NonSelfadjoint Operator
Algebras and Operator Spaces
Wavelets, Frames, and Image Analysis
Operator Algebras, Topological Groups, and Quantum Groups
Operator Algebras, C*algebras, and Algebra
Quantum information theory and science, quantum manybody physics
Functional Analysis is a broad area of modern mathematics
that has grown out of, and maintains connections with, multiple diverse topics
in science, engineering, and technology. Our
group pursues several lines of investigation, including basic research in
pure mathematics to improve general understanding of the subject, as well
as development of mathematical results that lay the groundwork for
applications. A common theme among our group members' work is the
study of operators on Hilbert spaces, which may be though of as
infinitedimensional generalizations of Euclidean space.
Many important collections of operators have algebraic structure that can be exploited to study all the operators simultaneously. Among other benefits, this algebraic viewpoint allows for the spectral theory of a single operator to be extended to a collection, and it provides a way to generalize the study of continuous functions on a topological space to noncommutative algebras, earning the subject the name "noncommutative topology".
Wavelets may be visualized as brief wavelike oscillations that can be combined to produce more complicated waves. Decomposing signals into wavelets is useful for both analysis and transmission of the signals, and techniques from Harmonic Analysis and Fourier Analysis are often applied. As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including audio signals, images, and video.
Quantum groups include several kinds of noncommutative algebras (or the spaces they act on) that arise in quantum mechanics and theoretical physics. Quantum groups are often viewed as depending on an auxiliary parameter, such as h or q. As this parameter varies, it deforms a commutative algebra of functions into an algebra of functions on a "noncommutative space" (in the spirit of the noncommutative geometry of Alain Connes).
While classical information is stored in bits of 0 or 1, quantum information is stored in "qubits" operating under two key principles of quantum physics: superposition (meaning each qubit simultaneously represents a 0 and 1 with different probabilities for each), and entanglement (meaning one qubit's state affects the state of another). Using these principles, qubits can process information in ways that are difficult or impossible with classical methods.

Dr. David
Blecher
Ph.D., University of Edinburgh in Scotland, 1988 M.Sc., Cambridge University in England, 1985 B.Sc., University of the Witwatersrand, 1983 Homepage Publications 

Dr. Blecher is a professor at the University of Houston. He has published more than 80 research papers in his field and has been the recipient of the UH Award for Excellence in Research and Scholarship. He serves on the editorial boards of the Houston Journal of Mathematics and the Journal of Mathematical Analysis and Applications, and he has also been the recipient of several NSF grants. Dr. Blecher's research interests include Operator Algebras, Operator Spaces, Operator Theory, and Functional Analysis. He is the author of the the following books and monographs:


Dr. Bernhard Bodmann
Ph.D., University of Florida, 2001 Masters (Diplom), Universität Erlangen, 1997 Homepage Publications 

Dr. Bodmann is a professor at the University of Houston. He has been the recipient of grants from NSF and NSERC. His research interests include uncertainty principles in harmonic analysis, the design of frames for the coding of analog signals, wavelet and filter design, and mathematical physics. Selected papers:


Dr. Mehrdad Kalantar
Ph.D., Carleton University, Canada, 2011 M.Sc., Sharif University, Iran, 2006 B.Sc., Chamran University, Iran, 2004 Homepage Publications 

Dr. Kalantar is an assistant professor at the University of Houston. His research interests include operator algebras, topological quantum groups, noncommutative harmonic analysis, and noncommutative ergodic theory. Selected papers:


Dr. Mark Tomforde
Ph.D., Dartmouth College, 2002 M.A., Dartmouth College, 1999 B.A., Gustavus Adolphus College, 1997 Homepage Publications 

Dr. Tomforde is a professor at the University of Houston. He has been the recipient of research grants from the NSF, NSA, and Simons Foundation. He has also supervised an NSF postdoc and worked with several graduate and undergraduate students on research projects. Dr. Tomforde's research interests include C*algebras of graphs and dynamical systems, classification of C*algebras, Leavitt path algebras, symbolic dynamics, and the use of algebraic techniques in operator algebra. He received the Butler Teaching Award in 2015, and a University of Houston Teaching Excellence Award in 2016. Dr. Tomforde is also active in mathematics outreach and the encouragement of students: He is the founding director of CHAMP, a STEM outreach program for underserved high school and middle school students in Houston, which won the AMS Award for Mathematics Programs that Make a Difference (article) and a Phi Beta Kappa Award for Engaging Broad Audiences. He is also a mentor for the National Alliance for Doctoral Studies in Mathematics and the faculty advisor for the UH mathematics department's AMS graduate student chapter. He is author of the following books and monographs:


Dr. Anna Vershynina
Ph.D. University of California, Davis 2012 M.A. University of California, Davis 2012 Homepage Publications 

Dr. Vershynina is an assistant professor at the University of Houston. Her primary research interests lie in the area of quantum information theory. Additionally she does work on quantum computation and quantum manybody physics. Selected papers:


Dr. Alan Czuroń
Ph.D., Polish Academy of Sciences, 2017 M.S., University of Warsaw, 2013 B.A., University of Warsaw, 2011 Homepage Publications 

Dr. Czuroń is a postdoc at the University of Houston. His research interests include functional analysis, geometric and analytic group theory, coarse geometry, harmonic analysis, and applications of compressed sensing methods applications in NMR spectroscopy. Selected papers:


Ahmed Abouserie
Advisor: Undecided Homepage 


Tattwamasi Amrutam
Advisor: Dr. Mehrdad Kalantar Homepage 


Sarah Chehade
Advisor: Dr. Anna Vershynina Homepage 


Dylan DomelWhite
Advisor: Dr. Bernhard Bodmann Homepage 


Mozahid Haque
Advisor: Dr. Mark Tomforde Homepage 


Robert Mendez
Advisor: Dr. Bernhard Bodmann Homepage 


Worawit Tepsan
Advisor: Dr. David Blecher Homepage 


Zhenhua Wang
Advisor: Dr. David Blecher Homepage 

The Analysis Research Group runs a weekly seminar hosting talks by both internal and external speakers:
The seminar is open to anyone who wishes to attend. 

There are several regular Analysis courses offered each semester as well as frequent
topics courses covering advanced material.
Students with suggestions for topics courses should contact a professor in the Analysis Group. 

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