Analysis Research Group
At the University of Houston
At the University of Houston
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 thought of as infinite-dimensional generalizations of Euclidean space.
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 or measurable functions to noncommutative algebras, facilitating the "noncommutative topology and integration theory".
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.
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Later connections to random matrix theory, combinatorics, representations of symmetric groups, large deviations, quantum information theory and other theories were established. Free probability is currently undergoing active research.
This is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience.