» Center of Excellence in Complex and Hypercomplex Analysis

A group of mathematicians and physicists from the Schmid College of Science and Technology at Chapman University joined some of their national and international collaborators to form this Center of Excellence. The research conducted under the center's umbrella is mainly motivated by the latest results in Clifford and Hypercomplex Analysis and endeavors to find new ways in which this research can be applied in mathematics and physics. The members of this center of excellence draw on their research experience and use analytical and computational techniques to build sound differential and integral theories in this context, as well as possible applications in Quantum Physics and other interdisciplinary fields.

Complex Analysis is a classical branch of mathematics, having its roots in late 18th and early 19th centuries, which investigates functions of one and several complex variables. It has applications in many branches of mathematics, including Number Theory and Applied Mathematics, as well as in physics, including Hydrodynamics, Thermodynamics, Electrical Engineering, and Quantum Physics.

Clifford Analysis is the study of Dirac and Dirac type operators in Analysis and Geometry, together with their applications. In 3 and 4 dimensions Clifford Analysis is referred to as Quaternionic Analysis. Furthermore, methods and tools of Clifford Analysis are extended to the field of Hypercomplex Analysis.

  • 2017 Lecture Series
  • 2017 Conference
  • Members
  • Past Workshops
  • Library

  • An Overview of Wavelet Theory, With an eye on Superoscillations
    Lectures by Professor David Walnut, George Mason University
    November 6-10, 2017,
    Chapman University, ORANGE, CA

     

    SCHEDULE:
    November 6th from 4 p.m. to 5:30 p.m.
    Lecture 1: Some Time-Frequency Transforms

    • What is a Time-Frequency distribution/representation?
    • The Short-Time Fourier Transform
    • The Ambiguity Function and Wigner's Distribution
    • Uncertainty Principles

    November 7th from 4 p.m. to 5:30 p.m.
    Lecture 2: Gabor Theory

    • D. Gabor's notion of information area
    • Discretizing the Short-Time Fourier Transform
    • Stability in time-frequency representations
    • The Balian-Low Theorem
    • The Amalgam BLT (Heil)
    • The Zak transform and proofs.

    November 8th from 4 p.m. to 7 p.m.
    Lecture 3: Theory of Frames

    • Finite Frames
    • Abstract Frames
    • The Frame operator
    • Riesz bases
    • Some historical remarks
    • The BLT from the perspective of frames

    Lecture 4: Structure Theorems for Gabor Frames

    • Existence of Gabor frames
    • Zak transform methods
    • Density theorems for Gabor frames
    • The Wexler-Raz and Ron-Shen duality
    • Wilson bases

    Lecture 5: Continuous and Discrete Wavelets

    • The Continuous Wavelet transform of Grossman and Morlet
    • The CWT as a time-frequency (time-scale) transformation
    • Relation to the Calderon Reproducing Formula
    • Discrete Wavelet decompositions of Frazier and Jawerth
    • Relation to Littlewood-Paley theory

    November 9th from 4 p.m. tp 7 p.m.
     Lecture 6: Orthonormal Bases of Wavelets

    • The Haar Wavelet basis
    • The Shannon Wavelet basis
    • The Meyer Wavelet basis
    • Local Cosine bases (Coifman-Weiss)
    • Wavelet Sets (Weiss et. al)

    Lecture 7: Multiresolution Analysis (Mallat-Meyer)

    • Definition of MRA in one dimension
    • Finding the wavelet from the scaling function
    • The Daubechies wavelets
    • MRA in higher dimensions

    Lecture 8: Coorbit Spaces (Feichtinger-Grochenig)

    • Representations of the Heisenberg group and the Affine group
    • Co-orbit spaces from irreducible group representations
    • Co-orbit spaces as reproducing kernel Hilbert spaces
    • Frames and sampling in co-orbit spaces

    November 10th from 2 p.m. to 3 p.m.
    Lecture 9: Wavelets in Functional Analysis

    • Modulation spaces
    • The Feichtinger Algebra
    • Pseudodifferential operators and Gabor frames
    • Wavelets as unconditional bases for Banach spaces
    • Wavelets and operators

    About Professor David Walnut:
    David F. Walnut received his Ph.D. degree in mathematics from the University of Maryland, USA in 1989 under the direction of John Benedetto. He has been on the faculty at George Mason University in Fairfax, Virginia, USA since 1990, where he currently holds the rank of Professor. His mathematical interests include Euclidean harmonic analysis, time-frequency analysis and sampling theory.

  • Mathematics, Signal Processing and Linear Systems: New Problems and Directions

    November 14-19, 2017

    Chapman University, Orange, CA

    The topic of the conference is the intersection between mathematical analysis (in a wide sense), signal processing and applications to electrical engineering problems. The research presented here is at the crossroad of mathematics (in particular complex analysis, functional analysis and stochastic processes), the theory of linear systems, signal processing and electrical engineering. The conference fits well in the general program of a future engineering school. 

    Organizers:

    Conference Schedule, list of speakers and abstracts will be published soon. For preliminary schedule and lodging information, please visit the personal web page of Mihaela B. Vajiac

  • Director

    Mihaela Vajiac Mihaela B. Vajiac, Director
    Research Interests: Quaternionic and Hypercomplex Analysis, Computational Algebra, Differential Geometry, PDEs, Integrable Systems
    Contact: Associate Professor of Mathematics, Chapman University, Orange, California, USA
    Email: mbvajiac@chapman.edu

    Chapman Faculty Members

    Yakir Aharonov Yakir Aharonov
    Research Interests: Quantum Physics
    Contact: Professor of Theoretical Physics, James J. Farley Professor of Natural Philosophy, Chapman University, Orange, California, USA
    Email: aharonov@chapman.edu
    Daniel Alpay Daniel Alpay
    Research Interests: Operator theory, Schur analysis, Rational functions and applications to linear system theory and wavelets
    Contact: Foster G. and Mary McGaw Professorship in Mathematical Sciences, Chapman University, Orange, California, USA
    Email: alpay@chapman.edu
    Daniele Struppa Daniele C. Struppa
    Research Interests: Quaternionic and Hypercomplex Analysis, Microlocal Analysis
    Contact: Chancellor of Chapman University, Orange, California, USA
    Email: struppa@chapman.edu
    Jeff Tollaksen Jeff Tollaksen
    Research Interests: Quantum Physics
    Contact: Associate Professor of Physics, Chapman University, Orange, California, USA
    Email: tollakse@chapman.edu
    Adrian Vajiac Adrian Vajiac
    Research Interests: Quaternionic and Hypercomplex Analysis, Computational Algebra, Quantum Field Theory, Foundations of Geometry
    Contact: Associate Professor of Mathematics, Chapman University, Orange, California, USA
    Email: avajiac@chapman.edu

    External Faculty Members

    Fabrizio Colombo Fabrizio Colombo
    Research Interests: Functional Calculus, Applications to Physics, Compatibility Conditions, Symmetries, PDEs.
    Contact: Associate Professor, Politecnico di Milano, Italy
    Email: fabrizio.colombo@polimi.it
    Hendrik De Bie Hendrik De Bie
    Research Interests: Hypercomplex Analysis.
    Contact: Associate Professor, Department of Mathematical Analysis, Universiteit Gent
    Email: hendrik.debie@ugent.be
    Maria Elena Luna Elizarraras Maria Elena Luna Elizarraras
    Research Interests: Clifford Analysis, Hypercomplex analysis.
    Contact: Professor, Holon Institute of Technology, Holon, Israel 
    Email: eluna_10@hotmail.com
    Graziano Gentili Graziano Gentili
    Research Interests: Quaternionic Analysis
    Contact: Professor, Dipartimento di Matematica "U.Dini", Universita di Firenze, Italy
    Email: graziano.gentili@math.unifi.it
    Irene Sabadini Irene Sabadini
    Research Interests: Syzygies, Free Resolutions, Hilbert function, Duality Theorems.
    Contact: Associate Professor, Politecnico di Milano, Italy
    Email: irene.sabadini@polimi.it
    Ahmed Sebbar Ahmed Sebbar
    Research Interests: Complex Analysis, Number Theory, Differential Operators
    Contact: Professor, Exceptional Class, University of Bordeaux
    Email: Ahmed.Sebbar@math.u-bordeaux1.fr
    Michael Shapiro Michael Shapiro
    Research Interests: Clifford Analysis, Hypercomplex Analysis.
    Contact: Professor, Holon Institute of Technology, Holon, Israel
    Email: shapiro1945@outlook.com
    Franciscus Sommen Franciscus Sommen
    Research Interests: Clifford Analysis.
    Contact: Professor, Department of Mathematical Analysis, Universiteit Gent
    Email: fs@cage.ugent.be

  • 2016: The “International Conference on Complex Analysis and Operator Theory - Celebrating Daniel Alpay’s 60th birthday” took place at Chapman University, Orange, California, November 4-7, 2016.

    The main topics of the conference were complex analysis, operator theory and other areas of mathematics which have been touched by Professor Daniel Alpay during his fertile scientific activity. This meeting was the perfect occasion to celebrate his 60th birthday among friends and collaborators.

    Committees:
    • Scientific: F. Colombo, D.C. Struppa, M.B. Vajiac, A. Vajiac
    • Organizing: I. Sabadini, M.B. Vajiac

    List of speakers: J. Behrndt, V. Bolotnikov, F. Colombo, J. Gantner, W. Helton, T. Hempfling, P. Jorgensen, H.T. Kaptanoglu, E. Kvaalen, D. Levanony, I. Lewkovicz, M. Mboup, A. Pinhas, I. Sabadini, B. Schneider, A. Sebbar, D.C. Struppa, M.B. Vajiac, A. Vajiac, D. Volok, A. Yger

    Schedule of events

    Book of Abstracts

    2016: Series of 12 lectures on Operators of Infinite Order and Exact WKB Analysis, Professor Takashi Aoki, Kindai University

    Von Neumann Hall (545 W Palm), 9-11 a.m., August 22-26 and August 29th.

    Part I Differential operators, microdifferential operators and pseudodifferential operators of infinite order

    1. Introduction
    2. Algebraic definitions of pseudodifferential operators in complex analytic category
    3. Kernel functions
    4. Symbols of pseudodifferential operators and symbolic calculus
    5. Exponential calculus
    6. Applications

    Part II Exact WKB analysis

    1. Introduction
    2. WKB solutions of ODE of second order with a large parameter
    3. Borel sums of WKB solutions and connection formulas
    4. WKB solutions and microdifferential operators
    5. Higher-order equations and infinite-order equations
    6. Applications to special functions

    2014: Workshop on Integral transforms, boundary values and generalized functions

    Continuing the tradition established in the past two years, the Center of Excellence in Complex and Hypercomplex Analysis (CECHA) at Chapman University organized the Workshop during the period of October 17th to October 21st, 2014 with the topic: "Integral transforms, boundary values and generalized functions".

    The Organizing Committee and members of the CECHA:

    Schedule of the events and talks.

    List of speakers: D. C. Struppa, M.B. Vajiac, A. Vajiac, D. Alpay, M. Martin, P. Cerejeiras?, D. Clahane, I. Sabadini, F. Colombo, G. Gentili, U. Kahler, M. Libine, C. Nolder, L. Cnudde, M. Shapiro, ?M.E. Luna-Elizarraras, A. Sebbar.

    2013: Workshop on Results in Mathematics and Physics obtained by Methods of Hypercomplex Analysis

    The Center of Excellence in Complex and Hypercomplex Analysis (CECHA) at Chapman University organized a Workshop during the period of November 7th 2013 to November 11th 2013, with the topic: "Results in Mathematics and Physics obtained by Methods of Hypercomplex Analysis".

    Schedule of Events

    The Organizing Committee and members of the CECHA:

    List of confirmed speakers:Daniele C. Struppa, Mihaela Vajiac, Adrian Vajiac, Maria Elena Luna Elizarraras, Michael Shapiro, Daniel Alpay, Craig Nolder, Irene Sabadini, Fabrizio Colombo, Graziano Gentili, Joao Morais, Matvei Libine, Thomas Hempfling, Dominic Rochon, Lander Cnudde?.

    2012: Workshop on Function Theories for Bicomplex and Hyperbolic Numbers

    The Center of Excellence in Complex and Hypercomplex Analysis (CECHA) at Chapman University organized the workshop entitled Function Theories for Bicomplex and Hyperbolic Numbers, which took place at Chapman University, October 22-26, 2012, in the Sandhu Conference Center.

    Function Theories for Bicomplex and Hyperbolic numbers have recently gained momentum in the Mathematics and Physics communities. The goal of this workshop has been to provide and stimulate interaction and communication among the researchers whose scientific interests include these two areas of Mathematics.

    The talks and discussions were directly related to the Analysis of Bicomplex and Hyperbolic Numbers and their applications.

    The Organizing Committee and members of the CECHA:

    List of confirmed participants:Daniele C. Struppa, Adrian Vajiac, Mihaela Vajiac, Maria Elena Luna Elizarraras, Michael Shapiro, John Ryan, Dominic Rochon, Daniel Alpay, Franciscus Sommen, Uwe Kahler, Paula Cerejeiras, Matvei Libine, Sebastien Tremblay, Craig Nolder, Irene Sabadini, Fabrizio Colombo, Dana Clahane

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