- Vitali Vougalter, University of Cape Town, will talk on "Existence of
Stationary Solutions for Some Systems of Integro-differential Equations
with Superdiffusion" at 4:00p.m. in N638 Ross.
ABSTRACT:

*We establish the existence of solutions of a system of integro-differential equations arising in populationd dynamics in the case of anomalous diffusion. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are being used.*Refreshments are served after the talk.

- Alexey Kuznetsov, York University, will speak on "Double Sine Function
and its Applications in the Analysis of the Fractional Laplacian" at
4:00p.m. in N638 Ross Building.
ABSTRACT:

*I will discuss two closely related problems: The first one concerns the eigenfunctions of the fractional Laplacian on the positive half-line (with zero boundary conditions on the negative half-line). The scond problem is related to the distribution of extrema of a stable provess and the spectral expansion of its semigroup. I will show that the solution to these problems can be given rather explicitly in terms of a curious special function, called "the double sine function". As applications of these results I will present a new integral transform with generalizes the classical trigonometric sine-tranform and I will also explain what role Diophantine approximations play in the study of extrema of stable processes.*Refreshments will be served after this talk.

- Xiaoxi Duan, York University, will give a talk entitled
"
**Pseudo-Differential Operators for Weyl Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Pseduo-differential operators with operator-valued symbols for Weyl transforms are introduced. We give suitable conditions on the sympbols for which these operators are in the trace class and give a trace formula for them.*Refreshments will be served after the talk>

- Vitali Vougalter, University of Cape Town, will speak on "
**Existence of Stationary Solutions for Some Nonlocal Reaction-Diffusion Equations**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The paper is devoted to the existence of solutions of a nonlocal reaction-diffusion equation arising in population dynamics. The proof is based on a fixed point technique. We use solvability conditions for elliptic operators in unbounded domains which do not satisfy the Fredholm property.*Refreshments will be served after the talk.

- Vitali Vougalter, University of Cape Town, will speak on
"
**Solvability Conditions for Some Non Fredholm Operators in Higher Dimensions**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We establish solvability in H^2(R^d) of certain linear elliptic equations involving the sums of second order differential operators without Fredholm property using the spectral and scattering theory for Schroedinger type operators, generalizing the results obtained in the early work.*Refreshments will be served following the talk.

- Shaosai Huang, Stony Brook Univeristy, will speak on "Local Splitting
Theorem on Singular Spaces" at 4:00p.m. in N638 Ross.
ABSTRACT:

*In this talk, I will recall the basics of the Cheeger-Colding local splitting theorem and show key steps of its generalization to manifolds with certain co-dimension two singularities. Then I will outline the proof of the compactness of certain classes of intrinsically Ricci flat metric measure spaces under the measured Gromov-Hausdorff topology.*Refreshmens will be served after the talk.

### Friday, August 29, 2014

### Friday, August 22, 2014

### Friday, August 15, 2014

- Shahla Molahajloo, Institute for Advanced Studies in Basic Sciences,
will speak on "Pseudo-Differential Operators on Non-Isotripic Heisenberg
Groups with Multi-Dimensional Centres" at 4:00p.m. in N638 Ross.
ABSTRACT:

*We first introduce non-isotropic Heisenberg groups with multi-dimensional cntres and the corresponding Schr/"odinger represenations. The Wigner and Weyl transforms are then defined. Using the Stone-von Neumann theorem, the Fourier transform is defined in terms of these representations and the Fourier inversion formula is given. Pseudo-differential operators with operator-valued symbols are introduced and can be thought of as noncommutative quantizations. We give necessary and sifficient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. We also give a characterization of trace class pseudo-differential operators and a trace formula for these trace class operators.*Refreshments will be swerved after the talk.

### Friday, August 1, 2014

- Vitali Vougalter, University of Cape Town, will give a talk
entitled "On the Existence of Stationary Solutions for Some Nonlinear Heat
Equations" at 4:00p.m. in N638 Ross.
ABSTRACT:

*We establish the existence of H^3(R^5) of stationary solutions of certain nonlinear heat equations using the Fixed Point Technique. The equation for the perturbed solution involves the second order differential operator without Fredholm propery.*Refreshments will be served after the talk.

### Friday, July 18, 2014

- Mateusz Kwasnicki, Wroclaw University of Technology, Poland, will
speak on "Spectral Theory for the Fractional Laplace Operator in an
interval and a half-line" at 4:00p.m. in N638 Ross.
ABSTRACT:

*The talk will start with a breif introduction to the fractional Laplace operator in a domain. In particular, I will state the eigenvalue problem and explain why the soluitons are different from the eigenfunctions of the classical (non-fractional) Laplace operator. After showing some general results, I will move on to two examples in dimension one: an interval and a half-line. In the former case, surprisingly little is known about the eigenvalues and eigenfunctions, a two-term asymptotic formula for the eigenvalues is one of the most detailed results here. For the half-line, an explicit expression for the (generalised) eigenfunctions is known, which yields a Fourier-type diagonalisation of the fractional Laplace operator.*Refreshments will be served after the talk.

### Monday, April 21, 2014

- Paolo Boggiatoo, Univ4ersit/`a degli Studi di Torino, will speak on
"Local Uncertainity Principles in the Cohen Class" at 4:00p.m. in N638
Ross.
ABSTRACT:

*Time-frequency representations in the Cohen class are quadratic forms expressed as a convolution between the classical Wigner transform and a suitable kernel. We explore their behavior with respect to uncertainty principles of local type. More preceisely the results presented in this talk show that a "too large" amount of energy cannot be concentrated in a "too small" set of the time-frequency plane. In particular, for square integrable signals, the energy of a time-frequency representation contained in a measurable set M can be controlled by the standard deviations of the signal and its Fourier transform, and by suitable quantities measuring the size of M.Refreshments will be served following the talk.*

### Friday, March 7, 2014

- Damir Kinzebulatov, Fields Institute, will speak on "Unique
Continuation for Schr/"odinger Operators" at 4:00p.m. in N638 Ross.
ABSTRACT:

*The property of unique continuation (UC) of solutions of a PDE plays a fundamental role in analysis, as well as its applications to mathematical physics and geometry, e.g., for proving that Schr/"odinger operators do not have positive eigenvlaues. The UC property is that if a solution vanishes on an open subset (of its connected omain) thn it must be equal to zero everywhere. In this talk, we prove the UC property for solutions of the differential inequality |\Delta u| \leg ||Vu|| for V in a wide class of potentials including those locally in L^{n/2} and u in a space Y_V of solutions containing all eigenfunctions of the corresponding self-adjoint Schr\"odinger operator.*This is a joint work with Leonid Shartser extending the classical results by D. Jerison and C. Kenig [Ann. Math., 1984], and E.M. Stein [Ann. Math., 1984].

Refreshments will be served following the talk.

### Friday, January 31, 2014

- Xiaoxi Duan, York University, will give a talk entitled "
**On the Fourth-Order Twisted Bi-Laplacian and the Associated Pseudo-Differential Operator on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*After a presentation of the Green function of the twisted bi-Laplcian, we introduce the second order pseudo-differential operator associated to the twisted bi-Laplacian and touch base on its spectral theory and number theory.*Refreshments will be served following the talk.

### Friday, January 17, 2014

- Xiaoxi Duan, York University, will speak on "
**The Heat Kernel and Green Function of a Fourther-Order Hypoelliptic Operator on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The twisted Laplacian is the quantum Hamiltonian of a particle moving on an infinite horizontal plane under the influency of a magnetic field acting perpendicular to the plane. The product of the twisted laplacian and its conjugate is a fourth-order operator and it has arithmetic information that the twisted Laplacian does not have. The fourth-order operator so obtained is dubbed the twisted bi-Laplacian. The aim of this talk is to construct the heat kernel and Green function of the twisted bi-Laplacian, which can then be lifted to the heat kernel and Green function of the product of the sub-Laplacian and its conjugate on the Heisenberg group.*Refreshements will be served after the talk.

### Friday, December 6, 2013

- Vitali Vougalter, University of Cape Town, will speak on "Solvability
Conditions for Some Non-Fredholm Operators in a Layer in Four Dimensions"
at 4:00p.m. in N638 Ross.
ABSTRACT:

*We study solvability in H^2 of certain linear nonhomogeneous elliptic problems involving the sum of the periodic Laplacian and a Schroedinger operator without Fredholm property and prove that under reasonable technical conditions, the convergence in L^2 of the right sides implies the existence and the convergence in H^2 of the solutions. We generalize the methods of spectral and scattering theory for Schroedinger-type operators from our preceding work.*Refreshments will be served after the talk.

### Friday, October 11, 2013

- Xiaoxi Duan, York University, will speak on "
**The Dirichlet Divisor Problem, Traces and Determinants for Complex Powers of the Twisted Bi-Laplacian" at 4:00p.m. in N638 Ross.**ABSTRACT:

*Estimating the counting function for the eigenvalues of th4e twisted bi-Laplacian leads to the Dirichlet divisor problem, which is then used to compute the trace of the heat semigroup and the Dixmier trace of the inverse of the twisted bi-Laplacian. The zeta function regularizations of the traces and determinants of complex powers of th4e twisted bi-Laplacian are computed. A formula for the zeta function regularizations of determinants of heat semigroups of complex powers of the twisted bi-Laplacian is given.*Refreshments will be served after the talk.

### Friday, August 16, 2013

- Vladimir Vinogradov, Ohio University, will speak on "New Properties of
the General Wright Special Functions and Related Topics in Probability
Theory and Mathematical Statistics" at 4:00p.m. in N638 Ross.
ABSTRACT:

*We discuss joint works with Richard B. Paris of University of Abertay Dundee on interesting connections between general Wright special functions and some classes of the univariate probability distributions. Such connections enable one to construct refined saddlepoint-type approximations for exponential dispersion models, establish new fluctuation properties of some stochastic processes, and also derive some previously unknown representations of the Theory of Special Functions. We review some subtle analytical methods used. This is based on our recently published joint papers with Olga Yanushkevichiene (Vilnius University) as well as our two unpublished manuscripts.*Refreshments will be served after the talk.

### Wednesday, August 14, 2013

- Alip Mohammed, Petroleum Institute, Abu Dhabi, UAE, will speak on
"Eigenvalue Problems for Some Elliptic Equations" at 4:00p.m. in N638
Ross.
ABSTRACT:

*Eigenvalue problems for holomorphic and harmonic functions are studied. The third boundary value problem for these functions is a collection of more general problems than the Robin or the Steklov types of problems and they have remained open for study. For the unit disc, explicit solutions to these problems can be provided. The Steklov spectrum is discrete for both functions. Inspired by this method, the eigenvalues and corresponding solutions of the Steklov problem for some elliptic equations in an n-dimensional ball can be constructed.*Refreshments will be served following the talk.

### Friday, June 7, 2013

- Vitali Vougalter, University of Cape Town, will speak on "Solvability
in the Sense of Sequences for Some Non-Fredholm Operators" at 4:00p.m. in
N638 Ross.
ABSTRACT:

*We study solvability of certain linear nonhomogeneous elliptic problems and show that under reasonable technical conditions, the convergence in L^{2}({R}^{d}) of the input implies the existence and the convergence in H^{2}({R}^{d}) of the solutions. The equations involve second order differential operators without Fredholm property and we use methods from the spectral and scattering theory for Schr\"odinger type operators.*Refreshments will be served after the talk.

### Friday, May 24, 2013

- Luigi Riba, Universit\`a di Torino, will speak on "
**Phases of Multi-Dimensional Stockwell Transforms and Multi-Dimensional Instantaneous Frequencies**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The instantaneous frequency of a multi-deminsional signal can be looked at as the gradient (derivative) of the phase of the signal in polar form. Multi-dimensional Stockwell transforms with phases built into the multiscale structure are shown to the ideal for computing the instantaneous frequencies of multi-dimensional signals.*Refreshments will be provided after the talk.

### Friday, May 17, 2013

- Jiawei Li, York University, will speak on "
**Localization Operators for Ridgelet Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Ridgelet transforms, developed by Emannuel Cand/'es in his Ph.D. Thesis in the Department of Statistics at Stanford, were intended to be a key to higher-dimensional intermittency, and higher-dimensional wavelet transforms. Yet th estructure of ridgelet transforms is richer in view of the underlying Radon transforms inthe genesis. In this talk, we introduce the localization operators for the ridgelet transforms, prove the L^2-boundeness of the localization operators corresponding to L^p symbols on the phase space, introduce a class of trace class operators related to these localization operators and obtain a trace formula for these operators. These trace class operators are reminiscent of the Landau-Pollak-Slepian operators and the more general wavelet multipliers.*Refreshments will be served after the talk.

### Friday, May 10, 2013

- Luigi Riba, Universit\'a di Torino, will speak on "
**Resolution of the Identity Formulas for Stockwell Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have been studied extensively. We introduce multi-dimensional Stockwell transforms that include multi-dimensional Gabor transforms as special cases. Resolution of the identity formulas for multi-dimensional Stockwell transforms are proved. Obstructions to the constructions of the formulas beyond dimensions 1, 2, 4 and 8 due to parallelizability fo the sphere of R. Bott and J. Milnor are explained.*Refreshements will be served after the talk.

### Friday, May 3, 2013

- Luigi Riba, Universit\'a di Torino, will speak on "
**Group Representations and Stockwell Transforms" at 4:00p.m. in N638 Ross.**ABSTRACT:

*After a brief recapitulation of how the gabor transforms and the wavelet transforms arise from the Weyl-Heisenberg group and the affine group, respectively, we show how Stockwell transforms come from a section of the homogeneous space obtained by taking the quotient of the affine Heisenberg group by a subgroup.*

- Vitali Vougalter, University of Cape Town, will speak on
"

- Shahla Molahajloo, Queen's University, will speak on "
**The Riemannian Geometry of the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We compute the Riemannian curvature of the Heisenberg group and then contract it to the sectional curvature, Ricci curvature and the scalar curvature. The main result is that the Heisenberg group is a manifold of constant positive curvature.*Refreshments will be served after the talk.

- Vitali Vougalter, University of Cape Town, will speak on "
**Sharp Semiclassical Bounds for the Moments of Eigenvalues for Some Schrödinger Type Operators with Unbounded Potentials**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we obtain sharp lower bounds for the moments of positive powers not exceeding one for such eigenvalues. When considering a Schrödinger operator with the relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the sharp Lieb-Thirring estimate for the moments of some negative powers of its eigenvalues.*(This talk was originally scheduled for July 13, 2012 and postponed.)Refreshments will be served after the talk.

- Alessandro Oliaro, Universit\`a di Torino, will speak on
"
**Regularity of a Class of Differential Operators**" at 4:00p.m. in N638 Ross.ABSTRACT:

*In this talk we deal with the problem of regularity for non-hypoelliptic partial differential equations with polynomial coefficients. An operator $A$ on the space of tempered distributions $\mathcal{S}^\prime$ is regular if $u$ belongs to the Schwartz class $\mathcal S$ whenever $Au\in \mathcal S$. By using tensor products of topological vector spaces and a kind of Wigner transform we reduce the study of regularity of a non-hypoelliptic differential operator, like the twisted Laplacian, to a hypoelliptic one, like the harmonic oscillator.*Refreshments will be served after the talk.

- Vitali Vougalter, University of Cape Town, will give a talk entitled
"
**Sharp Semiclassical Bounds for the Moments of Eigenvalues for some Schr\"odinger Type Operators with Unbounded Potentials**" at 4:00p.m. in N638 Ross..ABSTRACT:

*We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we obtain sharp lower bounds for the moments of positive powers not exceeding one for such eigenvalues. When considering a Schr\"odinger operator with the relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the sharp Lieb-Thirring estimate for the moments of some negative powers of its eigenvalues.**Refreshments will be served after the talk.*

- Xiaoxi Duan, York University, will give a talk on "
**Microlocal Analysis for the Geodesics of the Laplacians on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The analysis and geometry of a sub-bundle of the tangent bundle of the Heisenberg group generated by the horizontal vector fields lead to a study of the sub-Laplacian on the Heisenberg group. If the sub-bundle is the whole bundle, then we get the (full) Laplacian on the Heisenberg group. The Laplacian is an elliptic operator, but the sub-Laplacian is nowhere elliptic. This drastic difference in ellipticity is reflected in the nature of the geodesics, which are obtained as solutions of the Hamilton differential equations for the symbols of the Laplacians. As such, the method is a prototype of a technique known as microlocal analysis, also known as analysis on the cotangent bundle.*Refreshments will be served after the talk.

- Xiaoxi Duan, York University, will give a talk entitled "
**On a Lie Algebra for the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The aim of this seminar is to show that the real Lie algebra generated by the building blocks of the twisted Laplacian, also know as the Schr\"odinger operator of a charged partcle on an infinite plane with a constant magnetic field perpendicular to it, is isomorphic with the Lie algebra of the one-dimensional Heisenberg group. We begin with a review of the building blocks of the twisted Laplacian, the corresponding real Lie algebra, the Heisenberg group and its Lie algebra of left-invariant vector fields. Then we set up the isomorphism between the two Lie algebras.*Refreshments will be served after the talk.

- Shahla Molahajloo, Queen's University, will speak on "
**Geodesics on the Hierarchical Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Horizontal curves on the hierarchical Heisenberg group are defined. All the geodesics are obtained by solving the corresponding Hamiltonian system of partial differential equations. It is shown that the image of a geodesic on "certain" planes is a circle or a line, and the area of a circle is proportional to the difference of the missing directions. Conservation of energy along geodesics is also presented.*Refreshments will be served after the talk.

- Luigi Riba, Universit\`a di Torino, will speak on
"Continuous Inversion Formulas for Multi-Dimensional Stockwell
Transforms" at 4:00p.m. in N638 Ross.
ABSTRACT:

*Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have been studied extensively. We introduce in this talk multi-dimensional Stockwell transforms that include multi-dimensional Gabor transforms as special cases. Continuous inversion formulas for multi-dimensional Stockwell transforms are proved.*Refreshments will be served in Grad Lounge after the talk.

- Vitali Vougalter, University of Cape Town, will give a talk entitled
"
**On the Solvability Conditions for a Linearized Cahn-Hilliard Equation**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We derive solvability conditions in H^4 (R^3) for a fourth order partial differential equation which is the linearized Cahn-Hilliard problem using the results obtained for a Schroedinger type operator without Fredholm property in our preceding work.*Refreshments will be served in Grad Lounge after the talk.

- Jiawei Li, York University, will give a talk entitled "
**Localization Operators for Curvelet Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Wavelets have been a powerful tool for signal analysis for the past two or three decades and have been moved from the center by sparsity required of new gadgets. Curvelets as developed by Emmanuel Candes and David Donoho are particularly spectacular with regard to the sharp localization in the Fourier domain and the flexibility of the non-isotropic dilations. They are best used for analyzing singularities of functions and distributions lying on curves, and for almost diagonalization of pseudo-differential operators, Fourier integral operators and localization operators. The focus of this talk is on the theoretical underpinnings of the matrix representations of localization operators for curvelet transforms. A glimpse into the discrete matrix representations, which is work in progress, is offered if time permits.*Refreshments will be served in Grad Lounge after the talk.

- Toshinao Kagawa, Tokyo University of Science, will speak on "
**Weyl Transforms and Solutions to Schr\"odinger Equations for Time-Dependent Hermite Operators**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Using the diagonalization of Weyl transforms with radial symbols in L^\infty(R^2) in terms of explicit formulas for the eigenvalues with respect to the Hermite basis for L^2(R), exact solutions to Schr\"odinger equations governed by time-dependent Hermite operators are analyzed in detail. Formulas for the Schr\"odinger kernels of these time-dependent Hermite operators are derived.*Refreshments are served in Grad Lounge after the talk.

- Shahla Molahajloo, Queen's University, will speak on
"
**Metaplectic Equivalence of the Hierarchical Twisted Laplacian**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The hierarchical twisted Laplacian on L^2(R^n), n)1, can be considered as a higher dimensional analog working in all dimensions of the quantum mechanical Hamiltonian of an electron moving on an infinite plane under the influence of a magnetic field perpendicular to the plane - quantum Hall effect. Recent works have added evidence that this is indeed the correct interpretation by showing that the hierarchical twisted Laplacian is unitarily equivalent to the tensor product of the one-dimensional Hermite operator and the identity operator on L^2(R^n) by means of a suitable metaplectic operator, which is a Fourier integral operator more general than a pseudo-differential operator. (This is joint work with Luigi Rodino of Universit\`a di Torino and M. W. Wong at York University.)*Refreshements will be served in Grad Lounge after the talk.

- Julio Delgado, Universidad del Valle, Colombia, will speak on "
**The Trace and the Hardy-Littlewood Maximal Function**" at 4:00p.m. in N638 Ross.ABSTRACT:

*In this talk we present some results concerning the analysis of the trace of operators on Lebesgue spaces. We start with some preliminaries on the concept of trace and then we show a way to study traceable kernels using general versions of the Hardy-Littlewood maximal function on a class of measure spaces.*Refreshments will be served in Grad Lounge after the talk.

- Vital Vougalter, University of Cape Town, will speak on "
**On the Existence of Stationary Solutions for Some Integro-Differential Equations**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We show the existence of stationary solutions for some reaction-diffusion type equations in the appropriate H^2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.*Refreshments will be served in the Grad Lounge following the talk.

Kenro Furutani, Tokyo University of Science , will speak on "" at 4:00p.m. in N638 Ross.Action Function for a Higher Order Oscillator and the Heat Kernels of Higher Step Grushin Operators ABSTRACT:

*The purpose of this talk is to discuss how we can construct the heat kernels for sub-Laplacians in an explicit integral form. Of course such cases will be highly limited. Here we only treat with a typical operator, called Grushin operator. So, first we construct the heat kernel of the two step Grushin operator by two methods, one is the eigenfunction expansion which leads us to an integral form for the heat kernel, then we explain a method called complex Hamilton-Jacobi method invented by Beals-Gaveau-Greiner. The main content is a construction of an classical action function for a higher order oscillator. Until now, we have no explicit expression of the heat kernels for higher step Grushin operators. Here we show a phenomenon that our action function will play a role for the construction of heat kernels for higher step cases.*Refreshements will be served in the Grad Lounge following the talk.

- Luigi Riba, Università di Torino, will give a talk on "
**Localization Operators on the Stockwell Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*It is well-known that the Gabor transform and the wavelet transform have as their underspinning Lie groups given by, respectively, the Weyl-Heisenberg group and the affine group. Recent studies have revealed the structure of the Stockwell transform as a hybrid of the Gabor transform and the wavelet transform. More recently, the underlying Lie group for the Stockwell transform has been shown by Boggiatto, Galbis and Fernández to be a hybrid of the Weyl-Heisenberg group and the affine group, which we call the Stockwell group. Using the fact that the Stockwell transforms are induced by square-integrable representations on the Stockwell group, it is shown by Molahajloo and Wong that localization operators on the Stockwell group belong to the Schatten-von Neumann classes. After a review of the above-mentioned results, we show that with further conditions on the symbols, the corresponding localization operators are paracommutators, which include Fourier multipliers and, more interestingly, paraproducts first envisaged by J.-M. Bony in his study of nonlinear PDEs. These results are then compared with those obtained by Wong on the affine group.*Refreshments will be served in the Grad Lounge after the talk.

- Huaihui Chen, Nanjing Normal University, will give a talk entitled
"
**On the Bloch Constant**" at 4:00p.m. in N638 Ross.ABSTRACT:

*This talk contains a survey of the Bloch constant problem in geometric complex analysis. In particular, some new results on the harmonic Bloch constant are reported.*Refreshments will be served in Grad Lounge after the talk.

- Spyridon Alexakis, University of Toronto, will give a talk entitled
"
**The Space of Complete, Embedded Minimal Surfaces in Hyperbolic Space: Renormalized Area and Bubbling**" at 4:00p.m. in N638 Ross.ABSTRACT:

*I will discuss some older and some very recent work on a natural energy functional on the space of complete minimal surfaces in hyperbolic manifolds, with a boundary at infinity that can be freely prescribed. This functional was introduced by Graham and Witten via a renormalization of the (infinite) area of such surfaces. We show that it can be seen as an analogue of the well-studied Willmore energy for closed surfaces. We also discuss bubbling phenomena for this energy. This is the first study of bubbling in the context of surfaces whose boundaries are free. Joint work with R. Mazzeo.*Organizer's Note: Professor Alexakis obtained his PhD in Mathematics in 2005 from Princeton University with a thesis "Local and Global Aspects of Conformal Geometry" written under the direction of Charles Fefferman.

Refreshements will be served after the talk in the Grad Lounge.

- M.W. Wong, York University, will speak on "
**Geometric Analysis on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*This series of three lectures is intended for Ph.D. students who are interested in entering the field of geometric analysis on the Heisenberg group. In Lecture 1, the basics of the Heisenberg group, its Lie algebra of left-invariant vector fields and the sub-Laplacian are given. The conversion of the sub-Laplacian to a family of twisted Laplacians is explained. Then we introduce the Fourier-Wigner transforms, which are used to diagonalize the twisted Laplacians. In Lecture 2, the heat kernels and Green functions of the twisted Laplacians are constructed. They are then used to construct the heat kernel and Green function of the sub-Laplacian on the Heisenberg group. The procedure is akin to Feynman's sum over all histories in quantum mechanics. The classic formulas for the heat kernel and Green function of the sub-Laplacian are recaptured. Rudiments of sub-Riemannian geometry on the Heisenberg group are presented to illuminate these formulas. The last lecture is devoted to some recent not-yet-published results on the spectrum of the sub-Laplacian and the solution of the Schrödinger equation on the Heisenberg group.*Refreshments will be served in the Grad Lounge after the talk.

- Farzad Fathi Zadeh, York University, will speak on "
**The Gauss-Bonnet Theorem for Noncommutative Tori with a General Conformal Structure**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori T^2 with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number in the upper half plane, representing the conformal class of a metric on T2 , and a Weyl factor given by a positive invertible element C ( T2 ), the value (0) at the origin of the spectral zeta function of the Laplacian attached to ( T2 , , ) is independent of and .*Refreshments will served in the Grad Lounge after the talk.

- Farzad Fathi Zadeh, York University, will give a talk entitled
"
**The Gauss-Bonnet Theorem for Noncommutative Tori with a General Conformal Structure**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori $T^{2}_{\theta}$ with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number $\tau$ in the upper half plane, representing the conformal class of a metric on $T^{2}_{\theta}$, and a Weyl factor given by a positive invertible element $k \in C^{\infty}(T^{2}_{\theta})$,the value $\Delta(0)$ at the origin of the spectral $\zeta$ function of the Laplacian $\Del$ attached to ($T^{2}_{\theta},\tau,\kappa$) is independent of $\tau$ and $\kappa$. This is joint work with M. Khalkhali.*Refreshments will be served in Grad Lounge after the talk.

- Abdol-Reza Mansouri, Queen's University, will give a talk entitled
"
**Small-Time Asymptotics for a Class of Hypoelliptic Operators Arising from Optimal Control Theory**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Problems of optimal control of non-holonomic control-affine systems naturally lead to the definition of sub-elliptic Laplacians. One problem of interest in control theory is the precise relation between the value function of the optimal control problem and the heat kernel of the associated sub-elliptic Laplacian. In this talk, we will discuss the sub-elliptic Laplacian on a generalization of the Heisenberg group arising from an optimal control problem, and we will relate the value function of that optimal control problem to the small-time asymptotic behavior of the corresponding heat kernel.*Refreshments will be served in Grad Lounge after the talk.

- Vitali Vougalter, University of Toronto, will give a talk entitled
"
**On the Solvability Conditions for Some Non-Fredholm Operators**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We obtain solvability conditions for some elliptic equations involving non-Fredholm operators, which are sums of second order differential operators with the methods of spectral theory and scattering theory for Schr\"odinger-type operators.*Refreshments will be served in Grad Lounge after the talk.

- Geni Gupur, Xinjiang University, will be speaking on "
**Functional Analysis Methods for Reliability Models**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Using functional analysis, we analyze the dynamics for reliability models formulated by techniques of supplementary variables and described by semigroups of partial differential operators with integral boundary conditions. After some motivating remarks, we introduce the ideas in the study of (1) the well-posedness of reliability models, (2) the asymptotic behavior of time-dependent solutions of reliability models, and (3) the asymptotic behavior of indices of reliability systems. We conclude the lecture with some open problems.*Refreshments will be served in Grad Lounge after the talk.

- Geni Gupur, Xinjiang University, will speak on "
**Functional Analysis Methods for Queueing Models**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Using functional analysis, we analyze the dynamics for queueing models formulated by techniques of supplementary variables and described by semigroups of partial differential operators with integral boundary conditions. After some motivating remarks, we introduce the ideas in the study of (1) the well-posedness of queueing models, (2) the asymptotic behavior of time-dependent solutions of queueing models, and (3) the asymptotic behavior of indices of queueing systems. We conclude the lecture with some open problems.*Refreshments will be served in the Grad Lounge after the talk.

- Shahla Molahajloo, York University, will speak on "
**Ellipticity, Fredholmness, Spectral Invariance and Essential Spectra of Pseudo-Differential Operators on the Unit Circle**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Analogues of pseudo-differential operators of the H"ormander class for the unit circle S1 centered at the origin are studied. We prove that elliptic pseudo-differential operators are Fredholm on L^p(S^1), 1(p(ìnfinity. Then we prove the spectral invariance in L^2(S^1) and we use the spectral invariance to prove that ellipticity is a necessary condition for Fredholmness on L^2(S^1). We associate the minimal and maximal operators on L^p(S^1) to every pseudo-differential operator with symbol of positive order. If the pseudo-differential operator is elliptic, then we prove that the minimal and maximal operators are equal and we compute the domain of the minimal operator explicitly. The essential spectra of elliptic pseudo-differential operators with positive order and bounded pseudo-differential operators of order 0 are computed.*Refreshments will be served after the talk in the Grad Lounge.

- Alip Mohammed, York University, will give a talk entitled "
**The Poisson Equation with the Third Boundary Condition**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The inhomogeneous Robin/third boundary condition with general coefficient for the Poisson equation on the unit disc is studied in terms of holomorphic functions using Fourier analysis. It is shown that against the usual expectations this problem cannot have a unique solution unless the coefficient of the first order term in the boundary condition is a constant. For the case of general coefficient, it is actually a problem with essential singularity in the domain, but is still well-posed under proper assumptions and the unique solution can be given explicitly.*Refreshments will be served in Grad Lounge after the talk.

- Mohamad Pouryayevali, University of Isfahan, will speak on
"
**Generalized Gradients on Riemannian Manifolds**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Nondifferentiable functions are often considered on finite or infinite-dimensional Banach spaces, where the linear structure plays a central role. However, in various aspects of control theory, matrix analysis and optimization, nonsmooth functions arise naturally on smooth manifolds. In this talk we introduce a notion of generalized gradient of locally Lipschitz functions onRiemannian manifolds. Then, its relation with other subdifferentials will be discussed.*Refreshments will be served in Grad Lounge after the talk.

- Amram Meir, York University, will give a talk on "
**D.J. Newman's Short Analytical Proof of the Prime Number Theorem**" at 4:00p.m. in N638 Ross.ABSTRACT:

*As the problem of the frequency of primes among the positive integers was of interest already much earlier, Gauss expressed his belief (based on theoretical reasons and also on manual counting) that the "density" of the primes "on average" should be about 1/log n. His meaning was likely that the number of primes < x, for large x, is approximately about the value of the definite integral of 1/log t over [2,x]. The first significant result, due to Chebyshev, established that the number of primes < x is between 89% and 111% of the value of the above integral.**Riemann, in his landmark paper (1859) introduced new, complex analytical methods in attempting to solve the problem. In the decades that followed Riemann's paper several solutions were obtained by such major figures of mathematics of the time as Hadamard, de la Vallee Poussin, von Mangoldt and others. Later refinements were due to Hardy, Littlewood, Landau and many others. The solutions usually involve intricate and complicated calculations. In 1980, in a paper of less than 4 pages, D.J. Newman proved an analytic (Tauberian type) theorem from which, as he has shown, the Prime Number Theorem easily follows. The analytic theorem is exceptionally clever and involves not more than Cauchy's integral formula for analytic functions.*Refreshments will be served in Grad Lounge after the talk.

- Tadahiro Oh, University of Toronto, will give a talk entitled "
**KdV with Measures as Initial Data, and Stochastic KdV (SKdV) with Additive and Multiplicative Noise**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Bourgain '97 proved global well-posedness of the periodic KdV with measures as initial data, assuming that the total variation is sufficiently small. His argument was based on the nonlinear analysis on the second iteration of the integral formulation, assuming an a priori bound on the Fourier coefficients. With the complete integrability of KdV, he then proved such an a priori control.**In this talk, we first discuss the nonlinear analysis on the second iteration without the complete integrability or smallness assumption on the total variation. This answers a question posed by Bourgain at least in the local-in-time setting. This also provides a proof of the invariance of the white noise for KdV. Then, using a stochastic version of such nonlinear analysis, we discuss local well-posedness of SKdV with additive space-time white noise. Finally, we consider SKdV with multiplicative noise in $L^2(\mathbb{T})$. By a sequence of transformations, we reduce it to a system of mean-zero SKdV and a SDE (for the mean of the original solution), which is then shown to be globally well-posed.*Refreshments will be served in the Grad Lounge after the talk.

- Frederic Rochon, University of Toronto, will give a talk on "
**The Heat Kernel on Some Singular Spaces**" at 4:00p.m. in N638 Ross.ABSTRACT:

*After reviewing the construction of the heat kernel on a compact manifold from the point of view of Melrose, we will indicate how this approach, as was shown by Vaillant, admits nice generalizations to singular spaces like a surface with cusps. From this construction, we will show it is relatively easy to describe the short time asymptotic of various quantities associated to the heat kernel. If time permits, we will indicate some recent applications in index theory.*Refreshments will be served in Grad Lounge after the talk.

- Yu Liu, York University, will speak on "
**L^p Boundedness and Compactness of Localization Operators for Stockwell Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*L^p-boundedness is obtained for the localization operators for Stockwell transforms with respect to certain types of symbols and admissible wavelets. L^p Compactness of these operators is then discussed under certain conditions of the symbols and the admissible wavelets.*Refreshment is served in Grad Lounge after the talk.

- Brian Street, University of Toronto, will speak on "
**Multi-Parameter Carnot-Carathéodory Balls**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Abstract: We discuss multi-parameter Carnot-Carathéodory balls. In particular, we discuss questions motivated by multi-parameter singular integrals. These results generalize results due to Nagel, Stein, and Wainger in the single parameter setting.*Refreshments will be served in the Grad Lounge following the talk.

- Mehran Ebrahimi, University of Toronto, will speak on
"
**Hyper-Demons: Image Registration under Varying Illumination**" at 4:00p.m. in N638 Ross.ABSTRACT:

*In recent years, contrast-enhanced Magnetic Resonance (MR) imaging has emerged as a powerful screening tool for identifying carcinoma of the breast. Accurate registration (i.e. alignment) of dynamic contrast-enhanced breast MR images is valuable for proper identification of the lesions.**To address the problem, we propose and apply an extension of the well-known "demons" image registration algorithm. This novel generalization is derived directly from the optical-flow constraints in a variational formulation. The extension provides a new mathematical interpretation of the demons algorithm in a consistent and improved framework. (This is joint work with Anne T. Martel.)*Refreshments will be served after the talk.

- Tadahiro Oh, University of Toronto, will speak on "
**Multiplier Problem for the Ball**" at 4:00p.m. in N638 Ross.ABSTRACT:

*In this expository talk, we will discuss C. Fefferman's disproof of the Disc Conjecture "the characteristic function for the unit ball is an L^p multiplier in R^n for 2n/(n-1) < p < 2n/(n+1)." First, we will show that the Fourier multiplier operator T corresponding to the characteristic function for the unit ball is unbounded in L^p for the values of p outside the range described in the Disc Conjecture using the asymptotic behavior of the Bessel functions. Then, using the construction of Besicovitch-type sets in R^2, we will show that T is bounded only in L^2 (R^2) (which immediately implies that T is bounded only in L^2(R^n).)*Refreshments will be served in the Grad Lounge after the talk.

- Shahla Molahajloo, York University, will speak on
"
**Square-integrable Group Representations and Localization Operators for Modified Stockwell Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Modified Stockwell transforms including the classical Stockwell transforms and a variant of the versatile wavelet transform have recently been studied in the context of time-frequency analysis. They are hybrids of the Gabor transforms and wavelet transforms. As such, they perform multi-scale analysis while maintaining the intuition and the properties offered by the Fourier transforms. It is well-known that the mathematical underpinnings of the Gabor transforms and the wavelet transforms are based on square-integrable representations of the Weyl-Heisenberg group and the affine group, respectively. The aim of this talk is to present the Lie groups and the corresponding square-integrable representations for the modified Stockwell transforms. These results are based on the recent works of Boggiatto, Fernandez and Galbis. Localization operators for the Stockwell transforms, playing the roles of time-varying filters in time-frequency analysis, are introduced. The Schatten-von Neumann classes of these localization operators are established and the traces of trace-class localization operators computed.*Refreshements will be served in the Grad Lounge after the talk.

- M.W. Wong, York University, will give a talk on "
**Liouville's Theorems for the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We give noncommutative versions of Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group. (This joint work with Aparajita Dasgupta will appear in a volume of the Banach Center Publications and will be presented at the London Analysis Seminar held at University College London this summer.)*Refreshments will be served in Grad Lounge after the talk.

- Yu Liu, York University, will speak on "
**L^p Boundedness and Compactness of Localization Operators**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Stockwell transforms are hybrids of Gabor transforms and wavelet transforms in time-frequency analysis. Localization operators are filters that are used in signal analysis and image processing. The mathematical underpinnings of Stockwell transforms and the corresponding localization operators have only been developed in the last few years. In this talk, we give the boundedness and compactness on L^p(R) of localization operators for Stockwell transforms. (This is joint work with Dr. Alip Mohammed.)*Refreshments will be served in Grad Lounge after the talk.

- Shahla Molahajloo, York University, will speak on "
**L^2 Boundedness and Compactness of Pseudo-Differential Operators on the Unit Circle**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We present sufficient conditions on the symbols to ensure that the corresponding pseudo-differential operators are bounded linear operators and compact operators on L^2(S^1). Results on the Schatten-von Neumann properties of these operators are also presented. The correponding results for the group of all integers are described if time permits.*Refreshments are served in Grad Lounge after the talk.

- Fariba Ershad, Payam-e-Noor University, will give a talk on "
**An Extension of the Gleason-Kahane-Zelazko Theorem**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Let A be a unital complex commutative Banach algebra and M be a one-codimensional subspace of A. Then M is an ideal if and only if M consists only of non-invertible elements. This is the Gleason-Kahane-Zelazko theorem. In 1987, K. Jarosz extended this theorem and defined a property for Banach algebras. We shall talk about this and prove that every unital self-adjoint commutative Banach algebra has this property. Also we recall 4 problems that Jarosz in 1991 mentioned in his paper. (Two of these have been solved, but the others are still open.)*Refreshments will be served in Grad Lounge after the talk.

- Paolo Boggiatto, Universita di Torino, will give a talk on
"
**Time-Frequency Representations and Classes of Pseudo-Differential Operators (I)**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We introduce the time-frequency analysis of signals with particular attention to the spectrograms and Wigner representations. Then we connect them with localization operators and Weyl pseudo-differential operators, and study their actions in the setting of Lebesque spaces.*Refreshments will be served in Grad Lounge at 6:00p.m.

- Alessandro Oliaro, Universita di Torino, will speak on
"
**Time-Frequency Representations and Classes of Pseudo-Differential Operators (II)**" at 5:00p.m. in N638 Ross.ABSTRACT:

*We consider a modification of the Wigner representation depending on a parameter $\tau$, showing its connection with some well-known classes of pseudo-differential operators. We obtain further a new sesquilinear form that improves both the Wigner and its $\tau$ modifications from the point of view of the time-frequency representation of a signal.*Refreshments will be served in Grad Lounge at 6:00p.m.

*Aparajita Dasgupta, York University, will give a talk on "***Ellipticity and Fredholmness on L^p(R^ n) of Pseudo-Differential Operators with Exit at Infinity**" at 4:00p.m. in N638 Ross.*ABSTRACT:*Based on the works of D. Grieme and B.-W. Schulze, we prove that ellipticity and Fredholmness are equivalent for pseudo-differential operators with exit at infinity on L^p(R^ n).Refreshments will be served in the Grad Lounge following the talk.

- Frederic Rochon, University of Toronto, will speak on "
**The Residue Determinant from a Topological Point of View**" at 4:00p.m. in N638 Ross.ABSTRACT:

*After reviewing various notions of determinants on spaces of pseudodifferential operators, we will give a topological description of the residue determinant recently introduced by Simon Scott. We will then indicate how this can be used to get a global existence result for this determinant.*Refreshments will be served in Grad Lounge after the talk.

- Alip Mohammed, York University, will give a talk on "
**Solutions of the Robin Problem for Overdetermined Inhomogeneous Cauchy-Riemann Systems**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The inhomogeneous Robin conditions with general coefficients for overdetermined Cauchy-Riemann systems of equations on the poly-disc are studied using Fourier analysis. It is shown that this problem for the case of general coefficients is actually a problem with essential singularity in the domain, but is still well posed under certain compatibility conditions. Under proper assumptions, the unique solution is given explicitly.*Refreshments will be served in Grad Lounge after the talk.

- Aparjita Dasgupta, York University, will speak on "
**Weyl Transforms and the Heat Equation for the Sub-Laplacian on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The sub-Laplacian on the Heisenberg group is a nowhere elliptic partial differential operator. Using Weyl transforms depending on Planck's constant, we give a new formula for the solution of the initial value problem governed by the sub-Laplacian on the Heisenberg group. Estimates for the solution in terms of the initial value are given.*Refreshments will be served in Grad Lounge after the talk.

- M.W. Wong, York University, will give a talk entitled
"
**Fourier-Wigner Transforms and Liouville's Theorems for the Sub-Laplacian on the Heisenberg Group**" at 4:00p.m. in N638 Ross.ABSTRACT:

*By decomposing the sub-Laplacian on the Heisenberg group into a family of twisted Laplacians parametrized by Planck's constant and using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.*Refreshements will be served in Grad Lounge after the talk.

- Alip Mohammed, York University, will speak on "
**Sampling and Computations of Pseudo-Differential Operators**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Discrete formulas for pseudo-differential operators based on the Shannon-Whittaker sampling formula and the Poisson summation formula are given. Computations of pseudo-differential operators using the fast Fourier transform on MATLAB are presented if time permits. (This is joint work with M. W. Wong.)*Refreshments will be served in Grad Lounge after the talk.

- M.W. Wong, York University, will speak on "
**Absolutely Referenced Phase Information and Spectra of Modified Stockwell Transforms**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Modified Stockwell transforms are introduced to include the classical Stockwell transforms as special cases. Among them are also transforms that are reminiscent of the versatile wavelet transforms. The extension of the absolutely referenced phase information of the classical Stockwell transforms to the modified Stockwell transforms is given in terms of Riesz potentials. Based on the resolution of the identity formula, the spectra of every modified Stockwell transform are shown to form a reproducing kernel Hilbert space.*Refreshments are served in Grad Lounge after the talk.

- Yu Liu, York University will speak on "
**The Stone-von Neumann Theorem**" at 4:00p.m. in N638 Ross.ABSTRACT:

*A holy grail in the study of Lie groups is to seek explicit characterizations of irreducible and unitary representations. The Stone-von Neumann theorem gives a description of all the irreducible and unitary representations of the Heisenberg group. Of particular interests in analysis and PDE are the Schr\"odinger representations, which are infinite dimensional. We sketch a proof of the Stone-von Neumann theorem that is attributed to von Neumann and entails a study of the Gaussians, twisted convolutions, integrated representations, Wigner transforms and Weyl transforms. We give also a description of the square-integrable representations of the Weyl-Heisenberg group, which is a variant of the Heisenberg group with a compact center. The significance of these square-integrable representations in the study of wavelet transforms and localization operators is highlighted.*Refreshments are served in the Grad Lounge after the talk.

- Aparajita Dasgupta, York University, will speak on "
**Weyl Transforms and the Inverse of the Sub-Laplacian on the Heisenberg Group**" at 4:00p.m. in N627 Ross.ABSTRACT:

*The sub-Laplacian on the Heisenberg group is the prototype of a degenerate elliptic partial differential operator on $R^n$. It is a subelliptic operator in the sense that there is a loss of one derivative in the regularity of the operator. By defining new Weyl transforms depending on Planck's constant, we give new formulas for the inverse of the sub-Laplacian on the Heisenberg group and have a new look at the subellipticity. (This is joint work with M. W. Wong.)*Refreshments will be served in the Grad Lounge after the talk.

- Alip Mohammed, York University, will give a talk on "
**Wavelet Multipliers on L^p(R^n)**" at 4:00p.m. in 1016 Vari Hall.ABSTRACT:

*We give results on the boundedness and compactness of wavelet multipliers on L^p(R^n), 1<_p<_oo.*Refreshments will be served in the Grad Lounge after the talk.

- Aparajita Dasgupa, York University, will give a talk entitled
"
**Global Hypoellipticity of the Twisted Laplacian**" at 4:00p.m. in 1016 Vari Hall.ABSTRACT:

*We use pseudo-differential operators of the Weyl type (Weyl transforms) and Fourier-Wigner transforms of Hermite functions to construct the heat kernel and Green function of the twisted Laplacian. The Green function can be used to prove the global hypoellipticity of the twisted Laplacian. Then we introduce a scale of Sobolev spaces to measure the global hypoellipticity. (This is joint work with M. W. Wong.)*Refreshments will be served in Grad Lounge after the talk.

- Yu Liu, York University, will give a talk on "
**Inversion Formulas for Two-Dimensional Polar Stockwell Transforms**" at 4:00p.m. in Vari Hall 1016.ABSTRACT:

*We begin with a definition of a two-dimensional polar Stockwell transform sith an arbitrary window. We establish a resolution of the identity formula and hence an inversion formula for it if and only if the window satisifes an admissibility condition.*This is joint work with M.W. Wong.

- Raphael Ponge, University of Toronto, will speak on "
**Hypoelliptic Pseudo-Differential Operators, Noncommutative Geometry and Fefferman's Program**" at 10:30a.m. in N638 Ross.ABSTRACT:

*In this talk I will try to give a (non-technical) survey on how using (hypoelliptic) pseudo-differential techniques in a variety of contexts, including noncommutative geometry, spectral theory of H\"ormander operators and more general hypoelliptic operators, a new heat kernel of the Atiyah-Singer index formula and Fefferman's program.*Organizer's Remarks: Professor Raphael Ponge received his doctorate in mathematics from University of Paris-Sud under the supervision of Alain Connes.

- Andrew Toms, York University, will give a talk entitled "
**K-Theory for Operator Algebras III**" at 10:30a.m. in N638 Ross.ABSTRACT:

*We introduce the K_0 functor for (unital) C*-algebras, and examine some of its basic properties. Several examples will be discussed, with emphasis on the commutative case and the connection with vector bundles.*

- Andrew Toms, York University, will give a talk entitled "
**K-Theory for Operator Algebras I**" at 10:30a.m. in N638 Ross.ABSTRACT:

*We introduce the K_0 functor for (unital) C*-algebras, and examine some of its basic properties. Several examples will be discussed, with emphasis on the commutative case and the connection with vector bundles.*

- Andrew Toms, York University, will give a talk entitled "
**K-Theory for Operator Algebras I**" at 10:30a.m. in N638 Ross.ABSTRACT:

*We introduce the K_0 functor for (unital) C*-algebras, and examine some of its basic properties. Several examples will be discussed, with emphasis on the commutative case and the connection with vector bundles.*

- Raphael Ponge, University of Toronto, will speak on "
**Hypoelliptic Pseudo-Differential Operators, Noncommutative Geometry and Fefferman's Program**" at 10:30a.m. in N638 Ross.ABSTRACT:

*In this talk I will try to give a (non-technical) survey on how using (hypoelliptic) pseudo-differential techniques in a variety of contexts, including noncommutative geometry, spectral theory of H\"ormander operators and more general hypoelliptic operators, a new heat kernel of the Atiyah-Singer index formula and Fefferman's program.*Organizer's Remarks: Professor Raphael Ponge received his doctorate in mathematics from University of Paris-Sud under the supervision of Alain Connes.

- Aparajita Dasgupta, York University, will give a talk entitled
"
**Newton Polyhedra and Pseudo-Differential Operators**" at 2:30p.m. in N638 Ross.ABSTRACT:

*This is the first of a series of lectures on the symbolic calculus and functional-analytic properties of pseudo-differential operators with weighted symbols. The connections of the weights with Newton polyhedra are explicated.*Refreshments will be served in the Grad Lounge after the talk.

- Yu Liu, York University, will give a talk entitled "
**Polar Wavelet Transforms and Localization Operators**" at 2:30p.m. in N638 Ross.ABSTRACT:

*The notion of a polar wavelet transform is introduced. The underlying non-unimodular Lie group and the associated square-integrable representations are described. The resolution of the identity formula (aka the Plancherel formula) for the polar wavelet transform is then formulated. Localization operators corresponding to the polar wavelet transforms are then defined. While these linear operators are rather complicated in general, it is shown that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators in the sense of Janson, Peetre, Peng and Wong, paraproducts in the sense of Coifman and Meyer, and Fourier multipliers in the usual sense.*Refreshments will be served in the Grad Lounge after the talk.

- M.W. Wong, York University, will give a talk entitled "
**A Brief Overview on Forty Years of Pseudo-Differential Operators**" at 11:00a.m. in N638 Ross.ABSTRACT:

*This is an account of some developments of pseudo-differential operators. The starting point is the class of pseudo-differential operators studied in the 1966 seminal paper of Kohn and Nirenberg published in "Communications on Pure and Applied Mathematics". The connections with quantization envisaged by Hermann Weyl in his classic "Group Theory and Quantum Mechanics", first observed by Grossmann, Loupias and Stein in the 1968 paper "Annales de l'Institute Fourier (Grenoble)", will then be described in the context of Wigner transforms. These connections give new insights into the role of pseudo-differential operators in the analysis of signals and images in the perspectives of Gabor transforms and wavelet transforms. From these come the Stockwell transform that has numerous applications in geophysics and medical imaging. The recently developed mathematical underpinnings of the Stockwell transform, which seem to be at odds with those of the Gabor transforms and the wavelet transforms, will be highlighted. (This is an expository talk in mathematical analysis with precise formulations of theorems, but all proofs are omitted.)*Dim sum lunch in a Chinese restaurant (TBA) will take place after the talk.

- Jingzhi Tie, University of Georgia, will give a talk on
"
**Sub-Riemannian Geometry on the Heisenberg Group and Engel Fields**" at 2:30p.m. in N501 Ross.ABSTRACT:

*The Heisenberg group and Engel fields are first introduced. They are the simplest nilpotent Lie groups of step 2 and 3 respectively. The sub-Riemannian geodesics on them are then computed by solving the associated Hamilton's equations.*Refreshments will be served in the Grad Lounge after the talk.

- Alip Mohammed, York University, will give a talk entitled
"
**Solutions of the Riemann-Hilbert-Poincare Problem for the Inhomogeneous Cauchy-Riemann Equation**" at 2:30p.m. in N501 Ross.ABSTRACT:

*The Riemann-Hilbert-Poicare problem with general coefficient for the inhomogeneous Cauchy-Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coefficient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.*This is joint work with M. W. Wong.

Refreshments are served in the Grad Lounge after the talk.

- Lalu Mansinha, University of Western Ontario, will be speaking on
"
**Towards an Image Segmentation System Based on Local 2-D S-Transforms**" at 2:30p.m. in N638 Ross.ABSTRACT:

*In the interpretation of time-series and two-dimensional images interest is often focused on segments or patches with a specific pattern, signifying a certain consequence. Examples are the appearance of a P-wave on a seismic record, signifying the occurrence of a distant earthquake, or a spot on an MR-image, implying a cancer tumor in the patient. Visual and manual detection, and identification of this type of segments by experts remain the most widely used method of sensitive time series and image analysis. Automated detection and identification would not only increase the processing throughout, but also improve detection success rate through the ability to identify patches too small for manual visual detection. Visual identification of a patch is based on the brain's ability to interpret local statistical properties and local patterns on an image. The S-transform is a method of determining the local frequency spectrum for time series and images. The local spot spectrum at every pixel of an image makes it possible to mathematically define the local patterns and so construct an identification property table for each type. Every pixel would be labelled according to the table. Neighbouring pixels with identical labels would then be merged to form segments or patches. The S-transform is finding applications in varied fields, from the analysis of images of minerals in microscopes to the study of lavaflows on the Earth and on moons.*Refreshments will be served in Grad Lounge after the talk

- Leonid Shartser, York University, will speak on "
**G\r{a}rding's Inequality and $M$-Elliptic Pseudo-Differential Operators**" at 2:30p.m. in N638 Ross.ABSTRACT: posted outside the Chair's Office.

Refreshments will be served in the Grad Lounge after the talk.

- Peter C. Gibson, York University, will give a talk on "
**Fast Iterative Gaussian Quadrature**" at 4:00p.m. in N638 Ross.ABSTRACT:

*We construct a family F of probability distributions on the real line for which iterated Gaussian quadrature becomes super-efficient. It is somewhat surprising that such distributions exist at all, and the results have implications not only in the framework of numerical quadrature, but also for the general theory of orthogonal polynomials.*Refreshments will be served in the Grad Lounge after the talk.

- Alip Mohammed, York University, will give a talk on "
**The Riemann Problem for Torus Domains**" at 4:00p.m. in N638 Ross.ABSTRACT:

*The higher dimensional analogues of the Cauchy formula and the Plemelj-Sokhotzki formula for analytic functions on torus domains are established. The possibilities of formulating the Riemann problem for torus domains are questioned and a well-posed formulation is found which does not demand more restrictions than in the one-dimensional case. There are no extra solvability conditions caused by the higher dimension. For the first time canonical solutions of the Riemann problem for truely higher dimensional torus domains are represented and applied. The solutions of the homogeneous problem and the inhomogeneous problem are constructed by the canonical solutions. The natural connection between the Riemann problem and the Riemann-Hilbert problem is given without any restriction on the form of the analytic functions on torus domains. Thus, not only the well-posed formulation but also the solutions of the Riemann-Hilbert problem for torus domains are included indirectly.*Refreshments will be served in the Grad Lounge after the talk.

- Huaihui Chen, Nanjing Normal University, will speak on "
**Composition Operators on Bloch Spaces of the Unit Ball in ${mathbb{C}}^n$**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Composition operators on Bloch spaces of the unit ball in ${\mathbb{C}}^n$ have been completely studied. However, for $\alpha$-Bloch spaces and more generally, $\mu$-Bloch spaces, the situation is more complicated because these spaces are not Mobius invariant. We give in this talk a necessary and sufficient condition for composition operators to be bounded/compact between two such spaces.*Refreshments will be served in the Grad Lounge after the talk.

- Alip Mohammed, York University, will speak on "
**The Riemann Problem for the Unit Disk in the Complex Plane**" at 4:00p.m. in N638 Ross.ABSTRACT:

*An introduction to the Riemann problem for the unit disk in the complex plane is considered first. The problem, the needed formulas and their simple proofs are presented. Related results and some proofs are given in detail. The solvability conditions and the construction of the solution are treated with special care. At the end some attention is paid to the natural connection between the Riemann problem and the Riemann-Hilbert problem. (This talk is totally accessible to beginning graduate students who know the Cauchy integral formula.)*Refreshments will be served in Grad Lounge after the talk.

**Notes from the Organizer:**This talk is the first of a series of talks by Dr Mohammed. Dr. Mohammed is a visitor teaching in our department and will stay with us until next summer. His research interests are in complex analysis and partial differential equations. He obtained his PhD from the Free University of Berlin in 2002.

Yin Chen, Lakehead University, will give a talk entitled
"**Borel-Caratheodory Inequalities and Schwarz' Lemma for Analytic
Multifunctions**" at 2:30p.m. in N638 Ross.

ABSTRACT: *In this talk we will establish some Borel-Caratheodory
inequalities for analytic multifunctions and give several applications of
these inequalities such as Schwarz' lemma.*

- Bert-Wolfgang Schulze, Universitat Potsdam, will speak on "
**Boundary Value Problems as a Special Case of the Edge Calculus**" at 4:00p.m. in N638 Ross.ABSTRACT:

*Manifolds with smooth edges belong to the category of manifolds with (regular) singularities. The pseudo-differential calculus on such configurations refers to operators with a degenaracy that reflects the geometric nature of the singularities. Ellipticity is formulated in terms of a principal symbolic hierarchy. In the case of edges the symbols consist of two components, the interior and the edge symbol; the second one is operator-valued. Parametrices require additional conditions along the edges; they may be of trace and potential type, depending on the weights in the corresponding spaces. We show that the pseudo-differential calculus of boundary value problems (which contains the parametrices of elliptic differential boundary value problems) is a substructure of the edge calculus. In this connection the manifolds with boundary are interpreted as special manifolds with edges, where the boundary plays the role of the edge and the inner normal of the model cone of local wedges.*Refreshments will be served in Grad Lounge after the talk.

- Dong Liang, York University, will speak on "
**PDE Models of Populuation Growth: Theory and Numerical Simulation**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Computational biology has been playing more and more important roles in discovering the relation between species and their environment and in understanding the dynamics involved in the corresponding biological and physical processes. Recently, of particular concern is the joint impact of individual motions, diffusion dynamics and non-local maturation delayed effects on population growth. In this talk, we consider the population growth for species living in a high-dimensional spatial field. PDE models with delayed non-local effects are developed for the maturation population on unbounded and bounded domains. Numerical simulations are taken for investigating numerically the occurrence of one-hump traveling waves, asymptotically stable steady state solutions and periodic waves. Meanwhile, we present some theoretical results of the existence of traveling waves, the existence of periodic waves and the existence of positive equilibrium solutions and their stability for the problems.*Refreshments will be served in Grad Lounge after the talk.

**Special Analysis Seminar**Jiri Patera, Universite de Montreal, will speak on "

**General Method for Interpolation of n-dimensional Digital Data**" at 11:00a.m. in N627 Ross.ABSTRACT:

*Digital 2D data, given on rectangular or triangular lattice of any density, are exactly represented by continuous functions smoothly interpolating between the points of the grid. The metod is based on exploitation of families of uncommon expansion functions (discrete or continuous ones). Compact semisimple Lie groups of rank n underlie the method.*

- Jinman Kim, York University, will give a talk on "
**Heat Kernels and Harmonic Functions on the Poincare Upper Half Plane (A Continuation)**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Using the Iwasawa decomposition, the upper half plane can be looked at as SL(2,R)/SO(2). The advantage of bringing the transformation group SL(2,R) to the foreground is that the upper half plane is then the symmetric space of SL(2,R) and has its own Laplacian. The geometry so obtained is known as hyperbolic geometry and the Laplacian here is called the hyperbolic Laplacian for convenience. The hyperbolic Laplacian is different from the standard Euclidean Laplacian on the upper half plane, but the harmonic functions in the two settings are identical. Using the heat kernel of the hyperbolic Laplacian, we give a characterization of harmonic functions on the upper half plane in terms of an average mean value property. This is joint work with M. W. Wong in progress and extends an earlier result by the same authors for the Euclidean Laplacian on $R^n$.*Beverages will be served in Grad Lounge after the talk.

- Jinman Kim, York University, will give a talk entitled "
**Heat Kernels and Harmonic Functions on the Poincare Upper Half Plane**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Using the Iwasawa decomposition, the upper half plane can be looked at as SL(2,R)/SO(2). The advantage of bringing the transformation group SL(2,R) to the foreground is that the upper half plane is then the symmetric space of SL(2,R) and has its own Laplacian. The geometry so obtained is known as hyperbolic geometry and the Laplacian here is called the hyperbolic Laplacian for convenience. The hyperbolic Laplacian is different from the standard Euclidean Laplacian on the upper half plane, but the harmonic functions in the two settings are identical. Using the heat kernel of the hyperbolic Laplacian, we give a characterization of harmonic functions on the upper half plane in terms of an average mean value property. This is joint work with M. W. Wong in progress and extends an earlier result by the same authors for the Euclidean Laplacian on $R^n$.*Beverages will be served in Grad Lounge after the talk.

- Zhuang Niu, University of Toronto, will give a talk entitled "
**An Introduction to C*-Algebras (Part II)**" at 2:30p.m. in N638 Ross.ABSTRACT:

*In this talk, we will discuss the (topological) K-theory of C*-algebras. We begin with the definition, review the basic properties and check some examples.*Beverages will be served in Grad Lounge after the talk.

- Zhuang Niu, University of Toronto, will give a talk on "
**An Introduction to C*-Algebras (Part I)**" at 2:30p.m. in N638 Ross.ABSTRACT:

*This is an introductory lecture. We will discuss the basic definitions and properties of C*-algebras, especially the positivity and the GNS construction, which give us the equivalence between the abstract description and the concrete realization of C*-algebras.*Beverages will be served in Grad Lounge after the talk.

*Organizer's Notes: Zhuang Niu is a Ph.D. student at the University of Toronto working on operator algebras under the supervision of George Elliott. This is the first of a series of lectures on C*-algebras building up from ground zero to the forefront of research. (Ground zero here means a basic course in functional analysis covering basic topics such as Banach and Hilbert spaces, bounded linear operators, compact operators and spectral theory.)*

- Krystyna Bugajska, York University, will speak on "
**Compact Riemann Surfaces, Pseudo-Differential Operators and Theta Functions**" at 3:00p.m. in N638 Ross.ABSTRACT:

*It will be shown how compact Riemann surfaces appear in the theory of pseudo-differential operators. More precisely, we will consider isospectral deformation of the corresponding pseudo-differential operators and show how the integrability of a wide class of nonlinear differential equations corresponds to the linearization through the appropriate theta functions. If time allows, we will represent these results in terms of infinite Grassmannians. This is the first of a series of lectures.*Beverages will be served in Grad Lounge after the talk.

- Ryuichi Ashino, Osaka Kyoiku University, will speak on "
**A Pre-Processing Design for the Discrete Multiwavelet Transform of Irregularly Sampled Data**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Unshifted and shifted multiscaling functions are used as mathematical models for curve fitting of irregularly sampled data. A pre-processing design for the discrete multiwavelet transform based on this curve fitting method is proposed. This pre-processing procedure combined with multiwavelet neural networks for data-adaptive curve fitting is shown to perform well in the case of high resolution. In the case of low resolution, it is more accurate than numerical integration and cheaper than matrix inversion. This is joint work with Akira Morimoto and Remi Vaillancourt.*Beverages will be served in Grad Lounge after the talk.

- Michael Lamoureux, University of Calgary, will give a talk on
"
**Seismic Imaging via Gabor Multipliers**" at 10:00a.m. in N638 Ross.ABSTRACT:

*Seismic images of the earth's subsurface are created by processing the gigabytes of data recorded from geophones planted on the earth, which measure the vibrations caused by seismic waves reflected off geological structures deep under the surface. An accurate image is crucial to the commercial detection and recovery of hydrocarbons.**Useful seismic images rely on accurate mathematical models of the propagation and reflection of seismic energy, and accurate numerical methods to implement and invert the physical model. Typically these involve linear operators on a Hilbert space. We discuss the use of pseudodifferential operators to model the physics of wave propagation, and implementation of robust numerical techniques derived from time-frequency analysis using Gabor multipliers.*

- Askold Khovanskii, University of Toronto, will speak on
"
**Linearization of Hilbert's 16th Problem for Hamiltonian Systems and Fewnomials**" at 2:30p.m. in N638 Ross.ABSTRACT:

*The ideology of the theory of fewnomials is that a real variety defined by a "simple" system of equations should have a "simple" topology. This theory is applicable for a large class of systems of transcendental equations. It gives new information on levels of real elementary functions and even on algebraic equations. Arnold has suggested a linearization of Hilbert's 16th problem in a neighbourhood of the Hamiltonian systems. The theory of fewnomials allows a proof of the existence of a uniform bound in Arnold's problem. No special knowledge is required for this talk.*Beverages will be served in Grad Lounge at 3:30 p.m.

Organizer's Notes: Askold Khovanskii is Professor of Mathematics at the University of Toronto. He obtained the Russian doctoral degree in mathematics from the Steklov Mathematical Institute in Moscow in 1988 with a thesis entitled "Newton Polyhedra and Fewnomials". His book "Fewnomials" was translated into English and published by the American Mathematical Society in 1991. This talk runs hand in hand with Huaiping's talk on Hilbert's 16th problem. It shows how ideas and techniques from algebra, geometry and the theory of singularities may bear on an important problem, which has been studied by many experts in ODEs for many years.

- Huaiping Zhu, York University, will give a talk on "
**The Finiteness Part of Hilbert's 16th Problem for Quadratic Vector Fields**" at 2:30p.m. in N638 Ross.ABSTRACT:

*In this talk, an introduction to Hilbert's 16th problem and its development will be given. There is an ongoing project aiming at proving the finiteness part of the problem for quadratic vector fields, namely, the existence of a uniform bound for the number of limit cycles for quadratic vector fields. This project and its latest progress will be presented.*Beverages will be served in Grad Lounge at 3:30 p.m.

- Peter C. Gibson, York University, will speak on "
**Analysis of general linear operators via the Gabor transform**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Time-frequency analysis, rooted in the short-time Fourier transform, provides a flexible tool with which to analyze linear operators. In this talk we will focus on the representation of general linear operators in terms of their action on phase space, using the continuous version of the Gabor transform and its adjoint. Formulae relating the pseudo-differential symbol of an operator to its so-called Gabor symbol will be discussed, as will various open problems and directions for future research.*Beverages will be served in Grad Lounge after the talk.

- Hongmei Zhu, York University, will give a talk entitled "
**Looking inside signals using the S-transform**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Measured signals are typically of finite duration. They are dynamic and non-stationary processes whose frequency characteristics vary over time or space. This often requires algorithms capable of locally analyzing and processing signals. The recently developed Stockwell transform or S-transform in short (ST) combines the time-frequency representation of the windowed Fourier transform with the multi-scale analysis of the wavelet transform. Applying this transform to a temporal signal reveals information on what and when frequency events occur. In addition, its multi-scale analysis allows more accurate detection of subtle signal changes while interpretation in a time-frequency domain is easy to understand. In this talk, we give an overview of the theory of the ST and illustrate its effectiveness in biomedical applications. This is joint work with Dr Ross Mitchell.*Beverages will be served in Grad Lounge at 3:30 p.m.

- Ivana Alexandrova, University of Toronto, will speak on "
**Structure of the Short Range Scattering Amplitude for General Scattering Relations**" at 2:30p.m. in N638 Ross.ABSTRACT:

*In this talk I will describe the structure of the semi-classical short-range scattering amplitude. In particular, I will prove that the scattering amplitude at suitable energies $\lambda>0$ quantizes, in the sense of semi-classical Fourier integral operators, the scattering relation. I will provide all the necessary definitions during the talk.*Refreshments will be provided in the Grad Lounge after the talk.

- Alexandru Tamasan, University of Toronto, will give a talk entitled
"
**On the Flouorescence Problem**" at 2:30p.m. in N638 Ross.ABSTRACT:

*In the fluorescence problem one tries to determine the source of light passing through an absorbing and scattering tissue, from far away radiation or, equivalently, from boundary measurements. When there is no scattering, inversion of the attenuated X-ray transform does the job. In the presence of small enough (anisotropic part of) scattering, this can still be done. For three dimensional models, only scatterings in the directions parallel to a fixed plane need be small. This is joint work with G. Bal.*Beverages will be served in the Grad Lounge after the talk.

- Kenro Furutani, Science University of Tokyo, will give a talk on
"
**The Infinite-Dimensional H/"ormander Index and Asymmetry of Cauchy Data Spaces**" at 2:30p.m. in N638 Ross.ABSTRACT:

*We begin by looking at definitions and some properties of the Maslov index and the H\"ormander index. Then we explain how to generalize them to infinite-dimensional settings using functional analytic methods. We conclude with an example of the infinite-dimensional H\"ormander index, which expresses the asymmetry of Cauchy data spaces of an elliptic PDE when we decompose a manifold into two components by a hypersurface.*Beverages will be served in the Grad Lounge after the talk.

- Jinman Kim, York University, will give a talk on "
**Harmonic Functions and Heat Kernels on Symmetric Spaces Part III: SL(n,R)/SO(n) and Compact Symmetric Spaces**" at 10:00a.m. in N638 Ross.ABSTRACT:

*This is the last of a series of lectures on characterizing harmonic functions on symmetric spaces in terms of an invariantaverage mean value property. The ubiquitous heat kernels on suchspaces, which play a fundamental role in this research, will be explained and developed from first principles.*

- Jinman Kim, York University, will speak on "
**Harmonic Functions and Heat Kernels on Symmetric Spaces of Rank One. Part II: The Poincare Upper Half Plane**" at 10:00a.m. in N638 Ross.ABSTRACT:

*This is the second of a series of lectures on characterizing harmonic functions on symmetric spaces of rank one in terms of an invariant average mean value property. The ubiquitous heat kernels on such spaces, which play a fundamental role in this research, will be explained and developed from first principles.*

- Jinman Kim, York University, will speak on "
**Harmonic Functions and Heat Kernels on Symmetric Spaces of Rank One**" at 11:00a.m. in N638 Ross.ABSTRACT:

*This is the first of a series of lectures on characterizing harmonic functions on symmetric spaces of rank one in terms of an invariant average mean value property. The ubiquitous heat kernels on such spaces, which play a fundamental role in this research, will be explained and developed from first principles.*

**Analysis Candidate**Peter C. Gibson, University of Karlsruhe, will give a talk entitled "

**Diagonalization of Linear Operators via the Short-Time Fourier Transform**" at 2:00p.m. in N638 Ross.ABSTRACT:

*The short-time Fourier transform and its various discretizations lie at the heart of so-called Gabor theory, or Gabor analysis. This is a relatively young branch of harmonic analysis that has a wide range of potential applications, from signal processing to operator theory. In this talk a brief introduction to Gabor analysis will be given, after which we will focus on a particular topic: the diagonalization of linear operators by means of the short-time Fourier transform. Some recent results that have helped to establish the basic theory will be presented; in this context the extreme value distribution emerges unexpectedly as an advantageous windowing function for the short-time Fourier transform. Time permitting, we will discuss in addition some promising avenues of current research aimed at solving linear equations, and in particular PDE with badly behaved coefficients.*Notes: Dr Gibson is a candidate for the Analysis Position in the Department of Mathematics and Statistics, Faculty of Arts.

Refreshments will be served in N620 Ross Building at 3:00p.m.

**Analysis Candidate**Jie Xiao, Memorial University, will be speaking on "

**Conformally Invariant Function Spaces**" at 2:00p.m. in N638 Ross.ABSTRACT:

*This talk will be a survey, from the plannar isoperimetric inequality perspective, of conformally invariant function spaces and their applications to Complex Variables, Harmonic Analysis, Operator Theory, PDEs and Differential Geometry. The focus will be on the most interesting results.*

**Analysis Candidate**Jingzhi Tie, University of Georgia, will give a talk entitled "

**The Heisenberg Group and Its Connections with Several Complex Variables and PDE**" at 10:30a.m. in N638 Ross.ABSTRACT:

*The Heisenberg group is the simplest, non-commutative, nilpotent Lie group. It arises in two fundamental but different settings in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, there is its genesis in the context of quantum mechanics. In this talk, I will introduce the Heisenberg group and its Lie algebra from the setting of complex analysis. The Laguerre calculus is the symbolic tensor calculus induced by the Laguerre function on the Heisenberg group. Then I will use the Laguerre calculus to solve the $\bar\partial$-Neumann problem in the non-isotropic Siegel Domain. I will also talk about the non-solvability of the Hans Lewy operator. If time permits, I will also talk about my recent joint work with Chang and Greiner about the embedding problems of the Heisenberg group.*Refreshments will be served in N620 Ross Building at 10:00 a.m.

**Analysis Candidate**Monica Ilie, Texas A&M University, will be speaking today at 10:30a.m. in N638 Ross.

ABSTRACT:

*Given a locally compact group G, the Fourier and Fourier-Stieltjes algebra are important objects in abstract harmonic analysis. They are defined as spaces of coefficient functions associated with continuous unitary representations of G. In the same time, they can also be looked at as preduals of certain von Neumann algebras and, consequently, they have a natural operator space structure.**It is known that, for abelian groups, there is a deep connection between Fourier algebra homomorphisms and piecewise affine maps between their underlying groups. We explore this fact from the point of view of operator spaces, for general locally compact groups.*Refreshments will be served before the talk in N620Ross. Everybody welcome!

**Analysis Candidate**Vladimir G. Troitsky, University of Alberta, will give a talk on "

**The Invariant Subspace Problem: some recent advances**" at 10:30a.m. in N638 Ross.ABSTRACT:

*The search for invariant subspaces of continuous operators on Banach spaces has long been one of the most exciting topics of Functional Analysis. In this talk, I am going to describe several directions where this search has recently produced interesting and important results. In particular, the following topics will be discussed:**extensions and limitations of Lomonosov's Theorem;*

examples of operators with no invariant subspaces;

invariant subspaces of adjoint operators on dual Banach spaces.*As for the algebraic version of the Invariant Subspace Problem, some recent results on transitive and strictly semitransitive algebras will be presented.*Notes: Dr Troitsky is a candidate for the Analysis Position in the Department of Mathematics and Statistics, Faculty of Arts.

Refreshments will be served in N620 Ross Building at 10:00a.m.

**Analysis Candidate**Razvan Anisca, Texas A&M University, will talk on "

**Banach spaces with few symmetries**" at 2:30p.m. in N638 Ross.ABSTRACT:

*We will discuss constructions of Banach spaces having "few" symmetries. We will present results showing that such constructions can be achieved as subspaces of general Banach spaces, or at least as subspaces of Banach spaces from certain natural (and large) classes of spaces. This supports the idea that phenomena of this type are not merely accidents but they reflect a common behavior.**Results will include a new isomorphic characterization of a Hilbert space in terms of unconditionality.*Refreshments will be served in N620 Ross at 3:30p.m.

**Analysis Candidate**David Kerr, University of Muenster, will give a talk entitled "

**Dynamical entropy and Banach space geometry**" at 2:30p.m. in N638 Ross.ABSTRACT:

*Entropy provides a numerical measure of the complexity of a dynamical system. Of particular interest is whether or not it is positive, i.e., whether the system is chaotic or deterministic. We introduce a notion of dynamical entropy for Banach spaces and show that chaotic behaviour in this case can be described both geometrically and topologically. This leads to a geometric characterization of positive entropy in topological dynamics as well as to applications in C*-algebra structure theory and the study of noncommutative geodesic flows. This is joint work with Hanfeng Li.*Refreshments will be served after the talk in N620Ross. Everybody welcome!

The speaker is a candidate for a position in the Department.

- Professor Daoxing Xia, Vanderbilt University, will speak on
"
**Spectral Analysis of Hilbert Space Operators of Finite Type**" at 3:30p.m. in N638 Ross.ABSTRACT:

*A bounded linear operator A on a Hilbert space H is said to be of finite type if there is a finite-dimensional subspace K of H such that K contains the range of the commutator AB-BA and is invariant with respect to B, where B is the adjoint of A. In this talk, the analytic model, the trace formula for the commutator and the formula for the eigenvectors of some classes of operators of finite type will be given.*

- Zhaohui Zhang, York University, will speak on "
**Trace Class Norm Inequalities for Two-Wavelet Multipliers**" at 10:00a.m. in N638 Ross.ABSTRACT:

*The Landau-Pollak-Slepian operator in signal analysis has prompted the study of wavelet multipliers, which are in fact localization operators associated to modulations on the additive group $\Rn$. Such a wavelet multiplier is defined in terms of one admissible wavelet and its spectral properties such as the trace and the trace class norm inequality have been studied in detail. Recent works have been focussed on two-wavelet localization operators on locally compact and Hausdorff groups. We give in this talk sharp lower and upper estimates for the trace class norms of two-wavelet multipliers.*

- Alex Kushpel, Ryerson University, will give a talk on
"
**Some Applications of Geometry of Banach Spaces in Approximation Theory**" at 11:00a.m. in N638 Ross.*Please see M.W. Wong for Abstract information.*

- Bolin Ma, Hunan University, will speak on "
**Weighted Norm Inequalities for Marcinkiewicz Integrals with Rough Kernels**" at 11:00a.m. in N638 Ross.ABSTRACT:

*The $L^p$ norms of Marcinkiewicz integrals with rough kernels of functions $f$ with respect to certain weights are estimated in terms of the $L^p$ norms of the functions $f$ with respect to maximal functions of the weights. Some applications will be described. (This is joint work with Professor Yong Ding of Beijing Normal University and M. W. Wong of York University.)*

- T. Chan, Department of Electrical and Computer Engineering,
Royal Military College of Canada, will speak on "
**A Fast Time-Scaling Method for Wavelet Transform Applications**" at 2:00p.m. in N638 Ross.ABSTRACT:

*In the application of the wavelet transform ($WT$) to signal detection, for example radar or sonar pulses, there is a requirement to time-scale a random sequence by a non-integer factor. Given a signal $s(t)$, its time-scaled version is $s(at)=s_a(t)$, where $a$ is the scaling factor. The scaling causes dilation of $s(t)$ when $a<1$, and compression when $a>1$. When $s(t)$ is in analytic form, such as $s(t)={\mbox {cos}}\omega t$, then it is easy to produce $s_a(t)={\mbox {cos}}\omega at}$. However, in practice, $s(t)$ is generally an unknown random signal, and only its samples $s(n),\, n=0,1,\dots, N-1$, are available. Now, time-scaling $s(n$ requires first the reproduction of $s(t)$ from $s(n)$ by interpolation, time-scaling by $a$, then re-sampling to give $s_a(n)$. This process requires $N^2$ operations and has been the bottle neck of real time computations in $WT$.**The interpolation of the samples $s(n)$ to produce $s(t)$ is by fitting ${\mbox {sinc}}x={\frac{{\mbox {sin}}\pi x}{x}}$ functions between the samples. Theoretically, a sinc function has non-zero values from $-\nifty$ to $+\infty$, but its amplitudes diminish rather quickly for values of $x$ that are far away from zero. Noting this property, by retaining only $L$ sinc coefficients, a fast time-scaling algorithm that requires only $NL$ operations is proposed.**This seminar will give a brief introduction to $WT$ and some of its applications, discuss the fast time-scaling method, and conclude with some numerical examples.*

- Jinman Kim, York University, will give a talk on "
**Heat Equations on Homogeneous Spaces**" at 11:00a.m. in N638 Ross.ABSTRACT:

*This is the third and the last, but not the least, of a series of talks on heat equations on Riemannian manifolds. We give in this talk an existence and uniqueness theorem for Cauchy problems of heat equations on homogeneous spaces such as $S^2=SO(3)/SO(2)$ with initial data in the space of hyperfunctions.*

- Jinman Kim will give a talk on "
**The Cauchy problems for the Heat Equations on Riemannian Manifolds**" at 11:00a.m. N638 Ross.ABSTRACT:

*We present two results in this talk. The first is on a relaxation of the growth condition in time for the uniqueness of solutions of the Cauchy problem for the heat equation on a complete Riemannian manifold $M$ with ${\rm{dim}}(M)=n$ and ${\rm{Ric}}(M)\geq -K$ for some $K\geq 0.$ Then we give an integral representation for every positive solution of the heat equation on the manifold. These results extend the corresponding ones obtained by Cheng, Li and Yau in [1] and Li and Yau in [2]. This is joint work with M. W. Wong.*[1] S. Y. Cheng, P. Li and S. T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), 1021-1063.

[2] P. Li and S. T. Yau, On the parabolic kernel of the Schr\"odinger operator, Acta Math. 156 (1986), 153-201.

- Jie Xia, concordia University, will speak on "
**Cesaro Transforms: Boundedness and Beyond**" at 3:30p.m. in N638 Ross.ABSTRACT:

*In this talk I will show that the Cesaro transform is a bounded linear operator from $L^\infty(o,\pi)$ into $BMO(0,\pi)$. In addition, I will present a couple of possible extensions to higher dimensions.*

- Zhaohui Zhang, York University, will speak on "
**Traces of Two-Wavelet Multipliers**" at 10:00a.m. in N638 Ross.ABSTRACT:

*We initiate the notion of a two-wavelet multiplier, which models a filter with two windows in time-frequency signal analysis. We prove that under suitable conditions on the windows and the symbol, a two-wavelet multiplier is in the trace class, and we give a trace formula for it.*

- M.W. Wong, York University, will speak on "
**The Special Hermite Semigroup**" at 10:00a.m. in N638 Ross.ABSTRACT:

*We give a formula for the one-parameter strongly continous semigroup generated by the special Hermite operator L on R^2 in terms of Weyl transforms and use it to obtain an estimate for the solution of the initial value problem for the solution of the initial value problem for the heat equation governed by L in terms of the initial data.*

**Analysis Seminar**Professor Sundaram Thangavelu, Indian Statistical Institute, Bangalore , will speak on "" at 11:00a.m. in N638 Ross.Injectivity Sets for Spherical Means on the Heisenberg Group ABSTRACT:

*In this talk we prove that cylinders of the form $ \Gamma_R = S_R \times \R $ where $ S_R $ is the sphere $ { z \in C^n : |z| = R } $ are injectivity sets for the spherical mean value operator on the Heisenberg group $ H^n $ in $ L^p $ spaces. We prove that this result is a consequence of a uniqueness theorem for the heat equation associated to the sub-Laplacian. A Hecke-Bochner type identity for the Weyl transform proved by D. Geller and expansions in spherical harmonics are the main tools.*

**Analysis Seminar**Professor Sundaram Thangavelu, Indian Statistical Institute, Bangalore , will give a talk on "" at 11:00a.m. in N638 Ross.An Analogue of Hardy's Theorem for the Heisenberg Group ABSTRACT:

*We observe that the classical theorem of Hardy on Fourier transform pairs can be reformulated in terms of the heat kernel associated to the Laplacian on the Euclidean space. This leads to an interesting version of Hardy's theorem for the sub-Laplacian on the Heisenberg group. We also consider certain Rockland operators on the Heisenberg group and related Schrodinger operators on R^n.*

- Professor Sid Scull, York University, will speak on "
**Localization Operators and the Bargmann Transform**" at 10:00a.m. in N638 Ross.ABSTRACT:

*The Bargmann transform intertwines the usual (Schrodinger) representation of the Heisenberg group with another realization of this group on the Fock space. We will review the construction of the Bargmann transform, and show that it maps certain localization operators to generalized Toeplitz operators on the Fock space. Connections with quantization procedures will also be discussed.*

- Jinman Kim, York University, will speak
on "
**The Paley-Wiener-Schwartz Theorem and the Bochner-Schwartz Theorem for the Bargmann Transform**" at 11:30a.m. in N638 Ross.ABSTRACT:

*We begin with an introduction to the Bargmann transform corresponding to positive values of Planck's constant. We prove an analoque of the Paley-Wiener-Schwartz theorem for the Bargmann transform. We also characterize positive definite distributions and hyperfunctions in terms of the Bargmann transform, which is a natural generalization of the Bochner-Schwartz theorem for the Fourier transform.*

- Man Wah Wong, York University, will give a talk entitled
"
**Localization Operators on the Weyl-Heisenberg Group**" at 10:00a.m. in N638 Ross.ABSTRACT:

*We introduce localization operators on the Weyl-Heisenberg group and show that they are the same as the filters studied by Daubechies in signal analysis. A formula for the product of the Daubechies operators, i.e., filters, is given in terms of a new twisted convolution of the symbols of the operators. We then show that this twisted convolution is not a binary operation on L^2(C^n). A subspace of L^2(C^n) on which the twisted convolution is a binary operation is constructed.*This talk is a self-contained survey of the results on localization operators obtained by Jingde Du and me in the context of the Weyl-Heisenberg group. New proofs of some results will be given if time permits.

- Zhaohui Zhang, York University, will speak on "
**Localization Operators with Two Admissible Wavelets**" at 2:30p.m. in N638 Ross.ABSTRACT:

*We generalize the notion of a square-integrable representation, associated with an admissible wavelet, of a locally compact and Hausdorff group on a complex and separable Hilbert space with infinite dimension to one associated with two admissible wavelets. Localizaion operators corresponding to the new square-integrable representations with two admissible wavelets are introduced and studied. The Weyl-Heisenberg group and the affine group are given as examples to illustrate these new constructions.*

- Dr Qingtang Jiang, University of Alberta, will give a two-hour talk on
"
**The Design of Multiwavelets with Optimum Time-Frequency Resolution**" from 11:30a.m. to 1:30p.m. in N638 Ross.ABSTRACT:

*In this talk we will first discuss the procedure to design orthogonal multiwavelets with good time-frequency resolution. Then we give formulas to compute the time-durations and the frequency-bandwidths of scaling functions and multiwavelets, and give the parameterization of symmetric orthogonal multifilter banks. Finally we give an application of the optimal multiwavelets to image compression.*

- Dr Qingtang Jiang, University of Alberta, will give a two-hour talk on
"
**Some Topics Related to Wavelets**" from 10:00a.m. to 12:00p.m. in N638 Ross.ABSTRACT:

*Refinable functions (scaling functions) are fundamental to wavelet theory and subdivision. In the context of wavelet theory, the key step to the construction of wavelets is to construct suitable refinable functions. In the context of subdivision, the limiting surface of a subdivision process is a linear combination of integer translates of the refinable function corresponding to the subdivision scheme. In this talk we will first give a review on the construction of wavelets based on multiresolution analysis. Then we will discuss the characterizations for the orthogonality, stability, approximation order and smoothness of the refinable functions. The convergence of the cascade algorithm associated with the refinable functions will also be discussed. All the characterizations are given in terms of the masks associated with the refinable functions.*

- Man Wah Wong, York University, will speak on "
**Traces of Wavelet Multipliers**" at 5:00p.m. in N638 Ross.ABSTRACT:

*Wavelet multipliers are defined, according to Zhiping He and M. W. Wong, as integral operators on L^2(R^n) in terms of a unitary representation of the additive group R^n on L^2(R^n), and as such, they are special pseudo-differential operators with very nice spectral properties. We begin the talk with a survey of wavelet multipliers. Then we give a trace formula for wavelet multipliers as bounded linear operators in the trace class from L^2(R^n) into L^2(R^n) and use it to compute the trace of the n-dimensional Landau-Pollak-Slepian operator arising in signal processing, image compression, etc. (This is joint work with Jingde Du.)*

- Christopher Heil, School of Mathematics, Georgia Institute of
Technology, will speak on "
**Boundedness of Pseudo-Differential Operators via Modulation Space Techniques**" at 5:00p.m. in N638 Ross.ABSTRACT:

*Modulation spaces are function spaces that quantify the joint localization in both time and frequency of functions or distributions. In light of the fact that pseudo-differential operators can be naturally defined in terms of time-frequency analysis, it is natural to consider whether boundedness or other properties of a pseudo-differential operators can be characterized in terms of modulation space properties of the associated symbol. We will show that this approach leads directly to sufficient conditions for the boundedness of a pseudo-differential operator that recover the classical Calderon-Vaillancourt theorem while also extending it to certain non-smooth symbols. Similar techniques can be used to obtain conditions for an operator to be trace class or Schatten- von-Neumann class. The results in this talk are based on joint work with Karlheinz Groechenig, Jay Ramanathan, and Pankaj Topiwala.*

- Demetrio Labate, Georgia Institute of Technology, will speak on
"
**Composition of Pseudo-Differential Operators**" at 5:00p.m. in N638 Ross.ABSTRACT:

*We use a time-frequency approach to study the Kohn-Nirenberg and Weyl pseudo-differential operators. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We obtain a time-frequency characterization of the twisted product $\sigma\sharp \tau$ of two symbols $\sigma$ and $\tau$, and we show that the modulation spaces provide a natural setting to control the time-frequency content of $\sigma \sharp \tau$ from those of $\sigma$ and $\tau$. Using some recent results from the literature, we discuss some implications on the boundedness and spectral properties of the pseudo-differential operator with symbol $\sigma\sharp \tau$.*

" at 5:00p.m. in N638 Ross.**Jingde Du, York University, will speak on "**Gaussian Functions and Daubechies OperatorsABSTRACT:

*The problem of finding a filter that has the same effect as two filters arranged in series in signal analysis is the same as the computation of the product (or composition) of two Daubechies operators. It has been proved by J. Du and M. W. Wong that the symbol of the product of two Daubechies operators is a new twisted convolution of the symbols of the given operators. Unlike the standard twisted convolution associated to the Heisenberg group, however, this new convolution of two functions in L^2(C^n) need not be a function in L^2(C^n). Thus, it is an interesting problem to seek a subspace M of L^2(C^n) that is closed with respect to this new binary operation. We show in this talk that M can be taken to be the subspace of L^2(C^n) spanned by the Gaussian functions.*

- M.W. Wong, York University, will speak on "
**Weyl Transforms, Pseudo-Differential Operators and Localization Operators**" at 10:30a.m. in N638 Ross.ABSTRACT:

*This is a non-technical talk that describes, for a general audience in analysis, the various facets of Weyl transforms, pseudo-differential operators and localization operators in, respectively, quantization, elliptic partial differential equations and signal analysis. The connections between the three classes of operators are then explained and used in the study of the spectra of these operators.*

**Analysis Seminar**Jingde Du, York University, will speak on "

**A Product Formula for Localization Operators**" at 10:00a.m. in N638 Ross.ABSTRACT:

*The product of two localization operators with symbols F and G in some subspace of L^2(C^n) is shown to be a localization operator with symbol in L^2(C^n) and a formula for the symbol of the product in terms of a new convolution of F and G is given. (This is joint work with M.W. Wong)*

**Analysis Seminar**Professor Anatoli F. Ivanov, Penn State University, will give a talk entitled "

**Symmetric Delay Differential Equations**" from 11:00a.m. to 12:00p.m. in N638 Ross.ABSTRACT:

*Scalar delay differential equation of the form (1) x'(t)=f(x(t),x(t-1)) is called symmetric if the function f satisfies the conditions f(-x,y)=f(x,y)=-f(x,-y). It is known to possess periodic solutions that are also symmetric in the following sense: x(t+p)=-x(t) for all t and some p>0. Questions of the existence, uniqueness, stability/instability of the symmetric periodic solutions are discussed. The analysis is reducible to studying systems of ordinary differential equations.*

**Analysis/Industrial Mathematics Seminar**Professor William Margulies, California State University at Long Beach, will speak on "

**Least Squares Best Fit Algorithms**" at 2:30p.m. N638 Ross.ABSTRACT:

*A common way to solve a minimization problem is to use least squares techniques. These methods lead to ill-conditioned numerical problems. We use least squares methods supplemented by the Newton iteration to solve a minimization problem.*

**Analysis Seminar**Professor Y.J. Leung, University of Delaware, will speak on "

**On an Isoperimetric Conformal Modulus Problem**" at 3:00p.m. in N638 Ross.ABSTRACT:

*For a doubly connected domain R, its conformal modulus may be defined as the ratio of the radii of an annulus conformally equivalent to R. In this talk, we start with the problem of finding a rectifiable Jordan curve of fixed length encircling a fixed line segment inside such that the resulting configuration has maximum modulus. We'll use Schiffer's variational method to identify the extremal curve. The result is tied to the Balayage process of sweeping mass on one curve to another. We'll also examine the connection to the Friedrichs' operator of a domain as defined by Harold Shapiro.*The talk is elementary in nature. Basic knowledge in a first year graduate course in complex analysis and a little bit of operator theory would be enough for an understanding of the talk.

- Professor Jayakumar Ramanathan, University of Michigan and Eastern
Michigan University, will speak on "
**Singular Values and the Weyl Calculus**" at 3:00p.m. in N638 Ross.ABSTRACT:

*The Gabor decomposition and the wavelet transform are both methods of decomposing a function according to a tiling of phase space. This talk will present an application of these techniques to the analysis of operators. I will discuss recent work with Heil and Topiwala on the asymptotic behavior of the singular values of operators defined by the Weyl correspondence. Sufficient conditions on the smoothness and decay of the symbols are sought in order to insure that the associated operator lies in a particular Schatten class. As an interesting special case, we obtain an improvement of a result of Daubechies on when an operator is in the trace class. A new development and improvement of the Calderon-Vaillancourt theorem in the context of the Weyl correspondence is given. The methods involve the systematic use of Gabor frames as a basis in which to expand the symbol of the operator. The coefficients of the expansion are then related to the decay of the singular values. In this way I hope to present an interesting case study of how recent ideas in harmonic analysis can be applied to problems in the analysis of operators.*

Professor Lizhong Peng, Peking University, will speak on "

**Analysis on Octonions**" at 3:00p.m. in N638 Ross.

**Analysis Seminar**Dr Jishan Hu, Hong Kong Univeristy of Science and Technology, will speak on "

**The Painleve Property and Regular Differential Equations**" at 3:30p.m. in N638 Ross.ABSTRACT:

*In this talk, we give a brief review on the study of the Painleve property for integrable equations. We show how regular differential equations are related to integrable equations and the Painleve property.*

- Dana Petrescu-Rakin will speak on "
**Wavelet Multipliers and Signals**" at 3:30p.m. in N638 Ross.ABSTRACT: EM>The compactness, and hence information on the spectrum, for a class of pseudo-differential operators is obtained by showing that the pseudo-differential operators in question are multipliers defined by means of admissible wavelets associated to a unitary representation of the additive group R (the real line) on the Banach algebra of all bounded linear operators from X into X, where X is the Hilbert space of all square integrable complex-valued functions on R. A bounded linear operator from X into X arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier presented in this seminar.

This talk is an abridged version of the results obtained by Zhiping He and M.W. Wong in "Localization operators associated to square integrable group representations, Panamer. Math. J. 6(1) (1996), 93-104" and "Wavelet multipliers and signals, J. Australian Math. Soc. Series B-Appl. Math., to appear."

Seminar requirement for Master's Students

Reminder: Master's Mathematics Students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.

- Professor Daoxing Xia, Vanderbilt University, will speak on "
**On a Class of Operators with Finite Rank Commutators**" at 11:30a.m. in N638 Ross.ABSTRACT:

*We are studying a class F of non-normal operators A on a Hilbert space X satisfying the condition that there is a finite-dimensional subspace of X which is invariant with respect to the adjoint B of A and its self-commutator BA-AB. All the subnormal operators with finite rank self-commutators are in F. For an operator in this class, the eigenvectors of its adjoint span the space X. A reproducing kernel Hilbert space model has been introduced for these operators. The spectra of these operators are related to some quadrature domains in Riemann surfaces.*ABOUT PROFESSOR XIA: Professor Xia is one of the very best mathematicians trained in mainland China. He is at the same calibre with the best mathematicians of the "first" generation in Chinese mathematics as Mingde Cheng, Chaohao Gu, Luokeng Hua, Buqing Su and Wentsun Wu, among others. He is a member of the Chinese Academy of Sciences, the analogue of a FRS in England, a FRSC in Canada and a member of the National Academy of Sciences of U.S.A. He has also produced many students who are currently members of the Chinese Academy of Sciences. He was a professor of mathematics until 1981 at Fudan University in Shanghai, the university in the south at the same rank as Peking University in Beijing. He visited Moscow State University collaborating with Academician I.M. Gelfand in 1957-59. He held positions at the University of California at Santa Barbara, Purdue and SUNY at Stony Brook in 1979-80; the University of Iowa in 1981-82; the Institute for Advanced Study at Princeton and Ohio State University in 1982-83; and eventually the Department of Mathematics at Vanderbilt University was lucky to lure him as a permanent professor of mathematics as of 1984.

- There will be three talks on nonlinear problems in N638 Ross Building.
11:30 a.m. -- Dr. Caryl Margulies (University of California at Riverside) will speak on "

**A Finite Dimensional Analogue of the Fucik Spectrum**"1:00 p.m. -- Dim Sum Lunch at Grand Yatt Chinese Restaurant, 9019 Bayview Avenue, Thornhill

2:30 p.m. -- Professor William Margulies (California State University at Long Beach) will give a talk entitled "

**On the Structure of the Fucik Spectrum**"3:30 p.m. -- G. Mihai Iancu (York University) will speak on "

**Analytic Semigroups and Semilinear Heat Equations in Hilbert Spaces**"5:00 p.m. -- Alcoholic or Non-Alcoholic Beverages at Grad Lounge, 7th Floor, Ross Building

7:30 p.m. -- Dinner at Rosa's Place (an Italian restaurant), 2201 Finch Avenue West

ABSTRACTS:

*C. Margulies' talk: We develop a finite dimensional analogue of the Fucik spectrum or resonance set. We then apply results about this set to gain information about the existence of solutions to certain types of nonlinear matrix equations. The results provide insights into similar nonlinear partial differential equations. They are applicable to numerical analysis. They open up an area of problems dealing with nonlinear matrix equations, some of which are accessible to undergraduates. The talk should be understandable to anyone who has completed an upper division course in linear algebra.**W. Margulies' talk: We discuss a nonlinear Dirichlet problem which depends on two parameters. We then consider the subset of the plane, called the Fucik spectrum, where nontrivial solutions of the equation exist. It turns out that these points affect solutions to associated nonlinear equations as eigenvalues affect solutions to corresponding Fredholm equations. The structure, then, of this set is rather important. In this talk, we present, along with relevant definitions and examples, some history and some results on the structure of this point set.**G. M. Iancu's talk: The existence, uniqueness, and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are presented by developing new results in the theory of analytic semigroups of bounded linear operators. (This is joint work with Professor M. W. Wong and will appear in Applicable Analysis.)*

- Dr. Jun Ling, SUNY at Buffalo, will give a talk entitled: "
**A Bound for the First Fundamental Gap**" at 11:30a.m. in N638 Ross.ABSTRACT:

*The recent research on the computation of the spectrum of a differential operator due to Shing Tung Yau dates from the early work of Weyl, Courant and Polya. In this talk, we first give a brief introduction to some work on the estimate for the gap of the first two Dirichlet eigenvalues of the Schrodinger operator and then present an improvement on the results of Li and Yau, and Yu and Zhong.*

- Mihai Iancu, York University, will speak on "
**Travelling Waves of Semilinear Pseudo-Differential Evolution Equations**" at 4:30p.m. in N638 Ross.ABSTRACT:

*We use the Leray-Schauder degree theory and a Sobolev compact embedding theorem obtained by Martin Schechter to prove the existence of travelling wave solutions of semilinear evolution equations governed by elliptic pseudo-differential operators and very general and tractable nonlinearities. (This is joint work with M. W. Wong.)*