# Research

## 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.
- Giacomo Micheli, who works on applied algebra (finite fields, cryptography, coding theory) and number theory (density questions over global fields).
- Dmytro Savchuk, who works in geometric and combinatorial group theory; automata theory; p-adic dynamics.

### 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; analyzing 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 between harmonic analysis and approximation theory and 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.
- 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.
- Seung-Yeop Lee, who works on Random Matrix Theory, 2D Coulomb gas, Orthogonal Polynomials, Asymptotic Analysis, and the Riemann-Hilbert problem.
- Wen-Xiu Ma, who works on soliton theory, orthogonal polynomials, and numerical analysis.
- Hemant Pendharkar, who works in \(C^*\)-Algebras and non-self-adjoint Operator Algebras, has also contributed in Theoretical Physics, Theoretical Computer Science (Data Mining and Algorithms), Cryptography and Cryptology, Pedagogy and Engineering.
- Evguenii Rakhmanov, who works on complex analysis, approximation theory, orthogonal polynomials, and potential theory.
- Joel Rosenfeld, who works on machine learning, reproducing kernel Hilbert spaces, approximation theory, cyber-physical systems verification, fractional order partial and ordinary differential equations, operator theory and functional analysis, optimal control theory, adaptive dynamic programming, densely defined operators, and the history of mathematics.
- 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.

### 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 levels, 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.
- Giacomo Micheli, who works on applied algebra (finite fields, cryptography, coding theory) and number theory (density questions over global fields).
- Dmytro Savchuk, who works in geometric and combinatorial group theory; automata theory; p-adic dynamics.
- 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.
- 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.
- Maria Leite, who works in mathematical modeling, dynamical systems, nonlinear dynamics of networks, and optimal control.
- Wen-Xiu Ma, who works on soliton theory, classic and quantum integrable systems, and symbolic computations.
- Joel Rosenfeld, who works on machine learning, reproducing kernel Hilbert spaces, approximation theory, cyber-physical systems verification, fractional order partial and ordinary differential equations, operator theory and functional analysis, optimal control theory, adaptive dynamic programming, densely defined operators, and the history of mathematics.
- Razvan Teodorescu, who works in integrable nonlinear differential equations and mathematical physics.

### 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.
- Greg McColm, who works in combinatorics and probability, logic and computer science, and particularly in geometry.
- Theo Molla, who works in extremal graph theory.
- 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 geometric group theory and mathematics education.
- Zhenkun Li, who works in low-dimensional topology, knot theory, gauge theory, and Floer homology.
- 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.
- Masahico Saito, who works in knot theory, low-dimensional topology, and related algebraic structures.

### Statistics

- Gangaram “Gan” S. Ladde, who works on Dynamic Reliability/Survival Analysis under the Influence of Internal and External Intervention Processes and Control; Stochastic Modeling, Methods and Analysis of Dynamic Processes in Biological, Chemical, Engineering, Financial, Medical, Military, Physical and Social Sciences; Time Series Analysis and Applications; State and Parameter Estimation Problems in Statistical and Modeling Analysis; Multivariate/Large-Scale Systems Analysis; Stochastic Modeling of Network Dynamic Processes; Multi-Agent, Multi-Cultural, and Multi-Market/Financial Dynamic Systems; Approximations, Statistical Analysis, Conceptual and Computational Algorithms; Deterministic, Hereditary and Stochastic Qualitative and Quantitative Analysis of Dynamic Systems; Stochastic Hybrid Dynamic Processes under Environmental Structural Perturbations.
- Lu Lu, who works on reliability analysis, design of experiments, response surface methodology, survey sample design and methodology, multiple objective optimization, statistical engineering.
- Kandethody M. Ramachandran, who works on stochastic control problems; approximate solutions using weak convergence or Martingale techniques; computational techniques to obtain optimal controls; learning algorithms, which may arise in the context of artificial intelligence, via stochastic approximation techniques; applied problems involving stochastic calculus and distributed parameter systems; Software reliability, Digital communications, Applications of Wavelet analysis in Statistics and Signal Processing.
- Christos “Chris” P. Tsokos, who works on AI-BIG DATA statistical analysis and modeling. Cybersecurity/vulnerability — Stochastic Analysis and Modeling of Network Systems. Machine Learning, Deep Learning Data Analytics, Neural Network, Random Forest in Analyzing and Modeling Health Systems — brain, lung, prostate, ovarian, breast cancers. Parametric, nonparametric, Bayesian Reliability and Survival Analysis. Nonlinear Statistical Models for Social Sciences. Global Warming — causes, forecasting models, world-wide comparisons, regional clusterings. Analysis and Clustering of Nonstationary signals in Environmental, Engineering and Health Sciences. Radar Ground Detection systems driven by nonparametric-Time series algorithms. AI-Machine Learning Models for Finance, Capital Asset Selection and Allocation Analysis, and cybersecurity issues. Structuring and evaluation of diversified financial portfolios.