Analysis Research Group

at the University of Houston



















Group Leaders


Dr. David Blecher

Non-Selfadjoint Operator Algebras and Operator Spaces
 

Dr. Bernhard Bodmann

Wavelets, Frames, and Image Analysis
 
 

Dr. Mehrdad Kalantar

Operator Algebras, Topological Groups, and Quantum Groups
 

Dr. Mark Tomforde

Operator Algebras, C*-algebras, and Algebra
 

Dr. Anna Vershynina

Quantum information theory and science, quantum many-body physics

Research Areas




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 infinite-dimensional generalizations of Euclidean space.


Operator Algebras

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 and Frames

Wavelets may be visualized as brief wave-like 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

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).

Quantum Information

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.

Bios of Group Leaders


           

            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:

  • Operator algebras and their modules---an operator space approach by David Blecher and Christian Le Merdy, London Mathematical Society Monographs. New Series, 30. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2004. x+387 pp.
  • Categories of operator modules (Morita equivalence and projective modules) by David Blecher, Paul Muhly, and Vern Paulsen, Mem. Amer. Math. Soc. 143 (2000), no. 681, viii+94 pp.
  • The calculus of one-sided M-ideals and multipliers in operator spaces by David Blecher and Vrej Zarikian, Mem. Amer. Math. Soc. 179 (2006), no. 842, viii+85 pp





           

            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:

  • Bernhard G. Bodmann and Peter G. Casazza, The road to equal-norm Parseval frames, J. Funct. Anal. 258, 397-420 (2010).
  • Bernhard G. Bodmann, David W. Kribs and Vern I. Paulsen, Decoherence-Insensitive Quantum Communication by Optimal C*-Encoding, IEEE Trans. Inform. Theory 53, 4738-4749 (2007).
  • Bernhard G. Bodmann, Optimal linear transmission by loss-insensitive packet encoding, Appl. Comput. Harmon. Anal. 22, 274-285, (2007).
  • Bernhard G. Bodmann, Manos Papadakis, and Qiyu Sun, An inhomogeneous uncertainty principle for digital low-pass filters, J. Fourier Anal. Appl. 12, 181-211, (2006).
  • Bernhard G. Bodmann, A lower bound for the Wehrl entropy of quantum spin with sharp high-spin asymptotics, Commun. Math. Phys. 250, 287-300, (2004).





           

            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:

  • M. Kalantar and M. Kennedy, Boundaries of Reduced C*-algebras of Discrete Groups, J. Reine Angew. Math. (Crelle's Journal), to appear.
  • M. Kalantar, M. Neufang and Z.-J. Ruan, Realization of Quantum Group Poisson Boundaries as Crossed Products, Bull. Lond. Math. Soc. 46 (2014), no. 6, 1267-1275.
  • J. Crann and M. Kalantar, An Uncertainty Principle for Unimodular Quantum Groups , J. Math. Phys. 55 (2014), 081704.
  • M. Kalantar, A Limit Theorem for Discrete Quantum Groups, J. Funct. Anal. 265 (2013), no. 3, 469-473.





           

            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:

  • One-sided shift spaces over infinite alphabets (with William Ott and Paulette Willis), New York Journal of Mathematics. NYJM Monographs 5. State University of New York, University at Albany, Albany, NY, 2014. 54 pp.
  • The structure of graph C*-algebras and generalizations, Chapter in the book "Graph Algebras: Bridging the gap between analysis and algebra", Eds. Gonzalo Aranda Pino, Francesc Perera Domenech, and Mercedes Siles Molina, Servicio de Publicaciones de la Universidad de Málaga, Málaga, Spain, (2006).





           

            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 many-body physics. Selected papers:

  • E. A. Carlen, A. Vershynina, "Recovery map stability for the Data Processing Inequality", arxiv:1710.02409, 2017 (to appear)
  • A. Vershynina, "Entanglement rates for bipartite open systems", Physical Review A, 92(2):022311, (2015)
  • E. H. Lieb, A. Vershynina, "Upper bound on mixing rates", Quantum Information and Computation, 13(11&12):0986, (2013)
  • B. Nachtergaele, A. Vershynina, V. A. Zagrebnov, "Lieb-Robinson bound and the existence of the thermodynamic limit for the class of irreversible dynamics", AMS Contemporary Mathematics, 552:161, (2011)



Postdocs


           

            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:

  • A. Czuroń, M. Wojciechowski, On the isomorphisms of Fourier algebras of finite abelian groups, (to appear).
  • A. Czuroń, Property Flq implies property Flp for 1 < p < q < ∞, Advances in Mathematics vol. 307 (2017), 715-726.
  • M. Urbańczyk, D. Bernin, A. Czuroń, K. Kazimierczuk, Monitoring polydispersity by NMR diffusometry with tailored norm regularisation and moving-frame processing, Analyst, vol. 141 (2016), 1745-1752

Students


           

            Tattwamasi Amrutam

      Advisor: Dr. Mehrdad Kalantar
      Homepage
     


           

            Qianfan Bai

      Advisor: Dr. Anna Vershynina
      Homepage
     


           

            Sarah Chehade

      Advisor: Dr. Anna Vershynina
      Homepage
     


           

            Dylan Domel-White

      Advisor: Dr. Bernhard Bodmann
      Homepage
     


           

            Mozahid Haque

      Advisor: Dr. Mark Tomforde
      Homepage
     


           

            Corinne Lane

      Advisor: Undecided
      Homepage
     


           

            Robert Mendez

      Advisor: Dr. Bernhard Bodmann
      Homepage
     


           

            Worawit Tepsan

      Advisor: Dr. David Blecher
      Homepage
     


           

            Zhenhua Wang

      Advisor: Dr. David Blecher
      Homepage
     


Analysis Seminar




                        The Analysis Research Group runs a weekly seminar hosting talks by both internal and external speakers:

Analysis Seminar Webpage


The seminar is open to anyone who wishes to attend.







Theses of Former Students



                        2017
2016
2015
2013
2012
2011
2010
2009
2008
2006
  • Soha Abdulbaki, Advisor: Vern Paulsen, Thesis: Generalized Sigma-Delta Quantization.
  • Damon Hay, Advisor: David Blecher, Thesis: Non-Commutative Topology and Peak Interpolation for Operator Algebras.
  • Deepti Kalra, Advisor: Vern Paulsen, Thesis: Equiangular Cyclic Frames.

2003
  • Roderick Holmes, Advisor: Vern Paulsen, Thesis: Optimal Frames.
  • Masayoshi Kaneda, Advisor: Vern Paulsen, Thesis: Multipliers and Algebrizations of Operator Spaces.

2000
  • James Solazzo, Advisor: Vern Paulsen, Thesis: Interpolation and Computability.

1997
  • Rajia Khoury, Advisor: Vern Paulsen, Thesis: Closest Matrices in the Space of Doubly Stochastic Matrices.

1996
  • Sarah H. Ferguson, Advisor: Vern Paulsen, Thesis: Ext, Analytic Kernels and Operator Ranges.

1995
  • Chun Zhang, Advisor: Vern Paulsen, Thesis: Representation and Geometry of Operator Spaces.

1992
  • Che-Chen "Peter" Chu, Advisor: Vern Paulsen, Thesis: Finite Dimensional Representations of Function Algebras.

1991
  • Terry Richard Tiballi, Advisor: Vern Paulsen, Thesis: Symmetric Orthogonalization of Vectors in Hilbert Space.

1983
  • Ching-yun Suen, Advisor: Vern Paulsen, Thesis: The Representation Theory of Completely Bounded Maps on C*-algebras.




Contact Information


Department of Mathematics

University of Houston


Mailing Address
3551 Cullen Blvd., Room 641
Philip Guthrie Hoffman Hall
Houston, TX 77204-3008


We are located on the 6th floor of Philip Guthrie Hoffman Hall,
listed as PGH on the UH Campus map.


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