My area of expertise is in a variety of aspects of Analysis. I am interested in Operator Algebras, in particular in the type of Algebras of Operators known as von Neumann algebras and C*-algebras. Recently, my work has emphasized the relationship between harmonic analysis, functional analysis, wavelets and operator algebras. I am interested in wavelets and frames associated to fractal systems and the operators and operator algebras that arise from their study. I remain interested in C*-algebras that can be formed from discrete groups and multipliers; most recently I have studied spectral triples on twisted nilpotent discrete group C*-algebras arising from length functions on the group having the property of 'bounded doubling'.
keywords
Functional and Harmonic Analysis, C*-algebras, Wavelet and Frame Theory
K-theory for the integer Heisenberg groups.
K-Theory: interdisciplinary journal for the development, application and influence of K-theory in the mathematical sciences.
201-227.
1999
MATH 2001 - Introduction to Discrete Mathematics
Primary Instructor
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Spring 2021
Introduces the ideas of rigor and proof through an examination of basic set theory, existential and universal quantifiers, elementary counting, discrete probability, and additional topics. Credit not granted for this course and MATH 2002.
MATH 3001 - Analysis 1
Primary Instructor
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Spring 2020 / Fall 2020 / Fall 2023
Provides a rigorous treatment of the basic results from elementary Calculus. Topics include the topology of the real line, sequences of numbers, continuous functions, differentiable functions and the Riemann integral.
MATH 3430 - Ordinary Differential Equations
Primary Instructor
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Fall 2021
Involves an elementary systematic introduction to first-order scalar differential equations, nth order linear differential equations, and n-dimensional linear systems of first-order differential equations. Additional topics are chosen from equations with regular singular points, Laplace transforms, phase plane techniques, basic existence and uniqueness and numerical solutions. Formerly MATH 4430.
MATH 4001 - Analysis 2
Primary Instructor
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Spring 2021 / Fall 2021
Provides a rigorous treatment of infinite series, sequences of functions and an additional topic chosen by the instructor (for example, multivariable analysis, the Lebesgue integral or Fourier analysis). Same as MATH 5001.
MATH 4330 - Fourier Analysis
Primary Instructor
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Spring 2019
The notion of Fourier analysis, via series and integrals, of periodic and nonperiodic phenomena is central to many areas of mathematics. Develops the Fourier theory in depth and considers such special topics and applications as wavelets, Fast Fourier Transforms, seismology, digital signal processing, differential equations, and Fourier optics. Same as MATH 5330.
MATH 5001 - Analysis 2
Primary Instructor
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Fall 2021
Provides a rigorous treatment of infinite series, sequences of functions and an additional topic chosen by the instructor (for example, multivariable analysis, the Lebesgue integral or Fourier analysis). Same as MATH 4001.
MATH 5330 - Fourier Analysis
Primary Instructor
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Spring 2019
The notion of Fourier analysis, via series and integrals, of periodic and nonperiodic phenomena is central to many areas of mathematics. Develops the Fourier theory in depth and considers such special topics and applications as wavelets, Fast Fourier Transforms, seismology, digital signal processing, differential equations, and Fourier optics. Department enforced prerequisite: MATH 4001. Same as MATH 4330.
MATH 6310 - Introduction to Real Analysis 1
Primary Instructor
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Fall 2018 / Fall 2022
Develops the theory of Lebesgue measure and the Lebesgue integral on the line, emphasizing the various notions of convergence and the standard convergence theorems. Applications are made to the classical L^p spaces. Department enforced prerequisite: MATH 4001. Instructor consent required for undergraduates.
MATH 6320 - Introduction to Real Analysis 2
Primary Instructor
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Spring 2019 / Spring 2020
Covers general metric spaces, the Baire Category Theorem, and general measure theory, including the Radon-Nikodym and Fubini theorems. Presents the general theory of differentiation on the real line and the Fundamental Theorem of Lebesgue Calculus. Recommended prerequisite: MATH 6310. Instructor consent required for undergraduates.
MATH 8330 - Functional Analysis 1
Primary Instructor
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Fall 2019 / Fall 2020 / Fall 2023
Introduces such topics as Banach spaces (Hahn-Banach theorem, open mapping theorem, etc.), operator theory (compact operators and integral equations and spectral theorem for bounded self-adjoint operators) and Banach algebras (the Gelfand theory). Department enforced prerequisites: MATH 6310 and MATH 6320. Instructor consent required for undergraduates. See also MATH 8340.
MATH 8370 - Harmonic Analysis 1
Primary Instructor
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Spring 2018 / Fall 2024
Examines trigonometric series, periodic functions, diophantine approximation and Fourier series. Also covers Bohr and Stepanoff almost periodic functions, positive definite functions and the L^1 and L^2 theory of the Fourier integral. Applications to group theory and differential equations. Department enforced prerequisites: MATH 5150 and MATH 6320. Instructor consent required for undergraduates.
