Analysis

Faculty members involved

Raphael Clouatre (link). Raphael Clouatre works in operator theory. In recent years, there has been a flurry of activity in multivariate operator theory, that is, the study of several operators simultaneously. At this level of generality however, not much can be said. Accordingly, concrete operators such as those acting on Hilbert spaces of analytic functions receive considerable attention. His research centres around the building of bridges betwen these concrete operators and general abstract ones via the very powerful idea of dilation. This has prompted him to approach operator theory from two distinct angles: either function theoretic through the study of function spaces and their multiplier algebras, or operator algebraic through the consideration of various natural algebras generated by a given operator.

Craig Cowan (link). Craig Cowan works in the area of elliptic partial differential equations. He is particularly interested in semilinear pde's and in questions related to regularity, existence and multiplicity of their solutions. A lot of his work has focused on the regularity of extremal solutions (stable solutions) and on a related topic of Liouville Theorems for nonlinear pde.

Eric Schippers (link) works in complex analysis and Riemann surfaces. His work in complex analysis is on classes of analytic functions on domains. The geometric, algebraic and analytic properties of such classes of functions is part of what is called geometric function theory''. He is particularly interested in conformal invariants and their relation to extremal problems.

Riemann surfaces are the fundamental objects of complex analysis (while analytic maps are the morphisms). The study of parametrized spaces of Riemann surfaces, up to a certain topological and conformal equivalence, is called Teichmuller theory. This theory had its beginnings in the work of Riemann, but the modern approach was initiated by Teichmuller and the foundations laid by Ahlfors and Bers (among others). Currently, his work is on applications of Teichmuller theory to two-dimensional conformal field theory and vice versa.

Yong Zhang (link) works in Banach algebras and harmonic analysis. Currently his interest is in amenability properties of Banach algebras associated to locally compact groups and semigroups. Studying a vector space equipped with a norm topology gives rise to the theory of Banach spaces. Many Banach spaces that arise naturally have also the ring structure. Studying a ring endowed with an appropriate norm topology leads to the theory of Banach algebras. In a Banach algebra the algebraic and the topological structures interplay enchantingly, defining profound properties of the space.

Harmonic analysis is an area dealing with functions and actions of a locally compact topological group. Amenability theory for a locally compact group concerns invariant means on bounded functions defined on the group. The theory settles the Banach-Tarski paradox. Amenability of a locally compact group can also be characterized by a kind of bounded linear mapping, called derivation, from the group convolution algebra of the group. This inspires the amenability theory for Banach algebras, which has recently been extended further to various generalized notions of amenability to study the structure of a Banach algebra. Amenability of a group/semigroup can amazingly determine common fixed point property of the group/semigroup acting on a subset of a locally convex space. Fixed point theorems have important applications in various areas of Mathematics.

Nina Zorboska (link) works in an area that connects operator theory and complex analysis. The operators involved are generated by a fixed function and operate on spaces of analytic functions such as the Hardy, Bergman, Besov, and Bloch type spaces. The classes of operators include composition, multiplication and Toeplitz operators. In discovering the operator theoretic properties such as boundedness, compacteness and different levels of Fredholmness of the operator, one uses many interesting geometric, analytical and measure theoretic properties of the inducing function and of the functions belonging to the corresponding spaces. The research in this area brings about a deeper understanding of several different areas of analysis and of their connections, uses many beautiful classical function theoretic results and has motivated many recent new developments and results. The research also has a number of applications in current emerging areas such as, for example, control theory or modern theoretical physics.

Current students and postdoctoral fellows

• Ievgen Bilokopytov (Ph.D.) - Nina Zorboska
• Osu Ighorodhe (M.Sc.) - Yong Zhang
• Tapiwa Maswera (M.Sc.) - Yong Zhang
• Mohammad Sadeghi (Ph.D.) - Nina Zorboska
• Mohammed Shirazi (Ph.D.) - E. Schippers

Representative recent publications

• R. Clouatre, Completely bounded isomorphisms of operator algebras and similarity to complete isometries. Indiana Univ. Math. J. 64 (2015), no.3, 825-846.
• R. Clouatre, Unitary equivalence and similarity to Jordan models fo weak contractions of class C_0. Canad. J. Math. 67 (2015), no. 1, 132-151.
• R. Clouatre, Spectral and homological properties of Hilbert modules over the disc algebra. Studia Math. 222 (2014), no. 3, 262-282.
• A. T.-M. Lau and Yong Zhang, Finite dimensional invariant subspace property and amenability for a class of Banach algebras, Trans. Amer. Math. Soc., to appear.
• Yong Zhang, Weak amenability of commutative Beurling algebras, Proc. Amer. Math. Soc. 142 (2014), 1649-1661.
• Yong Zhang, The existence of solutions to nonlinear second order periodic boundary value problems, Nonlinear Anal. 76 (2013), 140-152.
• A. T.-M. Lau and Yong Zhang, Fixed point properties for semigroups of nonlinear mappings and amenability, J. Funct. Anal. 263 (2012), 2949-2977.
• N. Zorboska, Univalently induced, closed range composition operators on the Bloch-type spaces, Canadian Mathematical Bulletin 55(2) (2012), 441--448.
• N. Zorboska, Schwarzian derivative and general Besov-type domains, Journal of Mathematical Analysis and Application 379 (2011), 48--57.
• D. Radnell and E. Schippers, Fiber structure and local coordinates for the Teichmueller space of a bordered Riemann surface, Conformal Geometry and Dynamics 14 (2010), 14--34.
• O. Roth and E. Schippers, The Loewner and Hadamard variations, Illinois J. Math 52(4) (2008) 1399--1415.
• Z. Wu, R. Zhao, N. Zorboska, Toeplitz operators on the analytic Besov spaces, Integral Equations and Operator theory 60(3) (2008), 435--449.
• E. Schippers, Conformal invariants and higher-order Schwarz lemmas, J. Anal. Math. 90 (2003), 217--241.

News

MathCamp 2017 information is online.

Manitoba Workshop on Mathematical Imaging Science Friday, May 5, all day, Robert Schultz Lecture Theatre, details.

In an effort to help students, the Math department has put together the LevelUp program. See details here. Video explaining registration process is here.

Events

Thursday, May 4th, 2017 at 15:30, 418 Machray Hall
Tommy Kucera
Fibonacci and The Liber Abaci
(Seminar series : Colloquium)

Friday, May 12th, 2017 at 15:30, 418 Machray Hall
John Dallon
Modeling Amoeboidal Cell Motion -- Force vs Speed
(Seminar series : Colloquium)

Friday, August 4th, 2017 at 15:30, 418 Machray Hall
Jose Aguayo
TBA
(Seminar series : Colloquium)