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 C^m-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 L^p 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).

2005/2006 Analysis Seminar

2004/2005 Seminars

2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems

2003/2004 Working Seminar in Mathematical Physics