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Mathematics & Statistics

Research Areas

Mathematics Research Areas

Algebra and Number Theory

Algebra is a major branch of mathematics that studies abstract systems endowed with operations. The objectives are to understand the intrinsic structure of those systems, their classifications, and to provide profound insight and effective methods for other areas of mathematics and science. Our research areas include Finite Fields, Ring Theory (Frobenius Rings, Formal Power Series), Finite-dimensional Representation Theory (of groups, algebras, and quantum groups), Linear Algebra, Planar Algebras, Number Theory (p-adic Fields, Exponential Sums), Algebraic Combinatorics (Association Schemes, Difference Sets, Algebraic Graph Theory), Coding Theory and Cryptography, and applications of Classical and Super Lie Algebras.

The faculty in our department who work in this area are:

• Jean-François Biasse, who works in number theory, quantum computing, and cryptography.
• Brian Curtin, who works in algebra, representation theory, and algebraic combinatorics.
• Xiang-dong Hou, who works in algebra, combinatorics, number theory, coding theory, and cryptography.
• Wen-Xiu Ma, who works on applications of classical and super Lie algebras.
• Dmytro Savchuk, who works in algebraic combinatorics, computational group theory, and cryptography.

Analysis

Mathematical analysis could also be termed “continuous mathematics”. It provides powerful methods for modeling real-life phenomena. The triumph of mathematics that started in the 16th century and is still in full force today is primarily due to the invention and application of analytic tools and methods. Analysis has evolved hand-in-hand with physics, and the interaction of mathematical analysis with the rest of the sciences continues to be lively and mutually productive.

Harmonic and complex analysis deal with the decomposition of objects (like sound waves, picture signals, etc.) into basic building blocks; analysing properties of such decompositions and also the reverse reconstruction process. Potential theory lies on the boundary of real and complex analysis with direct connections to electrostatics, quantum mechanics, and other parts of physics. Approximation theory offers methods to replace complicated objects/models with simpler ones that are easier to handle. The theory of orthogonal polynomials lies in between harmonic analysis and approximation theory and it is closely related to other branches of mathematics (stochastic processes, combinatorics, mathematical physics, etc.). Banach-space theory and operator theory are relatively new areas of mathematics which focus on the geometry of generalizations of our standard 3-space and on properties of mappings between such spaces.

The faculty in our department who work in this area are:

• Catherine Bénéteau, who works on complex analysis, interpolation theory, and nonlinear extremal problems.
• Thomas Bieske, who works on analysis on sub-Riemannian and general metric spaces.
• Arthur Danielyan, who works on complex analysis and approximation theory.
• Arcadii Grinshpan, who works on complex analysis, inequalities, mathematical modeling, and special functions.
• Dmitry Khavinson, who works on harmonic and complex analysis, potential theory, and approximation theory.
• Sherwin Kouchekian, who works on operator theory, complex analysis, and mathematical physics.
• Wen-Xiu Ma, who works on soliton theory, orthogonal polynomials, and numerical analysis.
• Evguenii Rakhmanov, who works on complex analysis, approximation theory, orthogonal polynomials, and potential theory.
• Jogindar Ratti, who works on real and complex analysis and graph theory.
• Boris Shekhtman, who works on approximation theory and Banach space theory.
• Lesɫaw Skrzypek, who works on approximation theory and Banach space theory.
• Razvan Teodorescu, who works in stochastic processes, harmonic analysis, biorthogonal polynomials, and approximation theory.
• Vilmos Totik, who works on approximation theory, orthogonal polynomials, and potential theory.

Cryptography and Coding Theory

The goal of cryptography is to keep the information private, guarantee its authenticity and more generally provide tools on which we can rely to secure the cyberspace. Coding theory on the other hand helps us recover information when it is transmitted via a noisy channel. They are the backbone of cybersecurity, which is a strategic priority at the University of South Florida. USF hosts the Florida Center for Cybersecurity (FC$$^2$$), created in 2014, which aims to position Florida as the national leader in cybersecurity through education; innovative, interdisciplinary research; and community outreach. The Mathematics Department offers courses in cryptography and coding theory for science majors at both the undergraduate and the graduate level, as well as an online course which is part of the online Masters' degree in Cybersecurity. The research of the department members is focused on post-quantum cryptography (the design of primitives that will resist attacks from quantum computers), Fully Homomorphic Encryption (schemes that enable the computation on encrypted data), and network security.

The faculty in our department who work in this area are:

• Jean-François Biasse, who works in number theory, quantum computing, and cryptography.
• Xiang-dong Hou, who works in algebra, combinatorics, number theory, coding theory, and cryptography.
• Dmytro Savchuk, who works in algebraic combinatorics, computational group theory, and cryptography.
• Kaiqi Xiong, who works in computer and network security.

Differential Equations and Nonlinear Analysis

Partial differential equations (PDEs) and ordinary differential equations (ODEs) constitute a core area of applied mathematics. This area is unique in two aspects: it interacts closely with almost all the other major areas of pure, applied, and computational mathematics, and PDE/ODE serve as mathematical models in all disciplines of the physical and social sciences, especially the rapidly expanding applications in dynamical systems, differential geometry, integrable systems, theoretical physics, econometrics, finance, and biology.

The modern theory of PDEs includes the local and global existence and behavior of a variety of types of solutions, with methodology ranging from functional analysis to complex analysis to numerical analysis. The inverse scattering transform in soliton theory is one of the most important developments in applied mathematics in the twentieth century. The global existence and regularity of solutions for the three-dimensional Navier-Stokes equations in fluid dynamics is defined by the Clay Institute of Mathematics as one of the new millennium problems of mathematics.

The faculty in our department who work in this area are:

• Thomas Bieske, who works on nonlinear PDE's and potential theory in sub-Riemannian and general metric spaces.
• A. G. Kartsatos, who works on nonlinear differential equations in Banach spaces and nonlinear analysis.
• Dmitry Khavinson, who works on holomorphic PDEs, potential theory, and applications to astrophysics.
• Sherwin Kouchekian, who works on PDE boundary value problems and applications of potential distributions in scanning probe microscopy and nano-rings.
• Wen-Xiu Ma, who works on soliton theory, classic and quantum integrable systems, and symbolic computations.
• Razvan Teodorescu, who works in integrable nonlinear differential equations and mathematical physics.
• Yuncheng You, who works on nonlinear PDEs, infinite-dimensional and stochastic dynamical systems, control theory, nonlinear and dispersive waves, pattern formation, mathematics of finance, mathematical biology, and interdisciplinary data analysis.

Discrete Mathematics

Discrete Mathematics captures many of the most active research fields today, from theoretical computer science to probabilistic methods, from graph theory to category theory, with applications to all the natural sciences, the social sciences, the professions of business, engineering, and medicine, and even the humanities. At USF, we have faculty exploring many of these frontiers. Discrete Mathematics at USF is rather inclusive, with places for algebra, combinatorics, computing, logic, number theory, topology, and related areas.

In combinatorics, discrete structures (like graphs) are assembled, dissected, (re)arranged, or counted. These structures can be studied individually or collectively using algebraic, analytic, combinatorial, logical, probabilistic, or topological methods. Theoretical computer science ranges from the analysis of algorithms to the analysis of informational processes, particularly processes analogous to physical or biological processes such as network computation or self-assembly.

The faculty in our department who work in this area are:

• Nataša Jonoska, who works in biomolecular computation, symbolic dynamics, and formal languages.
• Milé Krajčevski, who works in combinatorial and geometric group theory.
• Greg McColm, who works in combinatorics, logic, and probabilistic methods.
• Brendan Nagle, who works in extremal combinatorics and hypergraph regularity methods.

Geometry and Topology

In Geometry and Topology, properties and structures of spatial objects — either rigid (geometry) or flexible (topology) — are studied using algebraic, analytic, and combinatorial methods. Our focus areas include algebraic topology, analysis on Riemannian and sub-Riemannian manifolds, discrete and Euclidean geometry, combinatorial and geometric group theory, Hamiltonian systems, knot theory, low-dimensional manifolds, quantum topology, symplectic manifolds, and transformation groups. Our research has applications in Chemistry (crystals and self-assembly processes), Biology (recombinant DNA processes), Control Theory, and Mathematical Physics.

The faculty in our department who work in this area are:

• Thomas Bieske, who works on partial differential equations, potential theory and analysis in sub-Riemannian and general metric spaces.
• Mohamed Elhamdadi, who works in topology, quantum algebra, and knot theory.
• Nataša Jonoska, who works in biomolecular computation, symbolic dynamics, and formal languages.
• Milé Krajčevski, who works in combinatorial and geometric group theory.
• Wen-Xiu Ma, who works on Hamiltonian theory, conservation laws, and symmetries of differential equations.
• Greg McColm, who works in geometric representations of physical and chemical objects.
• Marcus McWaters, who works in topology, topological algebra, and algebraic topology.
• Masahiko Saito, who works in knot theory, low-dimensional topology, and related algebraic structures.

Statistics Research Areas

The research interests of the Statistics faculty members can be found at Statistics Faculty Research.