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. My work first emphasized the study of these objects via both functional analytic methods and methods from algebraic topology. More recently, I have conducted research on the relationship between harmonic 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 arising in wavelet theory; most recently I have studied twisted group C*-algebras corresponding to N-adic rational numbers, generalized wavelets that can be associated to higher rank graph algebras, and the relationship of these latter wavelets to the eigenspaces of certain Laplace-Beltrami operators.
keywords
Functional and Harmonic Analysis, Operator 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 3001 - Analysis 1
Primary Instructor
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Spring 2020
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 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 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
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
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
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.
