### Our Research Interests

Faculty in the Department of Mathematics are deeply committed to excellence in teaching and research. Many specialize in both current and emerging areas of pure and applied mathematics. Graduate students as well as undergraduate students have the opportunity to work with our esteemed faculty on research projects.

We encourage you to contact faculty members whose areas of interest match yours even before you arrive.

##### Algebraic Geometry

Matthew Ballard (Ph.D., University of Washington, 2008), Algebraic Geometry. Research Interests include: Derived categories, mirror symmetry, birational geometry, invariant theory.

Alexander Duncan (Ph. D., University of British Columbia, 2011), Algebraic Geometry. Research interests include: birational geometry, Galois cohomology, linear algebraic groups, rational surfaces, and toric varieties.

Jesse Kass (Ph.D., Harvard University, 2009), Algebraic Geometry. Research Interests include: Singular curves, Jacobians.

Xian Wu (Ph.D., Harvard University, 1986), Algebraic Geometry, Differential Geometry, Complex Manifolds.

##### Applied Mathematics

Peter Binev (Ph.D., Sofia University, 1985), Scientific Computing, Approximation Theory, Numerical Analysis. Research interests include: nonlinear approximation, learning theory, high dimensional problems, numerical methods for PDEs, computer graphics, image and surface processing.

Daniel Dix (Ph.D., University of Chicago, 1988), Analysis. Research interests include: initial value problems for partial differential equations governing the evolution of nonlinear waves, asymptotic behavior of solutions, solutions with special symmetry, completely integrable equations, and solitons.

Lili Ju (Ph.D., Iowa State University, 2002), Computational Mathematics. Research interests include: Scientific computation and numerical analysis. Exact boundary controllability problems for the wave equation. Parallel algorithms and high-performance computing. Human brain imaging.

Xinfeng Liu (Ph.D., State University of New York at Stonybrook, 2006), Scientific computing, high performance computing, interficial phenomena, multiphase flows, computational biology, cellular dynamics.

Douglas Meade (Ph.D., Carnegie Mellon University, 1989), Applied Mathematics. Current research interests include numerical methods for wave propagation on unbounded domains, non-overlapping domain decomposition methods, and computer algebra systems.

Changhui Tan (Ph.D., University of Maryland, 2014), Applied Mathematics. Research interests include: nonlinear partial differential equations, fluid dynamics, hyperbolic conservation laws and complex biological models.

Paula Vasquez (Ph.D., University of Delaware, 2007), Applied Mathematics. Current research interests include Multiscale Modeling and Simulation of viscoelastic fluid flows, Computational and mathematical biology

Hong Wang (Ph.D., University of Wyoming, 1992), Numerical Analysis and Differential Equations. Research interests include: Numerical approximation to differential/integral equations, scientific computations.

Qi Wang (Ph.D., Ohio State University, 1991), Applied and Computational Mathematics, Computational Fluid Dynamics and Rheology of Complex Fluids, Continuum Mechanics and Kinetic Theory, Multiscale Modeling and computation of soft matter and complex fluids of anisotropic Microstructures, Multiscale modeling and computation of biofluids and biomaterials, Parallel and high performance Computing.

Xiaofeng Yang (Ph.D., Purdue University, 2007), Scientific computation, mathematical modeling of liquid crystalline polymers. Numerical analysis, spectral methods and scientific computing with applications in fluid mechanics.

Yi Sun (Ph.D., Princeton University, 2006) Applied and Computational Mathematics.

Research interests include: Multiscale modeling and simulation in solids, fluid
mechanics, chemistry and biology; Mathematical modeling and computation of biomaterials,
biofluids, cellular dynamics and traffic and pedestrian flow; Mathematical and computational
neuroscience.

Zhu Wang (Ph.D., Virginia Tech, 2012), Applied and Computational Mathematics. Research interests include: numerical analysis, scientific computing, reduced-order modeling, climate modeling, and inverse problems.

George Androulakis (Ph.D., University of Texas, Austin, 1996), Quantum mechanics and quantum information.

##### Approximation Theory

Wolfgang Dahmen (Dr. rer. nat., RWTH Aachen, 1976), Numerical Analysis and Approximation Theory, especially adaptive solution concepts in Learning Theory or Computational Harmonic Analysis, and in interdisciplinary applications.

Pencho Petrushev (Ph.D., Sofia University, 1977), Approximation Theory, Harmonic Analysis, Numerical Methods. Research interests include: nonlinear approximation by rational functions, splines, and wavelets, approximation by ridge functions and neural networks, image processing.

Vladimir Temlyakov (Ph.D., Steklov Institute, 1978), Approximations of functions in one variable and multivariable cases (approximations by polynomials, n-widths, optimal cubature formulas). Integral operators (estimates of singular numbers, approximation numbers, bilinear approximation of kernels of these operators).

##### Commutative Algebra

Andrew Kustin (Ph.D., University of Illinois, 1979), Commutative Algebra and Algebraic Geometry. Research interests include: the study of Cohen-Macaulay and Gorenstein algebras, finite free resolutions, linkage, deformation theory, and differential graded commutative algebras.

Matthew Miller (Ph.D., University of Illinois, 1979), Commutative Algebra and Mathematical Biology. Research interests involve problems in commutative algebra mostly using homological techniques, and the relationships between betti numbers and Hilbert functions. Recent interests are in mathematical biology, especially modeling of animal behavior.

Adela Vraciu (Ph.D., University of Michigan, 2000), Commutative Algebra and Algebraic Geometry. Research interests include: tight closure theory, linkage, and homological properties of rings and modules.

##### Differential and Integral Geometry

Ralph Howard (Ph.D., California Institute of Technology, 1982), Differential and Integral Geometry with excursions into Analysis. Research interests include: global Lorentzian geometry, geometric inequalities, stochastic geometry and analysis related to differential equations arising in geometry.

##### Discrete Mathematics

Joshua Cooper (Ph.D., University of California, San Diego, 2003), Combinatorics and Number Theory. Research interests include: extremality, regularity, and quasirandomness of graphs and permutations; combinatorial number theory; universal cycles; coding theory; combinatorial algorithms.

Eva Czabarka (Ph.D., University of South Carolina, 1998), Discrete Mathematics and its Applications. Research interests include: extremal set theory, graph theory, crossing numbers, network science, bioinformatics.

Lincoln Lu (Ph.D., University of California, San Diego, 2002), Discrete Mathematics. Research interests include: large information networks, combinatorial probabilistic methods, extremal graph theory, algorithms, computational geometry, computational biology, and Internet computing.

Laszlo Szekely (Ph.D., Eötvös University, 1983), Combinatorics and Graph Theory. Research Interests include Extremal combinatorics, discrete geometry, graphs drawn on Surfaces, Reconstruction of Phylogenetic trees from genetic sequences.

##### Functional Analysis

George Androulakis (Ph.D., University of Texas, Austin, 1996), Quantum mechanics and quantum information.

Stephen Dilworth (Ph.D., University of Cambridge, 1985), Functional Analysis. Research interests include: finite-dimensional and infinite-dimensional Banach space theory; classical Banach spaces; approximation in Banach spaces.

Maria Girardi (Ph.D., University of Illinois, 1990), Functional Analysis. Research interests include: functional analysis, esp. classical and geometrical Banach space theory.

Anton Schep (Ph.D., University of Leiden, 1977), Functional Analysis, Operator Theory. Research interests include: the study of linear integral operators on Banach function spaces, positive operators and C0-semigroups of positive operators on Banach lattices, spectral properties, and compactness properties of special classes of operators.

##### Logic, Set Theory, Algebra and Topology

George F. McNulty (Ph.D., University of California, Berkeley, 1972), Logic, Algebra, and Discrete Mathematics. The central themes of Dr. McNulty's research lie at the confluence of algebra , logic and computer science. They include finite axiomatizability of equational classes of algebras, structural properties of the lattices of equational theories, and algorithmic computability in algebraic, logical, and combinatorial settings.

Peter Nyikos (Ph.D., Carnegie Mellon University, 1971), Topology. Research interests include: point-set topology, especially covering and base properties of regular spaces, and the structure theory of locally compact spaces; the application of special axioms from set theory to constructing examples and establishing consistency and independence results; and applications of point-set topology, especially to Boolean algebras and functional analysis.

##### Number Theory

Matthew Boylan (Ph.D., University of Wisconsin, 2002), Number Theory. Research interests include: Number theory. In particular, elliptic modular forms and Maass forms and their applications to algebraic number theory, elliptic curves, L-functions, partitions, and other topics in number theory.

Michael Filaseta (Ph.D., University of Illinois, 1984), Number theory, including analytic, classical algebraic, combinatorial, computational, elementary, and transcedence topics. Research interests include lattice points close to (or on) a curve or surface, the distribution of special sequences of integers in short intervals, applications of Pade approximations to Number Theory, the irreducibility of polynomials over the rationals, and computations with sparse or lacunary polynomials.

Frank Thorne (Ph.D., University of Wisconsin, 2008), Number Theory; distribution of primes and broadly related questions.

Ognian Trifonov (Ph.D., Sofia University, 1989), Analytic Number Theory and Approximation Theory with particular interests in the use of finite differences to determine information about lattice points close to a curve or surface. Interests also include the application of these results to gap problems in Number Theory.

##### Emeritus Faculty

Colin Bennett (Ph.D., University of Newcastle upon Tyne, 1971), Analysis. Research interests include: harmonic analysis and the theory of interpolation of operators and concurrent computation.

Ronald DeVore (Ph.D., Ohio State University, 1967), Research interests include: Approximation Theory and Numerical Analysis.

Jerrold Griggs (Ph.D., Massachusetts Institute of Technology, 1977), Combinatorics. Research interests include extremal set theory, extremal graph theory, graph coloring, and applications of discrete math to biology, number theory, analysis of algorithms, and communications.

George Johnson (Ph.D., University ofTennessee, 1971), Research interests include: Nonlinear optimization and elementary mathematics education.

Peter W. Harley, III (Ph.D., University of Georgia, 1966), Topology. Research interests include: the study of generalized metric spaces, particularly symmetrizability and related concepts.

Richard Hudson (Ph.D., Duke University, 1971), Number Theory. Elementary number theory, analytic prime number theory, quadratic forms, class number formulae, forms of higher order, quadratic and higher power residues, comparative prime number theory, Gauss and Jacobi sums, and computer results in number theory

Thomas L. Markham (Ph.D., Auburn University, 1967), Linear Algebra. Research interests include: non-negativity and its generalizations, M-matrix theory, LU decompositions with applications, special classes of matrices and Schur complements.

Charles Nicol (Ph.D., University of Texas, 1954), Number Theory.

Konstantin Oskolkov (Ph.D., Steklov Institute, 1972), Fourier Series, Approximation, Oscillatory Sums and Integrals, Schrödinger type equations, Wavelets and Bases.

James W. Roberts (Ph.D., Rutgers University, 1970), Functional Analysis. Research interests include: compact convex sets; the study of F-spaces and other spaces of analytic functions; applications of functional analysis to problems in measure and integration.

H. E. Scheiblich (Ph.D., University of Texas at Austin, 1966), Algebra and Algebraic Semigroups. Research interests are in the area of algebraic semigroups which are regular in the sense of Von Neumann.

Robert Sharpley (Ph.D., University of Texas, Austin, 1972), Classical Analysis and its applications. Research interests include Fourier analysis, approximation theory, multiresolution analysis, signal and image processing, numerical analysis, visualization, and autonomous navigation. Professor Sharpley has served as PI on research projects supported by NSF, DOD (ARO, AFOSR, ONR), NASA, and DOE as well as several large equipment and industrial technology transfer grants.

Robert M. Stephenson, Jr. (Ph.D., Tulane University ,1967), Topology. Research interests include: symmetrizable spaces, pseudocompact spaces, and minimal topologies.

Manfred Stoll (Ph.D., Pennsylvania State University, 1971), Function Theory, Potential Theory, Several Complex Variables. Research interests include: the study of holomorphic, harmonic, and plurisubharmonic functions of one and several complex variables; Hp spaces, Bergmann spaces, and other spaces of harmonic and holomorphic functions of one and several complex variables; boundary behavior of Green potentials in domains in Rn and Cn.

David Sumner (Ph.D., University of Massachusetts, 1971), Graph Theory

The department is also home to the internationally-recognized Interdisciplinary Mathematics
Institute (IMI) for advanced research with potential applications. The IMI carries
on cooperative projects with leading universities, including Princeton, Stanford,
and Rice, and also interacts with various University of South Carolina programs in
the fields of Nanoscience, Engineering and Science.