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

**Seminars 2017-2018**

**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.