirelson has
introduced the set of quantum correlation matrices, but depending on the
interpretation of independence, there are two plausible definitions
of it. Tsirelson's problem asks whether these definitions are equivalent.
It turned out that this problem in quantum information theory is in fact
equivalent to Connes's embedding conjecture, one of the most important
open problems in theory of operator algebras. I will talk some recent
progress on Tsirelson's problem.
Mathematical Physics/Algebra seminar
Wednesday, April 17, McGill, Burnside 920, 15:00-17:00
Narutaka Ozawa (RIMS, Kyoto university)
Dixmier's Similarity Problem
Abstract:
A group G is said to be unitarizable if every uniformly bounded
representation of G on a Hilbert space is similar to a unitary
representation. Sz.-Nagy, Dixmier and Day proved that amenability
implies unitarizability, and Dixmier posed a problem whether the
converse is also true. I will report on not so recent anymore
progress on Dixmier's Similarity Problem.
This is a joint work with N. Monod.
FALL 2012
Friday, September 7, 14:30-15:30,
Burnside 920
Nadia Sidorova (University College, London)
Localisation in the parabolic Anderson model
Abstract:
The parabolic Anderson problem is the Cauchy problem for the heat
equation on the d-dimensional integer lattice with random potential.
It describes the mass transport through a random field of sinks and
sources and is actively studied in mathematical physics. We discuss,
for a class of i.i.d. potentials, the intermittency effect occurring
for such potentials, which manifests itself in increasing localisation
and randomisation of the solution.
Friday, September 14, 14:30-15:30,
Burnside 920
Dmitry Jakobson (McGill)
Nodal sets and negative eigenvalues in conformal geometry
Abstract:
This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge.
We study conformal invariants that arise from nodal sets and negative
eigenvalues of conformally covariant operators, including the Yamabe
operator (conformal Laplacian), and the Paneitz operator. We give several
applications to curvature prescription problems. We establish a conformal
version of Courant's Nodal Domain Theorem. We prove that the Yamabe operator
can have an arbitrarily large number of negative eigenvalues on any
manifold of dimension $n\geq 3$.
Monday, September 17, 14:30-15:30,
Burnside 920
Brendan Farrell (Caltech)
The Jacobi Ensemble and Discrete Uncertainty Principles
Abstract:
Our starting point concerns a discrete uncertainty principle: how
small can the support sets of a vector and its discrete Fourier transform
be? By taking a probabilistic and geometric approach we relate this question
to the third ensemble of random matrix theory, the Jacobi ensemble. We
present the limiting empirical spectral distribution of a random matrix
arising in the discrete Fourier setting and the first universality result
for the Jacobi ensemble. We discuss the relationship between these two types
of random matrices, as well an unexpected instance of universality. This
talk is partially based on joint work with László Erdős.
Monday, September 24, 14:30-15:30,
Burnside 920
Gilles Lebeau (Nice)
The talk is CANCELLED
Friday, September 28, 14:30-15:30,
Burnside 920
Brian Seguin (McGill)
Evolving Irregular Domains and a Generalized Transport Theorem
Abstract:
Well-known examples of transport theorems include the
Leibniz integral rule and a result due to Reynolds for
three-dimensional regions that convect with the motion of a continuum.
Using Harrison's recently developed theory of differential chains, I
will outline how to prove a generalized transport theorem that holds
for evolving irregular domains that may, among other things, develop
holes, split into pieces, or whose fractal dimension may evolve with
time. This result is of potential value in the calculus of variations
and continuum physics.
Wednesday, October 3, 14:30-15:30,
Burnside 1234 (NB room change!)
Elijah Liflyand (Bar Ilan)
Integrability of the Fourier transform: functions of bounded variation
Abstract:
Certain relations between the Fourier transform of a function of
bounded variation and the Hilbert transform of its derivative are
revealed. The widest subspaces of the space of functions of bounded
variation are indicated in which the cosine and sine Fourier
transforms are integrable.
Monday, October 15, 14:30-15:30,
Burnside 920
Parasar Mohanty (IIT Kanpur)
Space of completely bounded Lp multipliers and its pre-dual.
Abstract
Friday, October 19, 14:30-15:30,
Burnside 920
Spyros Alexakis (Toronto)
The Willmore functional on the space of complete minimal surfaces in
hyperbolic space: Boundary regularity and Bubbles.
Abstract:
We consider the space of complete minimal surfaces in H3
with a (free) boundary at infinity. We study the renormalized area of
such surfaces (as defined by by Graham and Witten) and show its
equivalence with the well-studied Willmore energy. We then discuss the
possible loss of compactness in the space of such surfaces with this
energy bounded above. This question has been extensively studied for
various energies in the context of closed surfaces, starting with the
classical work of Sacks and Uhlenbeck on harmonic maps. We derive
analogues of epsilon-regularity and bubbling in this setting. A key
difference (and difficulty) compared to the classical picture is a lack
of energy quantization. This is a joint work with R. Mazzeo.
Monday, October 22, 14:30-15:30,
Burnside 920
Frederic Rochon (UQAM)
Hodge cohomology of iterated fibred cusp metrics on Witt spaces
Abstract: After introducing iterated fibred cusp metrics on a
stratified space and making a quick review on intersection cohomology,
we will explain how soft analytical methods can be used to study the
L2
cohomology of such metrics. More precisely, when the stratified space
satisfies the Witt condition, we will show that the
L2 cohomology is
naturally identified with the intersection cohomology of middle
perversity. This is a joint work with Eugenie Hunsicker.
Monday, October 29, 14:30-15:30, Burnside 920
Thiery Daude (Cergy-Pontoise)
Inverse scattering at fixed energy in black hole spacetimes.
Abstract: In this talk, we shall consider massless Dirac fields
evolving in
the outer region of Reissner-Nordstrom-de-Sitter and Kerr-Newmann-de-Sitter
Black Holes, classes of spherically symmetric (resp. cylindrically
symmetric), electrically charged, spacetimes with positive cosmological
constant, exact solutions of the Einstein equations. We shall first define
the corresponding partial wave scattering matrices S(k,n), objects that
encode the scattering properties of an incoming Dirac waves having energy k
and angular momentum n. We shall then show that the metric of such black
holes is uniquely determined by the knowledge of the partial scattering
matrices at a fixed energy and for almost all angular momenta. The main
tool used to prove this result consists in complexifying the angular
momentum and in using the particular analytic properties of the
"unphysical" scattering matrix S(k,z) where z belongs now to the complex
plane. This result was obtained in collaboration with Francois Nicoleau
(Universite de Nantes).
Monday, November 5, 14:30-15:30,
Burnside 920
J. Royo-Letelier (U. Paris-Dauphine and U. de Versailles)
Two-component Bose-Einstein Condensates
Abstract: pdf
Friday, November 9, 14:30-15:30,
Burnside 920
Alexander Shnirelman (Concordia)
On the analyticity of particle trajectories in the flows of
ideal incompressible fluid.
Abstract:
Consider the flow of the ideal incompressible fluid in a
bounded domain. The velocity field is described by the Euler
equations. If the initial velocity is regular enough, then solution
exists for some time (which in the 2-dimensional case is infinite),
and is as regular as the initial condition is. However, the particle
trajectories which are obtained as a result of integration of the
velocity field are real analytic curves, in spite of just final
spatial regularity of the flow. This theorem has a long history
beginning in the work of Lichtenstein of 1925. In fact, it was really
proved only in 2012 (one proof by myself and another one by
Zheligovsky and Frisch). There is a related result about analyticity
of flow lines of a STATIONARY (time independent) solution of 2-d Euler
equations (Nadirashvili, 2012). The ideas of these proofs, and some
immediate implications will be discussed in this talk.
Monday, November 12, 14:30-15:30,
Burnside 920
Roland Bauerschmidt (UBC)
Positive definite decomposition of Green's functions
Abstract: I will show a simple method to decompose the
Green's functions
of elliptic partial differential operators, and of elliptic finite
difference operators, into integrals over positive definite and finite
range (properly supported) kernels. The method uses the finite
propagation speed of the corresponding wave equation, for differential
operators, and related properties of Chebyshev polynomials, in the
discrete case.
Friday, November 30, 14:30-15:30,
Burnside 920
Renjie Feng (McGill and CRM)
Geodesics in the space of Kahler potentials
Abstract:
It's well-known in Kahler geometry that the space of smooth
Kahler metrics in a fixed Kahler class over a polarized Kahler manifold is
formally an infinite dimensional non-positively curved
symmetric space if we endow
it with some L^2 metric, this result is proved by Semmes, Mabuchi and
Donaldson independently. This space is well-approximated by finite
dimensional space of Bergman metrics by Tian-Yau-Zelditch Theorem. It's
natural to ask whether geodesics can be approximated by Bergman
geodesics. In this talk, I will prove that the approximation of geodesics
is in C^\infty topology over the principally polarized Abelian varieties.
Friday, December 7, 12:30-13:30,
Burnside 920
Nick Haber (Stanford)
Microlocal analysis of radial points
Abstract:
Microlocal analysis relies on correspondences between quantum physics
and classical physics to give information about certain PDEs --- for
instance, linear variable-coefficient PDEs on manifolds, interpreted as
quantum systems. Foundational works of Duistermaat and Hormander establish
this framework under assumptions in which the associated classical dynamics
are well-behaved. In this talk, I present analogous results (including
propagation of singularities and a normal form) in a common setting in
which the corresponding classical dynamics are less well-behaved (in the
presence of radial points). This has applications in scattering theory as
well as analysis on spaces which are asymptotically Minkowski, hyperbolic,
and de Sitter. This work is in part joint with Andras Vasy.
Joint Mathematical Physics/Analysis seminar
Tuesday, December 11, 15:30-16:30, UdeM, Pav. Andre-Aisenstadt,
Room 4336
Vincent Rivasseau (Paris-Sud)
Invitation a la geometrie aleatoire en dimension superieure a deux
Abstract:
Pour traiter de systemes desordonnes en dimension 3 ou pour quantifier
la gravitation les physiciens souhaitent disposer d'une theorie robuste de
geometries aleatoires en dimension superieure a deux. On presentera une
piste ouverte recemment dans cette direction, qui generalise la theorie
des matrices aleatoires, utiles pour comprendre la geometrie aleatoire
en dimension 2, en une theorie de tenseurs aleatoires et etudie les
graphes de Feynman et les theories des champs associees.
The talk is CANCELLED
DATE AND TIME CHANGE: Wednesday, December 19, 13:00-14:00,
Burnside 920
Philippe Poulin (United Arab Emirates University)
Weighted Paley-Wiener Spaces and MC-Spaces
Abstract:
In their study of weighted Paley-Wiener spaces, Lyubarskii and Seip
exhibited structural properties shared by a larger class of de Branges
spaces, which we will call the MC-spaces. In this talk we will re-state
their results in their full generality. If time permits, we will show how
their method can be used for getting concrete realizations of the MC-spaces.
ANALYSIS-REALTED TALKS ELSEWHERE, FALL 2012
McGill-UdeM Spectral Theory
Seminar
Thursday, September 6, 13:30-14:30, UdeM, Room 5183
Michael Levitin (Reading)
Graphene operator pencil
McGill-UdeM Spectral Theory Seminar
Thursday, September 13, 13:30-14:30, UdeM, Room 5183
David Sher (CRM/McGill)
Conic Degeneration and the Determinant of the Laplacian.
CRM-ISM colloquium
Friday, September 14, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O.,
salle SH-3420, 16:00-17:00
Robert McCann
(University of Toronto)
A glimpse at the differential topology and geometry of optimal transportation
Abstract:
The Monge-Kantorovich optimal transportation problem is to pair producers
with consumers so as to minimize a given transportation cost. When the
producers and consumers are modeled by probability densities on two
given manifolds or subdomains, it is interesting to try to understand
the structure of the optimal pairing as a subset of the product manifold.
This subset may or may not be the graph of a map. The talk will expose
the differential topology and geometry underlying many basic phenomena
in optimal transportation. It surveys questions concerning Monge maps
and Kantorovich measures: existence and regularity of the former,
uniqueness of the latter, and estimates for the dimension of its
support, as well as the associated linear programming duality. It
shows the answers to these questions concern the differential geometry
and topology of the chosen transportation cost. It establishes new
connections --- some heuristic and others rigorous --- based on the
properties of the cross-difference of this cost, and its Taylor expansion
at the diagonal.
CRM-ISM colloquium
Friday, October 12, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O.,
salle SH-3420, 16:00-17:00
Rupert Frank
(Princeton)
Symmetry and Reflection Positivity
Abstract:
There are many examples in mathematics, both pure and applied, in which problems with symmetric formulations have non-symmetric solutions. Sometimes this symmetry breaking is total, but often the symmetry breaking is only partial. One technique that can sometimes be used to constrain the symmetry breaking is reflection positivity. It is a simple and useful concept that will be explained in the talk, together with some examples. One of these concerns the minimum eigenvalues of the Laplace operator on a distorted hexagonal lattice. Another example that we will discuss is a functional inequality due to Onofri. The talk is based on joint work with E. Lieb
McGill-UdeM Spectral Theory Seminar
Thursday, October 25, 13:30-14:30, UdeM, Room 5448
Yaiza Canzani (McGill)
Distribution of random perturbations of propagated eigenfunctions.
McGill-UdeM Spectral Theory Seminar
Thursday, November 1, 13:30-14:30, McGill, Room 1214
Alexandre Girouard (Universite de Savoie)
Uniform spectral stability for rough perturbations of domains.
McGill Mathematical Physics Seminar
Thursday, November 1, 15:00-16:00, Burnside 920
Jurg Frolehlich (ETH Zurich and IAS Princeton)
(Do) we understand quantum mechanics - finally (?)!
CRM-ISM colloquium
Friday, November 2, U. de Montreal, Pav. Andre-Aisenstadt,
2920, chemin de la Tour, SALLE 6214, 16:00-17:00
Jurg Froehlich (ETH Zurich)
Dissipative motion from a Hamiltonian point of view
Abstract:
I will study the motion of a classical particle interacting with a dispersive wave medium. (Concretely, one may think of a heavy particle interacting with an ideal Bose gas at zero temperature, in the large-density or mean-field limit.) This is an example of a Hamiltonian system with infinitely many degrees of freedom that describes dissipative phenomena. I will show that the particle experiences a friction force with memory, which is caused by the particle's emission of Cherenkov radiation of sound waves into the medium. This friction force decelerates the particle until its speed has dropped to the minimal speed of sound in the medium (=0, for an ideal Bose gas). Various open problems that I suspect might be of interest to analysts will be described. (The results presented in this lecture have been found in joint work with Daniel Egli, Gang Zhou, Avy Soffer and Israel Michael Sigal.)
CRM-ISM colloquium
Friday, November 23, U. de Montreal, Pav. Andre-Aisenstadt,
2920, chemin de la Tour, SALLE 6214, 16:00-17:00
Alexander Gamburd (CUNY Graduate Center)
Expander Graphs, Thin Groups, and Superstrong Approximation
SUMMER 2012
Wednesday, June 6, 12:30pm, Burnside 1214
Daniel Ueltschi (University of Warwick)
Introduction to cluster expansions: applications, combinatorics, tree estimates
Friday, June 29, 13:30-14:30, Burnside 920
Junfang Li (University of Alabama)
A priori estimates of prescribing curvature measure problems in
Riemann spaces
Abstract:
In this talk, we will discuss the prescribing curvature measure
problem in Riemann spaces (space forms with constant sectional curvature, 0,
-1, or 1.) This will generalize the previous work in Guan-Lin-Ma and
Guan-Li-Li from Euclidean space to elliptic space and hyperbolic space. I
will focus on the a priori estimates since these are the key steps. We
propose a uniform approach for C^0, C^1 estimate. The crucial step is the
C^2 estimate. As a result, we will settle down the problem in elliptic space
and prove the a priori C^2 estimates for some special cases in hyperbolic
space. For example, we will show a uniform C^2 estimate for surfaces in 3
dimensional hyperbolic space. Moreover, the uniform gradient estimate will
yield the existence for the prescribing mean curvature measure in all the
three Riemann spaces.
Thursday, July 26, 13:30-14:30, Pav. Andre-Aisenstadt, Room 4336
Le Hai Khoi (Nanyang Technological University, Singapore)
Composition Operators on Dirichlet series
Abstract:
We consider some problems for composition operators on a class of
entire
Dirichlet series with real frequencies in the complex plane whose Ritt order
is zero and
logarithmic orders are finite. Criteria for action and boundedness of such
operators are
given.
Monday, July 30, 13:30-14:30, Burnside 920
Frederic Robert (Nancy)
Sign-changing blow-up for scalar curvature type equations
Abstract:
Complex Analysis seminar
Monday, July 30, 15:30-16:30, Pav. Andre-Aisenstadt, Room 4336
Raphael Clouatre (Indiana University)
Similitude pour les operateurs de classe C_0
Monday, August 13, 13:30-14:30, Burnside 920
M. del Mar Gonzalez (Barcelona)
A Discrete Bernoulli Free Boundary Problem
Abstract:
We consider a free boundary problem for the p-Laplace operator which is
related to the so-called Bernoulli free boundary problem. In this
formulation, the classical boundary gradient condition is replaced by a
condition on the distance between two different level surfaces of the
solution. For suitable scalings our model converges to the classical
Bernoulli problem; one of the advantages in this new formulation is that
one does not need to consider the boundary gradient.
We shall study this problem in convex and other regimes, and establish
existence and qualitative theory. This is joint work with M. Gualdani and
H. Shahgholian.
Friday, August 31, 13:30-14:30, Burnside 920
Marco Veneroni (Pavia)
On minimizers of the bending energy of two-phase biomembranes
Abstract:
We consider the problem to find the shape of multiphase biomembranes,
modeled as closed surfaces enclosing a fixed volume and having fixed
surface area. The observed shape is assumed to be a minimizer of the
sum of the Canham-Helfrich energy, in which the bending rigidities
and spontaneous curvatures depend on the phase, and of a line tension
penalization for the phases interface. By restricting attention to
axisymmetric surfaces and phase distributions, we prove existence of
a global minimizer.
This is joint work with Rustum Choksi (McGill) and Marco Morandotti
(Carnegie Mellon).
2011/2012 Seminars
2010/2011 Seminars
2009/2010 Seminars
2008/2009 Seminars
2007/2008 Seminars
2006/2007 Seminars
2005/2006 Analysis Seminar
2004/2005 Seminars
2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems
2003/2004 Working Seminar in Mathematical Physics
2002/2003 Seminars
2001/2002 Seminars
2000/2001 Seminars
1999/2000 Seminars