# Research in the Math Department

The Department of Mathematics is active in a variety of research areas across multiple mathematical concentrations, including

## Mathematical Biology

Mathematical Biology is an interdisciplinary area of research that lies at the intersection of significant mathematical problems and fundamental questions in biology. It is commonly joint work with biologists, physicists, chemists, engineers, and social scientists to construct models of phenomena in the life sciences. These models could confirm, explore, and predict the biological systems, and more importantly, identify and analyze the emergent structures in the biological processes. The implication from the mathematical results could diagnose, cure, prevent, and influence the system. The scale of the problems may range from the microscopic cellular level to the individual person to the population to the world-wide environmental level. Members of the Mathematical Biology group at UTSA include:

## Theory and Applications of Partial Differential Equations

Partial differential equations are relationships between a function and its derivatives. A partial differential equation (PDE) therefore models a relationship between physical variables and their changes in time and space, for example, the concentration of a drop of ink after it has been dropped into the ocean, or the electromagnetic fields belonging to radio waves that are emitted and received by our modern devices. The theory of partial differential equations is a field at the intersection of pure and applied mathematics and studies the question of how to solve these equations. Differential equations are hard to solve, but by using tools from many areas of Pure Mathematics, it is possible to understand their solutions without knowing them in detail. The members of this group are interested in partial differential equations arising from physical sciences, other branches of mathematics as well as from real-world modeling problems, with applications to image processing, phase transition, fluids, general relativity, biology, and ecology.

• Changfeng Gui—Nonlinear Partial Differential Equations and Applied Mathematics, Image Analysis and Processing
• Dung Le—Differential Equations, Partial Differential Equations, Mathematical Biology, Nonlinear Functional Analysis, and Population Dynamics.
• Fengxin Chen—Infinite dynamical systems
• Dmitry Gokhman—Differential Equations, Mathematical Biology
• Ivanka Stamova—Qualitative properties, Control
• Mostafa Fazly—Nonlinear Partial Differential Equations
• Vu Hoang—Nonlinear hyperbolic PDE, Einstein equations
• Gani Stamov—Qualitative Theory, Applications to real-world phenomena
• Walter Richardson

## Analysis, Operator Theory, and Operator Algebras

The main directions of research are: Non-commutative Multivariable Operator Theory, Classification of C*-algebras, Non-commutative Probabilities, and Random Matrices.

The work on non-commutative Multivariable Operator Theory aims at developing a free analogue of the Sz.-Nagy-Foias theory of contractions for non-commutative domains and varieties in several non-commuting variables and a theory of free holomorphic functions on domains admitting universal models. This is accompanied by the study of the universal operator algebras associated with non-commutative domains and varieties in connection with their representation theory, the harmonic analysis on Fock spaces, and their classification up to completely isometric isomorphisms. This study is anchored in the classical complex function theory in several variables and complex algebraic geometry. The study of non-commutative domains and free holomorphic functions has applications to free probability, spectral theory for non-commuting operators, interpolation, optimization and control, systems theory, and mathematical physics.

The C*-algebras can be seen as “non-commutative topological spaces”. The main direction of research in this area is the classification and the structure of separable nuclear C*-algebras by invariants including K-theory. Also of interest is to find connections with the theory of non-commutative dynamical systems (crossed product C*-algebras) and with the theory of “non-commutative topological spaces” of dimension zero.

Another area of research is that of free and non-commutative probabilities and analysis, with emphasis on the asymptotic behavior of random matrices and non-commutative functions.
A current problem in many areas of science and engineering is the analysis of large amounts of data (large numbers of variables and observations). It is common to use principal component analysis to concentrate the variance and this requires an eigenvalue analysis of the sample covariance matrix which brings the problem into the regime of random matrix theory. This domain (random matrix theory) draws a lot of attention today: it found applications way outside of pure mathematics, in areas from Telecommunications and Quantum Information Theory to Finance and Traffic Control. The general goal of the theory of non-commutative functions, still to be achieved is the construction of an analytic functional calculus for several non-commuting variables on a Banach space. The theory is expected to have a wide range of applications, from non-commutative spectral theory, free probability theory, to analysis of linear matrix inequalities in optimization and control.

• Cornel Pasnicu—Functional Analysis, Operator Algebras, Operator Theory
• Gelu Popescu—Noncommutative functions, Operator Theory, Operator Algebras
• Raj Wilson—Classical Complex function theory in several variables; C*-algebras, Quantum algebras
• Mihai Popa—Non-commutative probabilities, Random matrices

## Numerical Analysis and Scientific Computing

Fast and reliable numerical solutions to complex problems are driving many of the exciting technologies of today. Methods for providing these solutions are key to machine learning, robotics, aerospace engineering, space exploration, computer graphics, and the mathematical modeling of physical, biological and financial systems. Efficiency, robustness, and scalability of numerical algorithms delimit the frontiers of many applications.

• Weiming Cao—Finite element analysis, Numerical solution of differential equations
• Jose Morales—Computational Electronics, Inverse Problems, Quantum Computing & Information
• Zhuolin Qu—Numerical Methods for Nonlinear PDEs, Computational Biology, Scientific Computing
• Claire Walton—Polynomial approximation methods, Numerical linear algebra, Largescale computing
• Tong Wu—Numerical Methods for Nonlinear PDEs, Scientific Computing, Data Assimilation
• Ghassan Nasr

## Mathematics Education

Mathematics education research, broadly speaking, is the systematic study of the teaching and learning of mathematics. Mathematics education research aims to answer questions related to 1.) what and how learners of mathematics (any age or population) think about and do mathematics, 2.) what teachers think about and do as they teach mathematics, and 3.) what factors can influence the learning of mathematics (e.g., inclusion practices, curriculum). Mathematics education research also encompasses the study of teacher learning (both prospective and in-service teachers) in the context of their practice and in professional development. Further, it can include the study of the broader, systemic views of schooling and the role of mathematics in K-12 and higher education. We typically use quantitative data (e.g., survey data, data from eye-tracking software) and / or qualitative data (e.g., interviews, observations) to answer our research questions.

• Juan B. Gutiérrez—Adaptive learning, predictive analytics, interdisciplinary environments
• Sandy Norman—RUME, problem-solving, cognition
• Su Liang—Mathematics Teacher Education, Active Learning, Mathematical Thinking
• Priya V. Prasad—Teacher knowledge; teacher preparation; equity
• Jessica Gehrtz—Instructor responsiveness to student thinking, Evidenced-based instruction, Research in Undergraduate Mathematics Education (RUME)
• Raquel Vallines-Mira—Curriculum Development, professional growth, elementary math for teaching, lesson study, continuous improvement, teachers’ beliefs, math content knowledge for teaching teachers (MCKTT)
• Tina Vega—Professional development, active learning, developmental mathematics
• Carolyn Luna—Inquiry learning, inclusion, equitable practices

## Model Theory and Foundations

A model consists of a set (called the universe of the model) endowed with additional structure, typically in the form of distinguished elements, relations and functions in the universe. A framework for comparing models and capturing their properties is called a logic (the usual first-order logic is the best known example). Model theory a branch of mathematical logic concerned with relationships between models vis à vis a given logic. Many classes of discrete mathematical structures (e.g., the classes of groups, rings, fields, vector spaces, matroids, graphs, etc.) are models of suitable first-order theories. Model theory crystalizes a very general perspective of mathematical objects; historically, it has been intimately connected to foundational questions in set theory, discrete mathematics and algebra. The research group in model theory and foundations at UTSA focuses on model theory of continuous structures (e.g., probability spaces and Banach spaces), with applications primarily in analysis. Members of the UTSA research group on model theory and foundations include:

• José Iovino—Set theory, topology, combinatorics
• Eduardo Duenez—Measure and integration, metric convergence, continuous learning structures

## Probability, Optimization, Computing, and Geometry

The ability to identify optimization problems that can be solved efficiently in theory and in practice is what makes most modern applications and many results in theoretical computer science possible. The second mainstay of modern computational science is the ability to reason and compute with uncertain data and perform an uncertainty quantification, i.e. probabilistic computing and computational statistics. Finally, most modern computational problems are high dimensional: quite counterintuitively, geometry of high dimensions exhibits regularity patterns that do not exist in low dimensions (such as the concentration of measure phenomenon) and these patterns can be exploited for immense computational gains. The group is interested in the several aspects of the interplay between geometry, probability, optimization, and computing. Currently, the members conduct research on randomization in computational algebraic geometry, the power and limitations of convex programming hierarchies, applications of optimal control to robotics, empirical observability and real-time optimization of large-scale systems, applications of randomized scientific computing to engineering, uncertainty quantification methodologies such as Bayesian statistics, stochastic Galerkin, and stochastic collocation for computational electronics, and inverse modeling via multi-objective optimization and Bayesian estimation in the context of electrochemistry.

• Mihai Popa—Free Probability and Random matrices
• Alperen Ergür—Computational Algebraic Geometry, Convex Optimization, Theory of Computing
• Jens Forsgård—Tropical Geometry, Convex Geometry, Combinatorial Optimization
• Jose Morales—Bayesian Statistics – Stochastic Galerkin, Collocation – Optimization
• Claire Walton—Optimal control, estimation, uncertainty quantification
• Gerald Busald—Data Analysis, Lottery

## Mathematical Physics

Mathematics is the universal language of nature and the universe. It is the common ground of all physical theories such as Einstein’s General Relativity, Quantum Mechanics, Quantum Field Theory, Particle Physics, Cosmology, Kinetic Theory and Fluid Mechanics. Mathematical Physics is an interdisciplinary field studying the structures of these physical theories. Its aim is twofold: first, to provide a rigorous mathematical foundation for physics, by turning intuitive physical insight into mathematical proof. Second, to make predictions about the physical universe based on the mathematical features of the theories describing physical reality. Mathematical Physics touches upon and uses tools from many other fields, such as Group Theory, Harmonic Analysis, Differential Geometry, Topology, Operator Theory and Partial Differential Equations.

• Mostafa Fazly—PDEs of Mathematical Physics
• Vu Hoang—Relativity, Fluids, Field theory
• Jose Morales—Kinetic Theory, Boltzmann-Poisson, Wigner functions
• Reza Aghayan—Differential Geometry, Theory of Lie Groups, Computer Vision, and Image Analysis

## Principles of Urbanism

The structure of cities: how cities evolve in time, and what morphologies distinguish between the successful ones and others that have long-term problems. Mathematical tools that help to understand urban morphology and evolution include fractals, networks, geometrical symmetries, and vector spaces. There are also topological implications of how buildings envelop urban space. Formally planned cities are much less fractal than informal spontaneous cities.