Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia or at McGill
For suggestions, questions etc. please contact Galia Dafni (gdafni@mathstat.concordia.ca), Dmitry Jakobson (jakobson@math.mcgill.ca) or Alexander Shnirelman (shnirel@mathstat.concordia.ca)

WINTER 2010

Please note, that friday seminars at McGill in winter 2010 will often be held from 13:30-14:30 (1 hour earlier!) in Burnside 920 (same room as before).

Friday, January 22, 13:30-14:30, Burnside 920
Julien Vovelle (Lyon)
Height dependent mean curvature equation

Friday, February 12, 14:30-15:30, Burnside 920
Gantumur Tsogtgerel (McGill)
Partial regularity for some regularized Navier-Stokes equations

Friday, February 19, 13:30-14:30, Burnside 920
Eugene Kritchevski (Toronto)
The scaling limit of the critical one-dimensional random Schrodinger operator
Abstract: We study the one dimensional discrete random Schrodinger operator
(H_N ψ)_n= ψ_n-1+ψ_n+1+v_nψ_n,
ψ₀=ψ_N+1=0,
in the scaling limit Var(v_n)=1/N. We show that, in the bulk of spectrum, the eigenfunctions are delocalized and that there is a very strong repulsion of eigenvalues. The analysis is based on a stochastic differential equation for the evolution of products of transfer matrices. The talk is based on a joint work with Benedek Valko and Balint Virag.

Friday, March 5, 13:30-14:30, Burnside 920
Almut Burchard (Toronto)
Title TBA

Friday, March 12, 13:30-14:30, Burnside 920
Ilia Binder (Toronto)
Title TBA

Special seminar
Wednesday, March 17, 15:00-16:00, Burnside 920
Robert Seiringer (Princeton/McGill)
The critical temperature of dilute Bose gases
Abstract: The effect of interparticle interactions on the critical temperature for Bose-Einstein condensation has been a controversial issue in the literature. Various approximation schemes lead to different conclusions, concerning both the sign and the magnitude of the shift in the critical temperature. We shall examine this question from the point of view of rigorous bounds. While lower bounds seem to be out of reach of present methods, we show that a rigorous upper bound can be established rather easily. Our bound shows that in the presence of repulsive interactions the critical temperature can not increase by more than the square root of $a \rho^{1/3}$, with $\rho$ the density and $a$ the scattering length of the interaction potential. Our method also yields valuable information in the case of a two-dimensional Bose gas. While there is no Bose-Einstein condensation in this case, a Kosterlitz-Thouless type phase transition is expected at a definite critical temperature. We prove that off-diagonal correlations decay exponentially above this temperature. (This is joint work with D. Ueltschi.)

Friday, April 9, time and room TBA
Emmanuel Milman (Toronto)
Title TBA

ANALYSIS-REALTED TALKS ELSEWHERE, WINTER 2010

Friday, February 5, 10:30-11:30, Concordia, Library bldg, LB-921-04
Alexey Kokotov (Concordia)
From Spectral Theory of Polyhedral Surfaces to Geometry of Hurwitz Spaces
Abstract: In this talk we show how to make use of an important spectral invariant of polyhedral surfaces (Riemann surfaces with conformal flat conical metrics) - the determinant of the Laplacian - to study the geometry of moduli spaces. For a special class of polyhedral surfaces - the surfaces with trivial holonomy - we obtain a holomorphic factorization formula for the determinant of Laplacian. The holomorphic factor appearing in this formula is the so-called Bergman tau-function - a universal object arising in various areas: from isomonodromic deformations of Fuchsian ODEs to random matrices and Frobenius manifolds. The Bergman tau-function gives rise to a section of the Hodge line bundle over the space of admissible covers (the Harris- Mumford compactification of the Hurwitz space i.e. the moduli space of meromorphic functions on Riemann surfaces). Analysis of the asymptotics of the Bergman tau-function near the boundary of the Hurwitz space leads to an explicit expression for the Hodge class of the space of admissible covers in terms of boundary divisors. This expression generalizes previously known results of Lando-Zvonkine for spaces of rational functions and Cornalba-Harris for spaces of hyperelliptic curves.

CRM-ISM colloquium Friday, February 5, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Robert McCann (Toronto)
Optimal multidimensional pricing facing informational asymmetry
Abstract: The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE.

SUMMER/FALL 2009

Friday, July 17, 14:30-15:30, Burnside 920
Jerome Vetois (Cergy Pontoise)
Bubble tree decompositions for critical anisotropic equations
Abstract: We describe the asymptotic behavior in energy space of Palais-Smale sequences for an anisotropic problem on a domain in the Euclidian space. This description is well-known in the isotropic case. In the general case, we emphasize the crucial role played by the geometry of the domain.

Friday, September 11, 14:30-15:30, Burnside 920
Andreas Seeger (Wisconsin)
On radial and conical Fourier multipliers
Abstract: This talk is about recent joint papers with Y. Heo and F. Nazarov. The goal is to characterize, for suitable $p$, the $L^p$ boundedness of convolution operators with radial kernels. There are connections with the so-called local smoothing problem for the wave equation and with questions on Fourier multipliers associated to cones.

Friday, September 18, 14:30-15:30, Burnside 920
Vitali Vougalter (Toronto)
On threshold eigenvalues and resonances for the linearized NLS equation
Abstract: We prove the instability of threshold resonances and eigenvalues of the linearized NLS operator. We compute the asymptotic approximations of the eigenvalues appearing from the endpoint singularities in terms of the perturbations applied to the original NLS equation. Our method involves such techniques as the Birman-Schwinger principle and the Feshbach map.

Joint seminar with applied mathematics
Friday, October 2, 14:30-15:30, Burnside 920
Renato Calleja (McGill)
Breakdown of Analyticity: Rigorous results and numerical implementations
Abstract: We formulate and justify rigorously a numerically efficient criterion for the computation of the analyticity breakdown of quasi-periodic solutions in Symplectic maps and 1-D Statistical Mechanics models. Depending on the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1-D models) and to the onset of global transport in symplectic twist maps. The criterion we propose here is based on the blow-up of Sobolev norms of the hull functions. The theorems that justify the criterion are based on an abstract implicit function theorems, which unifies several results in the literature. The proofs lead to fast algorithms, which we have implemented. We will show numerical implementations of the criterion.

Monday, October 5, 14:30-15:30, Burnside 920
Alberto Enciso (ETH)
Critical points and level sets of solutions to elliptic PDEs
Abstract: We will analyze some geometric properties of the solutions to the exterior boundary problem
\Delta u=0\quad {\rm in }\;\mathbf{R}^n\backslash\overline{\Omega},\quad u|_{\partial\Omega}=1,
with $u\to 0$ at infinity and $\Omega$ being a bounded domain with $C^{2,\alpha}$ connected boundary. We shall prove that the critical set of $u$ can be nonempty (in fact, of codimension $3$) even when $\Omega$ is contractible, thereby settling a question posed by Kawohl in 1988, discuss sufficient geometric criteria for the absence of critical points in this problem and analyze the properties of the critical set for generic domains. Time permitting, related problems on Riemannian manifolds will be discussed as well. Our results hinge on a combination of classical potential theory, transversality techniques and the qualitative theory of dynamical systems.

Friday, October 9, 14:30-15:30, Burnside 920
Roman Shterenberg (Alabama)
Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrodinger operator
Abstract: We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrodinger operator with a smooth periodic potential. This is a joint work with Leonid Parnovski.

Friday, October 16, 14:30-15:30, Burnside 920
Monika Ludwig (Poly NYU)
Valuations on Convex Sets and Sobolev Functions
Abstract

Monday, October 19, 14:30-15:30, Burnside 920
Yan Pautrat (Paris-Sud)
Central limit theorem for sums of non-commutative i.i.d random variables
Abstract: Our goal in this talk is to discuss non-commutative extensions of the central limit theorem. The first question regarding such extensions is the meaning of convergence in distribution in the non-commutative case, and the tools to prove this convergence in concrete models. We state a general non-commutative Levy-Cramer theorem. With the help of this theorem, we formulate and prove a general central limit theorem for sums of independent identically distributed non-commutative random variables. All results presented here are joint work with V. Jaksic and C.-A. Pillet.

Friday, October 23, 14:30-15:30, Burnside 920
Alex Furman (University of Illinois at Chicago)
Invariant and stationary measures for groups of toral automorphisms
Abstract: Joint work with J. Bourgain, E. Lindenstrauss, and S. Mozes. We study the dynamics of the action of a subgroup G of SL(d,Z) on the d-torus. Assuming G is rich (e.g. Zariski dense) we prove the only invariant or, more generally, stationary measures on the torus are combinations of Lebesgue and atomic measures. A quantitative equidistribution result is proven.

Friday, October 30, 14:30-15:30, Burnside 920
Dmitry Jakobson (McGill)
Curvature of Random Metrics
Abstract: This is joint work with Igor Wigman and Yaiza Canzani. We study the behavior of the scalar curvature for random Riemannian metrics close to metrics of constant scalar curvature. We next consider analogous questions for Branson's Q-curvature.

Friday, November 13, 13:30-14:30, Burnside 1205
Hans Christianson (MIT)
Eigenfunction concentration and non-concentration
Abstract: In this talk I will describe several results on eigenfunction concentration on compact manifolds. Specifically, when there is an unstable periodic geodesic, the eigenfunctions concentrate at most logarithmically as the eigenvalue tends to infinity, and when there is a stable periodic geodesic, there are highly localized approximate eigenfunctions. The proofs of both of these results follow from a very general framework of phase space analysis near a periodic orbit, which I will describe briefly. If there is time, at the end I will describe related work-in-progress with H. Hezari, J. Toth, and S. Zelditch on restrictions of quantum ergodic eigenfunctions.

Friday, November 20, 14:30-15:30, Concordia, Library building, LB 921-4
Serban Costea (McMaster)
Strong A-infinity weights and Sobolev capacities in metric measure spaces
Abstract: see pdf

Friday, November 27, 14:30-15:30, Burnside 920
Igor Gorelyshev (CRM)
On the adiabatic perturbation theory and the piston problem
Abstract: I will describe the methods of the adiabatic perturbation theory. I will also show how these methods can be applied to certain systems with impacts and in particular to the piston problem, which is an important problem in statistical mechanics.

Friday, December 4, Burnside 920, 14:30-15:30
Nikolay Dimitrov (CRM and McGill)
Rapid Evolution of Complex Limit Cycles
Abstract: Limit cycles of planar polynomial vector fields have long been a focus of extensive research. Analogous to the real case, similar problems have been studied in the complex plane where a polynomial differential one form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. Whenever the polynomial foliation comes from a perturbation of an exact one-form, one can introduce the notion of a multifold cycle. This type of cycle has at least one representative that determines a free homotopy class of loops in an open fibered subdomain of the complex plane. The topology of this subdomain is closely related to the exact one form, mentioned earlier. This talk will be an introduction to the notion of multifold cycles of a close to integrable polynomial foliation. We will explore the way they correspond to periodic orbits of certain Poincare maps associated with the foliation. We will also discuss the tendency of a continuous family of multifold limit cycles to escape from certain large open domains in the complex plane as the foliation converges to its exact part.

Friday, December 11, Burnside 920, 14:30-15:30
Domokos Szasz (Budapest and Toronto)
Billiard Models and Energy Transfer
Abstract

ANALYSIS-REALTED TALKS ELSEWHERE, FALL 2009

CRM-ISM Colloquium
Friday, September 25, 16:00-17:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Svetlana Katok (Penn State)
Structure of attractors for (a,b)-continued fraction transformations
Abstract: I will discuss one-dimensional maps related to a family of (a,b)-continued fractions, suggested for consideration by Don Zagier, and give a sufficient condition for validity of the Reduction theory conjecture that states that the associated natural extension maps have attractors with finite rectangular structure where every point of the plane is mapped after finitely many iterations. I will show how the structure of these attractors can be ``computed" from the data $(a,b)$, and give a dynamical interpretation of the ``reduction theory" that underlines these constructions. The set of parameter pairs $(a,b)$ for which the conjecture is not valid is also well-understood; in particular, the points for which the attractors do not have finite rectangular structure is a non-empty nowhere dense subset of the boundary $b=a+1$ of the set of parameters . If time permits, I will also explain how these continued fractions can be used for coding of geodesics on the modular surface. This is a joint work with Ilie Ugarcovici.

Nonlinear Analysis and Dynamical Systems seminar Wednesday, September 30, 14:00
CRM, Salle 4336, Pav. Andre Aisenstadt, 2920 Chemin de la Tour, Universite de Montreal.
Nikolay Dimitrov (CRM and McGill)
Rapid evolution of complex limit cycles
Abstract: Limit cycles of planar polynomial vector fields have long been a focus of extensive research. Analogous to the real case, similar problems have been studied in the complex plane where a polynomial differential one form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is dfined as a nontrivial element of the fundamental group of a leaf from the foliation. Whenever the polynomial foliation comes from a perturbation of an exact one form, one can introduce the notion of a multifold cycle. This type of cycle has at least one representative that determines a free homotopy class of loops in an open fibered subdomain of the complex plane. The topology of this subdomain is closely related to the exact one form, mentioned earlier. This talk will be an introduction to the notion of multifold cycles of a close to integrable polynomial foliation. We will explore the way they correspond to periodic orbits of certain monodromy (Poincare) maps associated with the foliation. We will also discuss the tendency of a continuous family of multifold limit cycles to escape from certain large open domains in the complex plane as the foliation converges to its integrable part.

Universite de Montreal Analysis Seminar
Thursday, October 1, 13:30-14:30
CRM, Salle 5340, Pav. Andre Aisenstadt, 2920 Chemin de la Tour, Universite de Montreal.
Manfred Stoll (University of South Carolina)
On generalizations of Littlewood-paley inequalities to domains in Rⁿ, n >=2.
Abstract can be found here

Graduate Seminar in Dynamical Systems
Monday, October 19, 12:30-13:30
Concordia University, Library building, 9th floor, room LB 921-4
Peyman Eslami (Concordia)
The Existence of the Lorenz strange attractor
Abstract: I will give a short overview of the Lorenz differential equations, the geometric Lorenz model and Tucker's proof of the existence of the Lorenz strange attractor.

CRM-ISM Colloquium
Friday, November 6, 16:00-17:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Christopher Sogge (Johns Hopkins)
Kakeya-Nikodym averages and Lp norms of eigenfunctions
Abstract: On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunctiongrowth. The problem which motivates us is to determine the $(M, g )$ with maximal eigenfunction growth. In an earlier work, two of us showed that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We strengthen this result here by showing that such a manifold must have a point where the set ${\mathcal R}_x$ of recurrent directions for the geodesic flow through x satisfies $|{\mathcal R}_x|>0$. We also show that if there are no such points, $L^2$-normalized quasimodes have sup-norms that are $o(\lambda^{n-1)/2})$, and, in the other extreme, we show that if there is a point blow-down $x$ at which the first return map for the flow is the identity, then there is a sequence of quasi-modes with $L^\infty$-norms that are $\Omega(\lambda^{(n-1)/2})$.

First Bavaria-Quebec Mathematical Meeting
November 30 - December 3, 2009
Meeting Room: CRM, Pav. Andre-Aisenstadt, Room 6214
Details: see conference page and schedule

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