# Analysis Seminars

Analysis seminars are typically held in Ross N627 on Friday afternoons, from 4:00 to 5:00pm.

Seminars 2018-2019

Friday, March 8, 2019

4:00p.m., 0008 DB

Shahla Molahajloo
Institute for Advanced Studies in Basic Sciences, Iran

Periodic Wigner and Weyl Transforms
ABSTRACT: A new time-frequency distribution for a periodic time signal is introduced by defining the periodic Wigner and Fourier-Wigner transform. We prove that most properties of the ordinary Wigner transform such as the Moyal identity, reconstruction formula, time-frequency marginal conditions and the resolution formula hold for the periodic case. Using the periodic Wigner transform, we define periodic Weyl transforms corresponding to suitable symbols. We give a necessary and sufficient condition for the self-adjointness of the periodic Weyl transform. Moreover, we give a necessary and sufficient condition for a periodic Weyl transform to be a Hilbert-Schmidt operator. Then we show how we can reconstruct the symbol from its corresponding Weyl transform.
Refreshments will be served after the talk.

Friday, February 15, 2019

4:00p.m., 0008 DB

Vitali Vougalter
University of Toronto

On the Solvability of Some Systems of Integro-differential Equations with Anomalous Diffusion in Two Dimensions
ABSTRACT: The work deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion with the negative Laplacian in a fractional power in two dimensions. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.
Refreshments are served after the talk.

Seminars 2017-2018

Friday, May 4, 2018

Professor Vitali Vougalter, University of Toronto

Existence in the Sense of Sequences of Stationary Solutions for Some non-Fredholm Integro-differential Equations

ABSTRACT: We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H^2 spaces. It is shown that, under reasonable technical conditions, the convergence in L^1 of the integral kernels implies the existence and convergence in H^2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property.

Refreshments will be served after the talk.

Friday, March 16, 2018

4:00p.m., N638 Ross Bldg.

On New Summation Theorem for Hypergeometric Functions and Some Open                   Problems of Analysis

ABSTRACT: We present a new summation theorem involving the Gauss hypergeometric function. Our results are motivated by some problems which originate in the Theory of Branching Processes as well as Queueing Theory. We delineate connections between the corresponding models which emerge in these two fields of the Theory of Stochastic Processes and their ‘special functions’ counterparts. We pose several open problems of Analysis. Partly based on joint work with R.B. Paris (Abertay University).

Refreshments will be served after the talk.

Friday, March 2, 2018

4:00p.m., N638 Ross Bldg.

Vitali Vougalter, University of Toronto

Solvability of some integro-differential equations with anomalous diffusion in two dimensions

ABSTRACT: The work is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains are used.

Refreshments will be served after the talk.

Friday, September 22, 2017

4:00p.m., N638 Ross

Bartek Ewertowski
The University of Auckland, New Zealand

An Introduction to Bernstein-Gelfand-Gelfane (BGG) Sequences of Differential Operators in Parabolic Cartan Geometry

ABSTRACT: Cartan geometry is a curved generalization of Klein's Erlangen program, and encompasses many geometries of interest, such as CR, semi-Riemannian, projective, and conformal geometry. One of the key goals of Cartan geometry is the study and classification of invariant differential operators. In particular, the so-called curved Bernstein-Gelfand-Gelfand (BGG) sequences of differential operators have generated a lot of interest in the past two decades. This talk will be an introduction to Cartan geometry and BGG operators, assuming only some basic familiarity with Lie groups, Lie algebras, and differential geometry.

Refreshments will be served after the talk.

Friday, September 15, 2017

4:00p.m., N638 Ross

Shahla Molahajloo
Institute for Advanced Studies in Basic Sciences, Iran

Wigner and Weyl Transforms on the Additive Group Z

ABSTRACT: The Wigner and Fourier-Wigner transform on $\Z$ are defined. Then we give an inversion formula for the Wigner transform and show that the Moyal identity, time and frequency marginal condition, shift invariant property and the modulation theorem for the Wigner transform are satisfied. Using the Wigner transform we define the Weyl transform corresponding to a suitable symbol on $\S1\times\Z$. Then we give a necessary and sufficient
condition for the self-adjointness of the Weyl transform on $\Z$. We prove that Weyl transforms on $\Z$ with $L^2$-symbols form an algebra on the
space of bounded operators on $L^2(\Z)$. Moreover, we give a necessary and sufficient condition for the trace class Weyl transforms on $\Z$. Then we
show how we can re-construct the symbol from its corresponding Weyl transform.

Refreshments are served after the talk.

Friday, August 25, 2017

4:00p.m., N638 Ross

Vitali Vougalter
University of Toronto

On the Solvability of Some Systems of Integro-Differential Equations with Anomalous Diffusion

ABSTRACT: The work deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion
with the Laplacian in a fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

Refreshments will be served after the talk.

Friday, August 4, 2017

4:00p.m., N638 Ross

Shahla Molahajloo
Institute for Advanced Studies in Basic Sciences, Iran

Pseudo-Differential Operators, Wigner Transforms and Weyl Transforms on the Poincare Unit Disk

ABSTRACT: Using the affine group and the Cayley transform from the unit disk D onto the upper half plane, we can turn D into a group, which we call the Poincare unit disk. With this construction, D is a noncompact and nonunimodular Lie group. We characterize all infinite-dimensional, irreducible and unitary representations of D. By means of these representations, the Fourier transform on D is defined. The Plancherel theorem and hence the Fourier inversion formula can be given. Then pseudo-differential operators with operator-valued symbols, operator-valued Wigner transforms and Weyl transforms on D are defined.

Refreshments are served after the talk.

Friday, July 28, 2017

4:00p.m., N638 Ross

Majid Jamalpour Birgani
Iran University of Science and Technology

The Heat Kernel and Trace Class Pseudo-Differential Operators on the Euclidean Space

ABSTRACT: We use the heat kernel of the Laplacian on the Euclidean space and some fairly well-known results on nuclear operators to give a characterization of trace class pseudo-differential operators on the Euclidean space. This gives another derivation of the trace formula for trace class pseudo-differential operators on the Euclidean space.

Refreshments are served after the talk.

Seminars 2016-2017

Friday, March 10, 2017

4:00p.m., N638 Ross

Vitali Vougalter
York University

Solvability in the sense of sequences for some non-Fredholm Operators Related to the Superdiffusion

ABSTRACT: We study solvability of some linear nonhomogeneous elliptic equations and show that under reasonable technical conditions the convergence in L^{2}(R^{d}) of their right sides yields the existence and the convergence in H^{1}(R^{d}) of the solutions. The problems involve the square roots of
the second order non-Fredholm differential operators and we use the methods of spectral and scattering theory for Schrodinger type operators.

Refreshments are served after the talk.

Friday, September 16, 2016

4:00p.m., N638 Ross

Shahla Molahajloo
Institute for Advanced Studies in Basic Sciences, Iran

Pseudo-Differential Operators on the Poincar\'e Group

ABSTRACT: Using the isomorphism between the affine group on the upper half plane and the corresponding group on the Poincar\'e unit disk, which we call for short the Poincar\'e group, we define the Fourier transform on the Poincar\'e group and give a Plancherel formula and a Fourier inversion formula. Then using the Fourier inversion formula, pseudo-differential operators on the Poincar\'e group are defined. The boundedness of pseudo-differential operators on the Poincar\'e group with operator-valued symbols given by Weyl  transforms is given.

Refreshments are served after the talk.

Friday, August 5, 2016

4:00p.m., N638 Ross

Jianxun He
Guangzhou University
Wavelet and Radon Transforms on Quaternion Heisenberg Groups
ABSTRACT
Let Q be a quaternion Heisenberg group. We give the decomposition
for the space of square-integrable functions associated with the
affine automorphism group of Q. In addition, the theory of continuous
wavelet transforms is investigated. Also, we study the Radon transform
and give several formulas for the inverse Radon transform using the
group Fourier transform, differential operators and the wavelet
transform.
Refreshments will be served after the talk.

Friday, July 29, 2016

4:00p.m., N638 Ross

Vitali Vougalter
University of Toronto

Existence of stationary solutions for some non-Fredholm
integro-differential equations with superdiffusion

ABSTRACT
We prove the existence of stationary solutions for some reaction-diffusion equations with superdiffusion. The corresponding elliptic problem contains the operators with or without Fredholm property. The fixed point technique in appropriate H^2 spaces is employed.

Refreshments will be served after the talk.

Monday, July 11, 2016

4:00p.m., N638 Ross

Liuchuan Zeng

Shanghai Normal University

Levitin-Polyak Well-Posedness of Completely Generalized Mixed
Variational Inequalities in Refexive Banach Spaces

ABSTRACT
Let X be a real reflexive Banach space. In this talk, we first introduce  the concept of Levitin-Polyak well-posedness of a completely generalized mixed variational inequality in X, and establish some characterizations of its Levitin-Polyak well-posedness. Under suitable conditions, we prove  that the Levitin-Polyak well-posedness of a completely generalized mixed variational inequality is equivalent both to the Levitin-Polyak
well-posedness of a corresponding inclusion problem and to the
Levitin-Polyak well-posedness of a corresponding fixed point problem. We also derive some conditions under which a completely generalized mixed variational inequality in X is Levitin-Polyak well-posed. Our results
improve, extend and develop the early and recent ones in the literature.
Refreshments will be served after the talk.

Seminars 2015-2016

April 1, 2016

Jingzhi Tie
University of Georgia
Yau's Gradient Estimate and Liouville Theorem for Positive
Pseudoharmonic Functions in a Complete Pseudohermitian manifold
I will introduce the basic notion of pseudohermitian manifold first
and derive the sub-gradient estimate for positive pseudoharmonic
functions in a complete pseudohermitian $(2n+1)$-manifold (M,J,\theta)
which satisfies the CR sub-Laplacian comparison property. It is served
as the CR analogue of Yau's gradient estimate. Secondly, we obtain
the Bishop-type sub-Laplacian comparison theorem in a class of complete  noncompact pseudohermitian manifolds. Finally we will show the natural  analogue of Liouville-type theorems for the sub-Laplacian in a complete  pseudohermitian manifold of vanishing pseudohermitian torsion tensors and nonnegative pseudohermitian Ricci curvature tensors. (This a joint project with Shu-Cheng Chang and Ting-Jung Kuo of National Taiwan University.)
Refreshments will be served after the talk.

November 21, 2015
Vitali Vougalter, University of Cape Town
Sharp Semiclassical Bounds for the Moments of Eigenvalues for some Schroedinger Type Operators with Unbounded Potentials
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 Schroedinger 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.
November 21, 2015
Vitali Vougalter, University of Cape Town
Sharp Semiclassical Bounds for the Moments of Eigenvalues for some Schroedinger Type Operators with Unbounded Potentials
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 Schroedinger 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.
September 25, 2015
Lizhong Peng
Peking University
The Helgason-Fourier Transform Associated to the Weighted
Laplace-Beltrami Operator on the Hyperbolic Unit Ball
The harmonic analysis is established for the weighted Laplace-Beltrami operator $\Delta_\theta$ on the hyperbolic unit ball. The associated  weighted Helgason-Fourier transform and the $\theta$-spherical transform are defined and studied. In particular, the inversion formula and the partial Plancherel theorem are obtained.
Refreshments are served after the talk.