## Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia or at McGill

## SUMMER 2010

Wednesday, September 8, Time and Room TBA
Elijah Liflyand (Bar Ilan)
Title TBA

## 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
(HN ψ)n= ψn-1n+1+vnψn,
ψ0N+1=0,
in the scaling limit Var(vn)=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)
A sharp rearrangement inequality for the Coulomb energy
Abstract: Balls have long been known to minimize capacity among bodies of a given volume; more generally, the Coulomb energy of a (positive) charge distribution will increase, if the distribution is rearranged to be symmetric decreasing. Corresponding inequalities hold for more general functionals involving multiple integrals. One may ask, whether a body whose capacity is close that of a ball must itself resemble a ball?
I will discuss this question in the context of recent results (due to Fusco, Maggi, Pratelli and others) that bound the "asymmetry" of a body in terms of its "isoperimetric deficit". Much less is known about inequalities for multiple integrals; in some cases, not even all optimizers have been identified. I will present a recent result on the Coulomb energy, sketch its proof, and mention open problems.
Wednesday, March 10, Burnside 1234, 13:00-15:00
Stanislav Molchanov (Univ. of North Carolina)
On the idea of ensemble
Abstract: In many cases when we are not able to study in detail a fixed mathematical object (a matrix, a function, an equation), we can conside a bit "softer" question: what are the properties of the "typical" object of this kind. The term "typical" one can understand either topologically or probabilistically. In the latter case one defines the set (or ensemble) of the similar objects and equips it by an appropriate probability measure P. Now the "typical" simply means P-a.s. We will illustrate this classical idea by the examples from the different areas of mathematics (random tomography, distribution of zeros of random polynomials and analytic functions, random pseudo prime numbers and zeta functions, quantum tunnelling in dimension 1).
Thursday, March 11, Burnside 1120, 13:00-14:30
Stanislav Molchanov (Univ. of North Carolina)
Non-random perturbation of Anderson hamiltonian

Friday, March 12, 13:30-14:30, Burnside 920
Ilia Binder (Toronto)
Solving Dirichlet problem, barriers with logarithmic growth, and boundary geometry
Abstract: Walk on Spheres is one of the classical algorithms for solving Dirichlet problem. I will discuss the relation of the rate of convergence of this algorithm and the geometric properties of a domain boundary. In particular, this rate of convergence depends on the existence of certain barrier functions with logarithmic growth at every boundary point. This is a joint work with Mark Braverman (Microsoft Research/University of Toronto).
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, March 26, 13:30-14:30, Burnside 920
The Mass critical fourth order Schrodinger equation
Abstract: We study the Mass critical fourth order Schrodinger equation in high dimension. We prove that solutions with L^2 initial data are global and converge to linear solutions at large time in high dimensions and when the equation is defocusing. If time permits we will discuss the focusing case and the case of lower dimensions.
Friday, April 9, 14:30-15:30, Burnside 920
Emmanuel Milman (Toronto)
Isoperimetric and Concentration Inequalities, and their applications
Abstract: The classical isoperimetric inequality in Euclidean space asserts that among all sets of given Lebesgue measure, the Euclidean ball minimizes surface area. Using a suitable generalization of surface area, isoperimetric inequalities may be investigated on general metric spaces equipped with a measure. A prime example is that of Euclidean space equipped with the standard Gaussian measure, in which case a classical result of Sudakov--Tsirelson and Borell asserts that minimizers of Gaussian boundary measure are given by half-spaces. One important reason for studying isoperimetric inequalities is that they easily imply concentration inequalities, which are very useful in applications. The latter do not provide infinitesimal information on boundary measure of sets, but are rather concerned with large-deviation information, bounding above the measure of sets separated from sets having half the total measure, as a function of their mutual distance in the large. In the case of a Gaussian measure for instance, the decay is like $\exp(-d^2/2)$, where $d$ is the mutual distance. In general, concentration inequalities cannot imply back isoperimetric inequalities. We will show that under a suitable (possibly negative) lower bound on the Bakry-\'Emery curvature tensor of a Riemannian-manifold-with-density (combining information from both the geometry of the space and the measure), completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds, which is crucial for applications. Contrary to previous attempts which could only produce dimension dependent bounds, our method is entirely geometric, following the approach set forth by M. Gromov and recently adapted by F. Morgan, which combines volume comparison theorems from Riemannian Geometry with results from Geometric Measure Theory on the existence and regularity of isoperimetric minimizers. We will try to set aside time to go over some applications of this result, ranging from Spectral Geometry to Statistical Mechanics. These include best possible dimension independent bounds on the first non-trivial eigenvalue of the (Neumann) Laplacian and the log-Sobolev constant on compact manifolds-with-density, and dimension independent stability results for the Poincar\'e and log-Sobolev inequalities under harsh perturbations of the underlying measure.
Monday, April 12, Burnside 920, 13:30-14:30
Victor Ivrii (Toronto)
2D Schrodinger with a strong magnetic field: dynamics and spectral asymptotics near boundary
Abstract: We consider magnetic Schrodinger operator
$$H=\bigl(h\nabla - \mu \mathbf{A}(x)\bigr)^2 +V(x), \qquad \mathbf{A}(x)=(-\frac{1}{2}x_2, \frac{1}{2}x_1)$$
(or more general one) and derive rather sharp asymptotics of $\int \psi (x)\, e(x,x,0)\,dx$ as $\mu\to\infty$ and $h\to 0$ where $e(x,y,\lambda)$ is a Schwartz kernel of spectral projector of $H$ and $\psi(x)$ is a cut-off function. Corresponding classical dynamics associaated with operator in question inside of domain is a cyclotron movement along circles of radius $\asymp \mu^{-1}$ combined with slow drift movement (with a speed $\asymp \mu^{-1}$) along level lines of $V(x)$. However near boundary dynamics consists of hops along it; this hop dynamics could be torn away from the boundary and become an inner dynamics and v.v. This classical dynamics has profound implications for spectral asymptotics (with remainder estimate better than $O(h^{-1})$ and up to $O( \mu^{-1}h^{-1})$). We consider also the case of superstrong magnetic field (as $\mu h\ge 1$) when classical dynamics is at least applicable but the difference between Dirichlet and Neumann boundary conditions are the most drastic.

The full and detailed analysis: Chapter 15 of My Future Monster Book

Monday, April 26, Burnside 920, 14:30-15:30
Stefano Bianchini (SISSA)
Seminar cancelled
Friday, April 30, Burnside 920, 10:00-11:00
Rasul Shafikov (University of Western Ontario)
What is the boundary value of a holomorphic function?
Abstract: A classical theorem of Fatou states that a bounded holomorphic function in the unit disc in C has radial limits almost everywhere on the boundary of the disc. Ever since, the problem of making sense of boundary values of holomorphic (or more generally, harmonic) functions (in one or more variables) has been an active area of research, often yielding far-reaching theories (think Hardy spaces). In this talk I will give an overview of two classical approaches to the problem, and will outline the idea of a new construction of boundary values of holomorphic functions of several variables for domains with non-smooth boundary.
Monday, May 3, 14:30-15:30, Burnside 920
Tayeb Aissiou (McGill)
Semiclassical Limits of eigenfunctions on flat n-dimensional tori
Friday, May 7, 11:00-12:00, Burnside 920
Wang Yi (Princeton)
Quasiconformal mappings, isoperimetric inequality and finite total Q-curvature
Abstract: Abstract: In this talk, we are going to prove the isoperimetric inequality on the noncompact conformally flat four manifold with totally finite Q-curvature and $\frac{1}{4\pi2}\int_{M}Q(x) dv_M(x)<1$. To do this, we will look at the construction of the quasiconformal mapping with suitable quantity of Jacobian. The affirmative answer to the existence of such mapping would imply the bilipschitz parametrization of the manifold as well as the isoperimetric inequality. Moreover, if the Q-curvature is non-negative, the conformal factor relates to $A_1$ weight.
Friday, May 21, 13:30-14:30, Burnside 920
Lauri Berkovits (Oulu University, Finland)
Parabolic BMO and A_p classes
Abstract: We present a parabolic or "one-sided" analogue of the classical BMO and A_p weight theory. Weighted inequalities, along with reverse Holder type and John-Nirenberg type inequalities will be discussed.
Friday, June 4, 13:30-14:30, Burnside 920
Felix Finster (Regensburg)
Do minimizers of causal variational principles have a discrete structure?
Abstract: The aim of the talk is to give an easy accessible introduction to causal variational principles and to illustrate the structure of the minimizers in a simple model example. Causal variational principles are defined, where for simplicity I will always restrict attention to the special case of prescribed eigenvalues and to spin dimension one. By further specialization one gets a minmizing problem on the two-dimensional sphere, which will serve as our model problem. After a short survey of the existence theory we shall analyze the structure of the minimizers. Numerical results are shown which indicate that the every minimizer is a weighted counting measure. This motivates a general theorem which states that the interior of the support of a minmizing measure is always empty. The proof of this theorem will be outlined. This result can be understood that when minimizing, an effect of spontaneous symmetry breaking is followed by the formation of a discrete structure.

## ANALYSIS-RELATED 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.
Applied Mathematics Seminar
Monday, April 12, 16:00-17:00, Burnside 920
Jianke Yang (U. Vermont)
Newton-conjugate-gradient methods for solitary wave computations
Abstract: In this talk, we present the Newton-conjugate-gradient methods for solitary wave computations. These methods are based on Newton iterations, coupled with conjugate-gradient iterations to solve the resulting linear Newton-correction equation. When the linearization operator is self-adjoint, the preconditioned conjugate-gradient method is proposed to solve this linear equation. If the linearization operator is non-self-adjoint, the preconditioned biconjugate-gradient method is proposed to solve the linear equation. The resulting methods are applied to compute both the ground states and excited states in a large number of physical systems, such as the two-dimensional Nonlinear Schrodinger equation with and without periodic potentials, and the fifth-order Kadomtsev-Petviashvili (KP) equation. Numerical results show that these proposed methods are faster than the other leading numerical methods, often by orders of magnitude. In addition, these methods are very robust and always converge in the examples that have been tested.
CRM-ISM colloquium
Friday, April 16, 16:00-17:00, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Surface Evolution under Curvature Flows - Existence and Optimal Regularity

## 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
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
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

2005/2006 Analysis Seminar