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• 1 media/Wave5 - Blue-SURF-New.jpg 2020-09-18T18:01:03+00:00 Dr. Daniel Alpay 10 Schmid plain 2020-10-16T20:03:39+00:00
  
  
  Email: alpay@chapman.edu
  Website:  http://www1.chapman.edu/~alpay/
  
  College: Schmid College of Science and Technology, Fowler School of Engineering
  Department: Mathematics, Electrical Engineering, Computer Science
  
  Overview of scholarly research/creative activity:  A dual formation in electrical engineering (Telecom Paris, 1978) and theoretical mathematics (Weizmann Institute of Sciences, Israel, 1986) leads to abstract mathematical problems, motivated by signal theory and engineering. I teach topics such as machine learning, statistical physics, information theory and complex variables.
  
  Specific projects working on:  My current research interests include:
  1. Infinite dimensional analysis and generalized stochastic processes. Using the theory of
  nuclear spaces and stochastic distributions we study models for stochastic processes and their derivatives, beyond the fractional Brownian motion.
  2. Superoscillations: The topic was introduced by Yakir Aharonov (Chapman) and Michael Berry, with a motivation from quantum mechanics and is related to  new problems in harmonic analysis.
  2. Hypercomplex analysis and quaternionic operator theory. We also consider analysis in the ternary algebra, the Grassmann algebra and study relationships with models in quantum mechanics. We also consider in this setting infinite order differential operators.
  3. Function theory on a compact Riemann surface, and relationships to over-determined linear systems and pairs of self-adjoint commuting operators.
  4. Linear system and matrix-valued rational functions. This line of research includes the study of inverse scattering problems with rational data.
  5. Reproducing kernel Hilbert spaces. We consider various extensions of the
  Bargmann-Fock-Segal space of quantum mechanics, and  characterize them in terms
  of operators in reproducing kernel spaces.
  6. Wavelets: A general setting for wavelets which goes beyond the Lebesgue space is under investigation.
  In each of these questions I can give to a prospective student a first elementary question, which can lead to a serious undergraduate, or graduate, research problem.
  
  Number of students looking to work with: 3-4
  
  Time commitment for students:  Meet once a week
  
  When students are needed: Spring 2021
  
  Requirements for students who work with you:  Willingness to study new topics and complete material not yet studied in class.
  
  What would students be expected to do​​​​​​: Research questions
  
  Additional Information: This is an introduction to research in theoretical mathematics, and requires a lot of work to study topics usually not studied at the undergraduate level. Part can be done under the framework of an individual study course.