The Department of Mathematics is active in a variety of research areas across multiple mathematical concentrations. Visit each area for a more detailed description and contact a research faculty member listed below for more information.

- Mathematical Biology
- Theory and Applications of Partial Differential Equations
- Analysis, Operator Theory, and Operator Algebras
- Numerical Analysis and Scientific Computing
- Mathematics Education
- Model Theory and Foundations
- Probability, Optimization, Computing, and Geometry
- Mathematical Physics
- Principles of Urbanism
- Systems and Control

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.

**Participating Faculty**

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.

**Participating Faculty**

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.

**Participating Faculty**

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.

**Participating Faculty**

Foundational mathematics is concerned with mathematical results about the nature of mathematics (hence, it is referred to as ‘metamathematics’). For example, due to the complexity and richness of models of the theory of arithmetic, there are elementary facts about numbers that cannot be proved or disproved. Model theory is the area of foundations concerned with the complexity of all mathematical theories and their models. Theories possess varying degrees of complexity. The most elementary example of a simple ("tame") theory is the theory of vector spaces: all of its models can be classified in terms of one simple invariant: dimension. In general, theories with an abstract notion of dimension (e.g., algebraically closed fields) are tame. More complex ("wild") examples are given by the theory of Banach spaces and the theory of C*-algebras. The paradigm of a non-tame theory is the Queen of Mathematics, number theory.

In recent years, model theory has proved to be powerful in solving difficult problems in a variety of areas of mathematics, including number theory, combinatorics, random graph theory, functional analysis, mathematical physics, and mathematical economics. The model theory team at UTSA combines methods from both discrete and continuous logic to address transdisciplinary problems.

**Participating Faculty**

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.

**Participating Faculty**

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.

**Participating Faculty**

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.

**Participating Faculty**

Dynamical systems and the control of these systems present challenges of both theory and computation. Dynamical systems are used as models in science and technology to provide a realistic and adequate mathematical description for many real-world phenomena. Throughout the sciences they are used to model diverse topics such as space travel, ecological systems, population dynamics, economics, weather, autonomous vehicles, and neural networks. Differential equations, difference equations, time-scale systems, and probability laws are used when modeling, which leads to the study of continuous and discrete types of dynamical systems. Predictive analysis of these systems can be difficult due to features like nonlinearity and parameter sensitivity. Hence, such systems are related to bifurcation and chaos. Control theory studies the ability to guide dynamical systems given available inputs into the system. This leads to additional challenges such as stability, estimation, controllability, and optimization. The research on the behavior of dynamical systems and their control theory is an intersection (cross-section) between several research topics and disciplines including engineering and physics.

**Participating Faculty**

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.

**Participating Faculty**