## 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), Ivo Klemes (klemes@math.mcgill.ca) or Alexander Shnirelman (shnirel@mathstat.concordia.ca)

## SUMMER 2007

Tuesday, May 29, 2pm-3pm
Burnside 1234
How to choose a Lagrangian

Monday, June 4, 2pm-3pm
Burnside 920
V. Roytburd (Rensselaer Polytechnic Institute)
From BS to KS and beyond: Dissipativity and pattern formation
Abstract. Attempts to capture salient features of cellular flame instabilities have led to a variety of beautiful mathematical models of a very geometrical nature. The most famous of them is the Kuramoto---Sivashinsky (KS) equation. Its lesser relative is the Burgers---Sivashinsky (BS) equation (which is just a linearly forced Burgers equation). The linear dispersion relations for both equations admit exponential mode growth for a range of long waves. Nonetheless the equations are dissipative due to the nonlinear mixing. For the purposes of this talk, dissipativity means that the eventual time evolution of solutions is confined to a bounded (actually compact) absorbing set. The principal subject of this talk is yet another, recently introduced model of quasi-steady development of cellular flames, the Quasi-Steady equation (joint work with M. Frankel, IUPUI). In a sense, QS is intermediate between BS and KS, as its dispersion relation coincides with that for BS for short waves, and is virtually identical to that of KS for long waves. Similarly to KS, QS demonstrate a very rich dynamical behavior (note that BS has more or less trivial dynamics). The proof of dissipativity and generalizations to elliptic pseudo-differential operators will be discussed.

Monday, June 11, 14:00-15:00
Burnside 920
New constructions of submanifolds of the sphere which are critical points of the volume functional
Abstract: If one searches for k-dimensional submanifolds with critical k-dimensional volume in a Riemannian manifold, then one is led towards elliptic partial differential equations involving the mean curvature vector of the submanifold. I will present new constructions of volume-critical submanifolds of the sphere in two contexts: hypersurfaces with constant mean curvature in spheres of any dimension; and Legendrian submanifolds in spheres of odd dimension that are stationary under variations preserving the contact structure. These are constructed by solving the associated elliptic PDE using singular perturbation theory. I will then highlight some of the analytic and geometric similarities between these two contexts.

Friday, June 15, 14:00-15:00
Burnside 920
Xinan Ma (Univ. of Science and Tech. of China)
A generalization of Brascamp-Lieb Theorem on log concavity to $\Sigma_{2}$ operator
Abstract: In this talk I shall consider the generalized of Brascamp-Lieb's theorem on (JFA76) the log convcavity of first eigenfunction of the Dirichlet eigenvalue problem on eulidean convex domain, more precisely we consider the operator of second elementry symmetric functon of the hessian in three dimension case.As an application we get the Brunn-Minkowski inequality and describe the equality case. This is a joint works with Xu Lu.

July 16-18, Burnside 920
Vojkan Jaksic and Eugene Kritchevski (McGill)
Preparatory lectures for F. Germinet's course on July 18-23.
Monday July 16, 14:00-16:00 Jaksic: The spectral theorem
Tuesday July 17, 14:00-16:00 Jaksic: The ergodic theorem
Wednesday July 18, 14:00-16:00 Kritchevski: Weyl sequences and Borel-Cantelli lemmas

July 18-23, Burnside 920
F. Germinet (Cergy-Pontoise)
Mini-course: A basic introduction to Random Schroedinger operators
Wendesday July 18, 10:00-12:30. Lecture 1
Thursday July 19, 10:00-12:30. Lecture 2
Friday July 20, 10:00-12:30. Lecture 3
Saturday July 21, 10:00-12:30. Lecture 4
Monday July 23, 10:00-12:30. Lecture 5

July 23, 14:00-15:00
Concordia, LB 921-4
I. Markina (U. Bergen, Norway)
Tangential Cauchy-Riemann operator on quaternionic Sigel upper half space
Abstract

July 23, 15:30-16:30
Concordia, LB 921-4
A. Vasiliev (U. Bergen, Norway)
Tangential slit solutions to the Loewner equation
Abstract: We consider the Loewner differential equation generating the unit disk or the upper half-plane onto itself minus a single slit. Marshall and Rohde recently proved that if a slit is a non-tangential quasiarc, then the corresponding Loewner equation contains a Lip(1/2) driving term. We prove that if the slit is first-order tangential, then the driving term is Lip(1/3).

## WINTER 2007

Friday, January 12, 2-3pm
Burnside 920
Michael Levitin (Heriot-Watt)
Calculating eigenvalues and resonances in domains with regular ends

Monday, January 29, 2:30-3:30pm
Burnside 920
Jason Metcalfe (Berkeley)
Global Strichartz and smoothing estimates for rough Schrodinger equations
Abstract I will talk about recent work with J. Marzuola and D. Tataru. We prove some frequency localized smoothing estimates for Schrodinger equations with C^2 asymptotically flat coefficients. In particular, we make no assumptions on the trapping and prove the said estimates outside of the region where the trapping may occur, modulo lower order error terms. We then use a long time parametrix construction of Tataru to prove global-in-time Strichartz estimates on this same region.
Friday, February 2, 2:00-3:00pm (Note time change!)
Burnside 920
Dimiter Vassilev (UC Riverside)
Conformal geometry and function theory on quaternionic contact manifolds
Abstract We shall present some results concerning Einstein structures on quaternionic contact manifolds and their conformal deformations. From the analytical point of view the study concerns the best constant in the Folland-Stein L2 embedding theorem and geometric function theory on quaternionic contact manifolds.
Joint Applied Mathematics/Analysis seminar
Friday, February 9, 1:30-2:30pm
Room TBA
Olaf Steinbach (Graz)
Boundary Integral Equations: Analysis and Applications
Abstract For a long time boundary integral equations were used to prove the unique solvability of second order boundary value problems. Using indirect boundary integral formulations with single and double layer potentials those considerations led to first and second kind boundary integral equations to be solved. In particular for the latter Neumann series are an appropriate choice. However, there was and there is still an ongoing discussion about the convergence estimates in suitable function spaces. In this talk we will present some results to prove the contraction property of the double layer potential in fractional Sobolev spaces. These estimates have an deep impact for the design of efficient boundary element methods, i.e. preconditioning and adaptivity. Moreover, there are strong relations with domain decomposition methods and the Dirichlet to Neumann map involved.

Monday, February 12, 2:30-3:30pm
Burnside 920
Majorization, Matrices and the Geometry of Polynomials
Abstract In this talk, we explore the connections between the majorization order, matrix theory and the geometry of polynomial root sets. We will show how majorization can be used to prove results about polynomials (such as the solution of the De Bruijn-Springer conjecture and an improvement to Mahler's inequality) and will discuss an application of majorization to the fastest mixing Markov chain problem of Boyd, Diaconis and Xiao. We will also explore a possible Hilbert space approach to the Bombieri norm on polynomials.

Friday, March 16, 14:30-15:30
Burnside 920
Julian Edward (Florida International U.)
Ingham-type inequalities for complex frequencies and applications to control theory
Abstract For complex valued sequences $\{ \omega_n\}_{n=1}^{\infty}$ of the form $\om_n=a_n+ib_n$ with $a_n\in {\bf R}$ and $b_n\geq 0$, we prove inequalities of the form $\int_0^T|\sum_{n=p}^{\infty}x_ne^{it\om_n}|^2dt\geq C \sum_{n=p}^{\infty}|x_n|^2/(1+b_n)$, for all sequences $\{ x_n\}$ with $\sum_{n=1}^{\infty}|x_n|^2/(1+b_n)<\infty.$ We apply these to prove exact null controllability for a class of hinged beam equations with mild internal damping.

Friday, March 23, 14:30-15:30
Burnside 920
Malabika Pramanik (UBC)
$L^2$ decay estimates for oscillatory integral operators in several variables with homogeneous polynomial phases
Abstract Oscillatory integral operators mapping $L^2(\mathbb R^{n_1})$ to $L^2(\mathbb R^{n_2})$ play an important role in many problems in harmonic analysis and partial differential equations. We will briefly discuss the applicability of these operators in various contexts and give an overview of the current literature. We also mention recent results (joint with Allan Greenleaf and Wan Tang) where, extending earlier work of Phong and Stein in the case $n_1 = n_2 = 1$, we obtain optimal decay rates for the $L^2$ operator norm of oscillatory integral operators in $2+2$ variables with generic phases. Some other higher dimensional situations are also addressed.

Friday, March 30, 14:30-15:30
Burnside 920
Alexander Koldobsky (Missouri)
Intersection bodies and L_p-spaces
AbstractIntersection bodies have become one of the central objects in convex geometry. We discuss a recently discovered connection between this class of bodies and $L_p$-spaces with $p<0.$ This connection allows to get new geometric results by extending to negative values of $p$ different results from the classical theory of $L_p$-spaces.

Thursday, April 5, 17:00-18:00
Concordia, LB 921-4 (Library building)
D. Dryanov (Concordia)
Kolmogorov Entropy for Classes of Convex Functions
Abstract Kolmogorov (or $\epsilon$-) entropy of a compact set in a metric space measures "how massive is it" and thus replaces its dimension (which is usually infinite). This notion is widely applied in the approximation theory. However, recently the need to estimate the Kolmogorov entropy emerged in the Fluid Dynamics, where some nontrivial compact sets appear as final states of the evolution of ideal incompressible flows. An example of such set is the set of plane-parallel flows in a periodic strip with parallel walls having convex velocity profile. In our talk we answer the following question asked by A.Shnirelman: What is the exact asymptotic of the $\epsilon$-entropy for uniformly bounded classes of convex function in the $L^p$-metric? We show that for $1<=p<=\infty$, the Kolmogorov $\epsilon$-entropy of the metric space of convex and uniformly bounded functions, equipped with $L^p$-metric has the precise asymptotic $\epsilon^{-1/2}$.

Friday, April 13, 14:30-15:30
Burnside 920
Yuri Safarov (King's College, London)
Asymptotic formulae for the spectral function
Abstract The talk will discuss asymptotic behaviour of the spectral function of the Laplace operator on a manifold without boundary for large values of the spectral parameter. It will give an overview of known results obtained with the use the wave equation technique and Fourier Tauberian theorems.

Friday, April 20, 14:30-15:30
Burnside 920
C.J. Xu (Univ. de Rouen)
Hypoellipticity of Boltzmann equation
Abstract In this talk, we prove the regularity(in Sobolev space and in Gevrey class) effet of Cauchy problem for Boltzmann equation without Grad's angular cutoff. We study full nonlinear Boltzmann eqution for spatially homogeneous space, and linear Boltzmann eqution in inhomogeneous case. We use the nonlinear microlocal calculus and Fefferman-Phong's uncertainty principle to get the regularity of weaks solutions. So it is quite different with classical H-theorem for Boltzmann equation.

Monday, April 30, 14:30-15:30
Burnside 920
G. Staffilani (MIT)
Nonlinear Schrodinger equation in Hyperbolic spaces

Tuesday, May 1, 11:00-13:00
Burnside 920
F. Germinet (Univ. Cergy-Pontoise, Paris)
1. Random Schrodinger operators and Anderson localization
2. A concentration inequality and applications
The third lecture
3. Single energy multiscale analysis and Anderson localization
will be given from 14:40-15:40 on May 2, during the Analysis Day at CRM (see below)
Abstract In these lectures, I will review standard and new results on Anderson localization for random Schrodinger operators. Such models describe the motion of an electron in disordered media. As discussed by Anderson in 1958, a disordered medium should have a trapping effect, at least at low energies, and one then refers to Anderson localization". In the first talk, I will present different models as well as standard results obtained over the last 30 years. In the second talk, I will state and prove a new concentration bound for particular functions of independant random variables. As an application, it will yield Anderson localization for an arbritrary (non degenerate) underlying probability measure. In the third talk, I will show how to extract exponentially decaying eigenfunctions out of the basic and standard single energy multiscale analysis introduced by Frohlich and Spencer in 1983.

ANALYSIS DAY: Wednesday, May 2, 11:00-16:00
CRM, Room 5340
11:00-12:00
Francois Germinet (Univ. Cergy-Pontoise, Paris)
Single energy multiscale analysis and Anderson localization
Abstract In the talk, I will show how to extract exponentially decaying eigenfunctions out of the basic and standard single energy multiscale analysis introduced by Frohlich and Spencer in 1983.
12:00-13:30 Lunch
13:30-14:30
Francis Clarke (Institut universitaire de France et Universite de Lyon)
Regularity of solutions in the calculus of variations
Abstract An overview of some classical and recent results on the regularity of solutions in the (single and multiple integral) calculus of variations. Along the way are reviewed the celebrated theorems of Hilbert-Haar and De Giorgi.
14:40-15:40
Igor Wigman (CRM and McGill)
Nodal lines for random eigenfunctions of the Laplacian on the torus
Abstract We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $4\pi^2\lambda$ with growing multiplicity $N\to\infty$, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is $const\sqrt{\lambda}$. Our main result is that the variance of the volume normalized by $\sqrt{\lambda}$ is bounded by $O(1/\sqrt{N})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace. This is joint work with Z. Rudnick (Tel Aviv university).

Thursday, May 3, 15:30-16:30
Burnside 920
J. Schenker (IAS, Princeton and Michigan State Univ.)
Estimating the real part of complex eigenvalues of non-self adjoint Schroedinger operators via complex dilations
Abstract This talk will focus on a lower bound for the real part of eigenvalues of dissipative operators L = H + i g F with H the 1D Harmonic oscillator, F an analytic function on the real line, and g the coupling parameter. When the coupling g is large the eigenvalues lie in a half plane {z: Re z > g^v} with v an exponent that depends on F. This cannot be seen directly by variational arguments since the perturbation i g F is skew-adjoint, but can be seen using complex dilations. As a result of the bound, one sees that the dissipation of the semi-group e^{-t L} is greatly enhanced compared to e^{-t H} although the difference of the generators is skew-adjoint. This is a linear manifestation of the phenomenon hypercoercetivity'' identified by C. Villani.

Monday, May 7, 14:30-15:30
Burnside 920
S. Roy (Laval)
Extreme Jensen measures.
Abstract.

## FALL 2006

Friday, September 29, 2:30pm
Burnside 920
Dan Mangoubi (CRM/McGill)
On inner radius of nodal domains

Friday, October 6, 2pm
Burnside 920
Simone Warzel (Princeton)
The Canopy Graph and Level Statistics for Random Operators on Trees
Abstract For random operators with homogeneous disorder, it is generally believed that the spectral characteristics of the infinite operators, which may vary over different energy ranges, are reflected also in the distributions of the energy gaps in finite volume versions of the operators. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which is the only example where both spectral types have been established, the level statistics is always Poisson. However, we also find that this does not contradict the common belief if that is carefully reviewed, as the relevant limit of finite trees is not the infinite tree graph but rather what is termed here the canopy graph. For this tree graph, the random Schoedinger operator is proven to have only pure-point spectrum at any strength of the disorder.
Friday, October 20, 2pm
Burnside 920
Steve Zelditch (Johns Hopkins)
Inverse spectral problem for analytic plane domains with one symmetry
Abstract We sketch the proof that simply connected real analytic plane domains with one symmetry are determined by their Dirichlet or Neumann spectral among other such domains.
Monday, October 23, 2:30pm
Burnside 920
Alexander Strohmaier (Bonn)
High Energy Limits of Laplace-type and Dirac-type Eigenfunctions and Frame Flows
Abstract We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. This is about a joint work with Dmitry Jakobson.
Friday, October 27, 2pm
Burnside 920
Y. Pautrat(Paris-Sud)
Quantum Central Limit Theorem
Abstract Following a joint project with Jaksic, Ogata and Pillet we study the linear response of coupled fermionic systems close to thermodynamic equilibrium. The cornerstones of linear response theory are the Kubo formula, the Onsager reciprocity relations and the fluctuation-dissipation theorem, which relates the fluctuations of observables around their equilibrium value, to the transport properties of the system. In this talk we describe a general central limit theorem, which allows us to construct an algebra of operators representing time fluctuations of, e.g. flux observables. Dynamical properties of the system appear in this algebra; in particular the fluctuation algebra is commutative if and only if the system is in equilibrium and in that case we recover the fluctuation-dissipation thorem.
Friday, November 3, 2pm
Burnside 920
Der-Chen Chang (Georgetown)
Fundamental Solutions for Hermite and Subelliptic Operators
Abstract In this talk, we first introduce a geometric method based on multipliers to compute heat kernels for Laplacian operators with potentials. Using the heat kernel, one may compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on the Euclidean spaces and Heisenberg groups. As a consequence, one may obtain the fundamental solutions for the sub-Laplacian $\Box_J$ in a family of quadratic submanifolds which was introduced by Nagel, Ricci and Stein.
Friday, November 10, 2pm
Burnside 920
Pavel Bachurin (Toronto)
Ergodicity and the size of a neighbourhood of singularity manifolds of dispersing billiards.
Abstract Ergodic theory of multi-dimensional dispersing billiards has been developed in 1980s. An important part of the theory is the analysis of the structure of the sets, where the billiard map is discontinuous. They were assumed to be smooth manifolds till recently, when a new pathological type of behaviour of these sets was found. Thus a reconsideration of earlier arguments was needed. I'll show that for a generic configuration of scatterers these sets behave relatively well and dispersing billiards are ergodic. The proof is base on an estimate of the volume of a neighborhood of a level set of a smooth function (joint with Ch. Fefferman), which has independent interest.

## SUMMER 2006

Monday, June 12, 2:30pm
McGill, Burnside 920
Alexander Vedenov (Kurchatov Institute, Moscow)
Bacterial cell from physical point of view
Abstract: Bacterial cell is considered as system of polymerizing molecular machines of three types - DNA polymerases, RNA polymerases and ribosomes. For an exponentially growing population of bacteria, we suppose synchronous operation of three subsystems of polymerizing molecular machines. Then it is possible to estimate the numbers of these machines in bacterial cell. The result is qualitatively consistent with experimental data for E.coli. The exponential growth on mediums with a various composition of deoxynucleotides triphosphates, ribonucleotide triphosphates and amino acids is discussed. The results compared with experimental data for cell size and for population doubling time.

Thursday, June 29, 3:00pm
McGill, Burnside 920
Eric Sawyer (McMaster)
Regularity of certain subelliptic Monge-Ampere equations
Abstract: Assuming that k is smooth and vanishes only at nondegenerate critical points, we prove that C^2 solutions u to the Monge-Ampere equation detD^2u=k are smooth if and only if the subGaussian principal curvature of u is positive. More general k are treated as well, and the proofs involve an n-dimensional partial Legendre transform, Calabi's identity for the square of third order derivatives of solutions, the Campanato method of Xu and Zuily, the Stein-Rothschild lifting and approximation of vector fields, Jerison's Poincare inequality, subelliptic DeGeorgi Nash Moser theory, Guan's commutator lemma and other subelliptic techniques. Hilbert's 17th problem also plays a role, and we discuss the ways in which these varied topics enter into the problem of detecting smoothness in subelliptic Monge-Ampere equations. In two dimensions, Guan had earlier obtained such results for C^1,1 solutions, but the extension to C^1,1 solutions in higher dimensions remains a challenging open problem.

2005/2006 Analysis Seminar