## Pure Mathematics

The department has strong research groups in several areas of pure mathematics including analysis, topology, number theory, and algebra.

C∗-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics. Faculty with this specialty include the following:

#### Valentin Deaconu

My area of research is Operator Algebras, which is part of functional analysis. Functional analysis is the study of spaces of functions and other Banach spaces, and is related to differential equations, linear algebra, topology, and abstract algebra. More precisely, I study groupoid C*-algebras and K-theory. Groupoids are similar to groups, except that they have many units, and one cannot compose just any two elements. Additional structure is necessary, like a topology and a family of measures, in order to define a groupoid C*-algebra, which sometimes looks like a set of (infinite) matrices with complex entries. K-theory is a generalized cohomology theory, which is used in algebraic topology in order to distinguish surfaces and other topological spaces. The methods in my research are also inspired from dynamical systems, and the applications are in quantum statistical mechanics.

#### Alex Kumjian

My field of research is a branch of analysis called Operator Algebras. It is an intriguing mixture of analysis and infinite-dimensional linear algebra. It is a relatively new field that has its origins in the mathematical formalism of quantum mechanics. I am particularly interested in operator algebras which arise from dynamical systems.

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

#### Chris Herald

My current research focus is low-dimensional manifolds and knot theory, especially invariants of 3-manifolds and knots that are constructed using gauge theory and/or character varieties, including Floer homology and the Casson invariant, and their connections to knot polynomials, signature, and Khovanov homology.

#### Stanislav Jabuka

My research interests are in low-dimensional topology and geometry and range from classical knot theory to the topology of 3-manifolds and smooth 4-manifolds. The tools I use in my research are various gauge theories including Donaldson and Seiberg-Witten theory but most prominently discovered Heegaard Floer theory. I am interested in question pertaining to knot concordance, specifically torsion in the smooth knot concordance group. With regards to smooth 4-manifolds, I am interested in better understanding the Heegaard Floer invariants of Lefschetz fibrations.

#### Tynan Kelly

My research interests are in low-dimensional topology and knot theory. My work uses both algebraic and geometric techniques to study knots and their invariants, ranging from Kirby calculus and handlebodies to classical knot invariants (such as the Alexander and Jones polynomials and Casson-Gordon invariants) to more modern techniques such as the Heegaard-Floer and Khovanov homology theories. I am particularly interested in knot concordance and the role that knots play in the topology of 3- and 4-manifolds.

#### Ed Keppelmann

I study the fixed point theory of continuous maps on compact spaces, such as a torus and generalizations called Nilmanifolds and Solvmanifolds. Examples of these include the famous Klein Bottle or the collection of n by n upper triangular matrices with 1s on the diagonal. Certain properties of the fixed points of a map on one of these spaces are homotopy invariant, i.e., they don't change when the map is deformed. These properties are studied using techniques from group theory, combinatorics, and lots and lots of Linear Algebra.

#### Jing-Jing Huang

My primary research interests lie in the interaction of analytic number theory, diophantine geometry and harmonic analysis. That is to say, I am interested in using analytic methods (complex analysis, fourier analysis, etc) to solve number theoretic problems (finding integral/rational solutions to diophantine equations, the distribution of prime numbers, etc). My current research project is to study the distribution of rational points near a curved manifold, which (if solved in a satisfactory manner) will have major applications to problems in metric diophantine approximation.

#### Chris Rogers

My research interests are in homotopy theory, homological algebra, and higher categorical algebra. Most of the problems that I work on are quite geometric in nature and have their origins in certain areas of mathematical physics (e.g., formal deformation quantization and topological field theory).

#### Jonathan Beardsley

My research interests are in category theory, algebraic topology, and homotopy theory. I am specifically interested in using homotopical or "derived" algebra, in the form of operads, spectra, and infinity categories, to understand and classify structures that naturally arise in geometry and topology. Some geometric structures that I have specifically studied in previous work or am currently interested in include: cohomological invariants of topological spaces, the stable homotopy groups of spheres, the relationships between cobordism rings as the structure group is varied, A-infinity categories, and singular knots and links.

#### Mehmet Gumus

My primary research areas are Matrix Theory and their Applications, and Matrix Computations. My main scientific interests lie in applying matrix-theoretic tools to matrix stability problems that arise in systems and control theory, biological sciences, and economics.

#### Pan-Shun Lau

My research interests lie in matrix analysis, operator theory and applications. I am particularly interested in numerical range and its generalizations such as joint numerical range and higher rank numerical range. A common theme in much of my work is the interplay between the geometric properties of the numerical range and the algebraic properties of operators and matrices. Recently, I am also interested in matrix and operator inequalities.

#### Tin-Yau Tam

My areas of specialization are Matrix Theory and their Applications, Multilinear Algebra, Numerical Ranges, and Lie Theory. I am particularly interested in matrix problems that have connection with Lie groups or Lie algebras. I am also interested in matrix problems, for example, matrix inequalities, that have geometric favor.

## Applied Mathematics

The applied mathematician conducts research in partnership with the fields of medicine, engineering, finance, industry and more with the goal of solving a specific problem in the real world. Pure mathematics often bleeds over into the application side of things—the fields are not siloed from one another.

#### Tom Quint

My primary research interest lies in game theory, specifically the theory of matching games. Matching games are n-player cooperative games, in which players care about the identity of players with whom they are "matched" -- examples include "the marriage game", labor markets, and the modeling of economic markets with indivisible goods. Other interests within game theory are the study of power in legislative bodies and the theory of money. Outside of game theory, I have collaborated on several research papers in graph theory.

#### Colin Grudzien

In physical applications, dynamical models and observational data play dual roles in uncertainty quantification, prediction and learning, each representing sources of incomplete and inaccurate information. In data rich problems, first-principle physical laws constrain the degrees of freedom of massive data sets, utilizing our prior insights to complex processes. Respectively, in data sparse problems, dynamical models fill spatial and temporal gaps in observational networks. However, many physical systems exhibit chaos and observations are thus required to update predictions where there is sensitivity to initial conditions and uncertainty in model parameters. Data assimilation broadly refers to the techniques used to combine the information from models and observations to produce an optimal estimate of a probability density or a test statistic. These techniques include methods from Bayesian inference, dynamical systems, numerical analysis and optimal control, among others. My research interests lie in this intersection, using dynamical and statistical tools to develop theory for, and study applications of, statistical learning algorithms in physical systems. My application interests include climate, geophysics and the electric grid.

#### Paul Hurtado

I use techniques from the fields of dynamical systems, probability and statistics to develop and analyze mathematical models of real world (often biological) systems. I use those models to address specific questions related to population ecology and evolution, epidemiology (infectious diseases) and immunology. I also use those models as a basis for studying the relationship between process and pattern in a more general context, and sometimes pursue interesting mathematical questions that arise in these applications. When working on specific applications, I use different mathematical tools as the need arises, however I primarily use methods from applied nonlinear dynamics and bifurcation theory, including computational methods.

#### Eric Olson

I study Navier-Stokes equations, dynamical systems, fractal dimensions and turbulence. Techniques used include functional analysis and large scale numerical simulation. I am currently working on the normal form of the Navier-Stokes equations, the Langrangian averaged Navier-Stokes alpha model of turbulence, numerically determining modes, the Bouligand dimension and the bioremediation of contaminated soil.

#### Mark Pinsky

My research centers on the modeling, simulation and control of complex nonlinear systems, multiscale computing, integration of asymptotic and numerical techniques, abstraction and reduction techniques for large nonlinear models, modeling and control of bifurcation and chaos phenomena, and robust, impulsive and observation control of nonlinear systems.

Specific current projects include a) Multiscale modeling and simulation of molecule systems, b) Modeling and simulation of flow of magnetized nanoparticles in blood vessels, c) fast algorithms for modeling of atmospheric chemistry phenomena, d) Image tracking and processing algorithms, e) Stability bounds for partially uncertain nonlinear systems, and f) Robust control of bifurcation phenomena.

#### Andrey Sarantsev

My research is in Stochastic Analysis, particularly stochastic differential equations, as well as long-term stability; and in Quantitative Finance and Actuarial Science, where I use Stochastic Analysis and Econometrics tools. I am also interested in other applications of Statistics and Probability, particularly Biology and Ecology.

#### Deena Schmidt

My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability and statistics, dynamical systems, and random graph theory to shed light on biological issues while generating new mathematical questions. In particular, I study stochastic processes on networks with applications in neuroscience and stochastic models in genetics.

Currently, I am working on optimal reduction techniques for complex ion channel gating models, which can be represented as a stochastic (Markov) process on a graph. I am also looking at the relative contributions of network structure and node dynamics in determining the collective dynamics of a network, thinking specifically about neuronal networks involved in sleep-wake regulation. Lastly, I am broadly interested in mathematical and statistical applications in population and evolutionary genetics.

#### Aleksey Telyakovskiy

My research interests include numerical analysis, approximate solution techniques, mathematical modeling, and mechanics of flows through porous media. Some specific applications in these topics are contaminant transport, modeling of underground flows in enhanced oil recovery and saltwater intrusion. In some cases analytical or approximate analytical solutions can be constructed as a result important qualitative information on nature of solutions can be obtained, but in general equations are solved numerically. There are many numerical methods that can be used for analysis, such as, finite differences, mixed finite elements and Eulerian-Lagrangian methods. In modeling of flows through porous media multiple physical phenomena occur that complicate the numerical solution. Here are some of the difficulties: sharp fronts, which traditional upwind numerical schemes do not resolve due to the excessive numerical diffusion; a grid orientation effect; equations may include chemical reactions, consequently nonlinear thermodynamic constraints must be solved taking significant computational time. All these make these problems interesting and challenging.

#### Pavel Solin

My research group is developing novel computational methods for challenging multiphysics problems in various areas of engineering and science, including nuclear engineering, civil engineering, electrical engineering, mechanical engineering, molecular design (quantum chemistry), climate modeling, and others. We collaborate with numerous researchers from national labs and universities both in the U.S. and in Europe. All our results are freely available online -- check out our flagship open source projects Hermes and FEMHub. Students working in my group have various backgrounds ranging from theoretical physics, mathematics, computer science, to various engineering areas. This is absolutely necessary for successful completion of challenging interdisciplinary projects. Working together with people from other fields and national labs is an excellent experience for our students. We are looking forward to working with outstanding students -- the best way to join our team is by getting active in the projects mentioned above.

## Statistics and probability

Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.

#### Mihye Ahn

My research focuses on developing novel methodology to solve statistical problems raised from neuroimaging data, including fMRI, sMRI, DTI, and EEG. Generally, the functional neuroimaging data are of spatially and temporally high-dimensions. Analyzing these data includes various statistical topics: time series analysis, dimension reduction, classification, variable selection, longitudinal data analysis, covariance estimation, etc. I am also interested in variable selection methods for repeatedly measured data. I have collaborated with many scientists in various fields, including veterinary science, psychiatry, radiology, neurology, immunology, and biomedical engineering.

#### Yinghan Chen

My research interests include Monte Carlo methods, Statistical computing, Bayesian analysis, Latent class models, Item response theory, and Longitudinal analysis. Currently, I'm working on developing sampling algorithms to conduct statistical inference in educational assessments and networks.

#### Paul Hurtado

I use techniques from the fields of dynamical systems, stochastic processes, probability and statistics to develop and analyze mathematical models of biological systems. I use those models to address questions that arise in population ecology, evolution, epidemiology (infectious diseases) and immunology. Recently, I've begun working with methods for fitting nonlinear dynamic models to time series data. Using these models as statistical models presents a number of challenges, as parameter estimators for these models are not guaranteed to be as statistically well-behaved as, for example, estimators for classical linear models. In addition to parameter estimation for dynamic models, I also use approximation methods that exploit the deeper connections between deterministic models and their stochastic counterparts, as these two modeling frameworks can both be useful in applications.

#### Tomasz Kozubowski

Professor Kozubowski works in the general area of stochastic modeling of natural phenomena in variety of fields, including climate research, geosciences, finance, and economics. His research interests include distribution theory, Laplace distribution and its generalizations, limit theory for random sums, heavy tailed distributions, statistics of extremes, mathematical statistics, financial and insurance mathematics, stochastic models for hydro-climatic phenomena, and fractal scaling processes. He has co-authored over 100 research publications, including two monographs.

#### Anna Panorska

My research interests include probability, statistics, stochastic modeling and interdisciplinary work. In particular, I study the limit theory for random and deterministic sums of random quantities and estimation for heavy tailed distributions. Stochastic modeling and interdisciplinary work cover finance and insurance, hydrology and water resources, atmospheric science and climate, environmental science and biostatistics. Current research projects include statistical estimation for heavy tailed hydrology data, climate and hydrological extremes in the US, and clean water issues in Nevada and California.

#### Andrey Sarantsev

My research is in Stochastic Analysis, particularly stochastic differential equations, as well as long-term stability; and in Quantitative Finance and Actuarial Science, where I use Stochastic Analysis and Econometrics tools. I am also interested in other applications of Statistics and Probability, particularly Biology and Ecology.

#### Deena Schmidt

My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability and statistics, dynamical systems, and random graph theory to shed light on biological issues while generating new mathematical questions. In particular, I study stochastic processes on networks with applications in neuroscience and stochastic models in genetics.

Currently, I am working on optimal reduction techniques for complex ion channel gating models, which can be represented as a stochastic (Markov) process on a graph. I am also looking at the relative contributions of network structure and node dynamics in determining the collective dynamics of a network, thinking specifically about neuronal networks involved in sleep-wake regulation. Lastly, I am broadly interested in mathematical and statistical applications in population and evolutionary genetics.

#### Ilya Zaliapin

My research focuses on theoretical and applied statistical analysis of complex (non-linear) dynamical systems, with emphasis on spatio-temporal pattern formation and development of extreme events. Specifically, I work on multiscale methods of time series analysis, heavy-tailed random processes, and spatial statistics. This choice is predicated by the essential common properties of the observed complex systems: they tend to evolve in multiple spatio-temporal scales; and have observables that exhibit absence of characteristic size, long-range correlations in space-time, and not-negligible probability of assuming extremely large values. The underlying methods of analysis include those of hierarchical aggregation and its inverse - branching processes.

Examples of the observed systems relevant to my research include the Earth's lithosphere which generates destructive earthquakes, its atmosphere that produces El-Ninos, stock-markets subject to financial crashes, etc. My current applications and ongoing collaborations are in Solid Earth geophysics (seismology, geodynamics), climate dynamics, computational finance, biology, and hydrology.

## Undergraduate research

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