My current interests focus on discrete groups and their role in algebraic geometry and other fields. Especially interesting are Coxeter groups and their related braid groups. Although group theory is officially "algebra", the part of it that I like is more a branch of geometry.

Todd Arbogast's areas of expertise include the development and analysis of numerical algorithms for the approximation of partial differential systems, high performance and parallel scientific computation, and mathematical modeling. He specializes in applications to fluid flow and transport in porous media, such as the Earth's subsurface. His research accomplishments include the development of: Eulerian-Lagrangian schemes for advective flow; cell-centered finite difference, mixed finite element, and mortar techniques for geometrically irregular problems; homogenization, modeling, and simulation of flow through multiscale fractured and vuggy geologic media; and variational multiscale methods for heterogeneous media and nonlinear problems.

Fourier Analysis - Sharp Inequalities and Geometric Manifolds Geometric inequalities provide insight into the structure of manifolds. The principal objective of my research is to develop a deeper understanding of the way that sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. Asymptotic arguments identify geometric invariants that characterize large-scale structure. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and reflect the dilation character of the manifold. Sharp estimates constitute a critical tool to determine existence and regularity for solutions to pde's, to demonstrate that operators and functionals are well-defined, to explain the fundamental structure of spaces and their varied geometric realizations and to suggest new directions for the development of analysis on a geometric manifold. Model problems and exact calculations in differential geometry and mathematical physics are a source of insight and stimulus, particularly conformal deformation, fluid dynamics, quantum physics, statistical mechanics, stellar dynamics, string theory and turbulence.

I study interactions between algebraic geometry, representation theory and mathematical physics. In particular I'm interested in applications of techniques from homotopy theory and topological quantum field theory to geometric representation theory and the geometric Langlands program.

I work in stable homotopy theory, focusing primarily on research problems arising from constructions made possible by the modern theory of structured ring spectra (and the homotopy theory of module categories over such "geometric" rings).

Spectral and dynamical problems in quantum field theory, random Schrodinger equations, renormalization group methods, mean field and macroscopic scaling limits of quantum dynamics, Hamiltonian dynamics.

My research at this point is focused on analyzing local and global points of elliptic curves as the field of definition changes. One of the tools that I use is Iwasawa Theory.

Currently doing administration full time.

My research focuses on different areas, related to both analysis and geometry. One of my main field of research is optimal transport: given a mass distribution, find the most effective way of moving this mass from one place to another, minimizing the transportation cost. This problem has found in recent years important applications to many other different areas, such as Monge-Ampère type equations, evolution partial differential equations, isoperimetric and functional inequalities, Riemannian geometry. I have been working in all of these areas. I also work in variational problems in the calculus of variations, as for instance studying geometric and regularity properties of minimizers of functionals modeling liquid drops and crystals. More recently, I also started to work on regularity theory for elliptic partial differential equations (both of local and non-local type) and free boundary problems.

I work on problems in geometry and topology, often with an eye towards quantum field theory and string theory. My collaboration with Michael Hopkins and Constantin Teleman began with the discovery that the Verlinde ring in conformal field theory may be expressed in terms of twisted K-theory. We are now investigating various extensions of these ideas and also aspects of topological quantum field theory. I am also currently collaborating with Jacques Distler and Gregory Moore on problems arising more directly from physics.

Applied and Computational Analysis, Mathematical and Statistical Physics, non-linear Kinetic and Partial Differential Equations.

Topology of 4-manifolds, symplectic and contact topology (especially in dimensions 4 and 3), topological construction of Stein surfaces and domains of holomorphy.

My general interests are in computational and applied mathematics with an emphasis on classical continuum mechanics. My current efforts are focused on modeling the mechanical properties of DNA at various length scales. Keywords: modeling, numerical analysis, differential equations, integral equations, geometry of curves and surfaces.

I develop and analyze numerical methods for multiscale problems in math and physics. These methods are designed to correctly capture asymptotic limits at the discrete level. I am currently working on problems related to multi-phase fluid flow and the Boltzmann transport equation.

I am trying to understand how geometry relates to Stiefel-Whitney numbers. That is to say, how does geometry relate to the obstructions to a closed manifold bounding a compact manifold?

I work in commutative algebra, principally dealing with questions concerning the homological properties of local rings. My specialty is rings of mixed characteristic - the ring itself has characteristic zero but its residue field has characteristic p.

I study homotopy theory and homotopical algebra in the sense of Quillen. Specifically, I'm interested in algebraic structures, especially monoidal structures, on (â^z, n)-categories and the algebraic structure of the category of (â^z, n)-categories itself. I am also interested in stable homotopy theory and the application of stable homotopy-theoretic methods to questions in algebra.

I am an algebraic geometer with particular interests in moduli spaces and birational geometry

I work on problems in geometric analysis. The broad motivations for my research program are to find and classify optimal geometries by using evolution equations like Ricci flow, mean curvature flow, and cross curvature flow. I use techniques from analysis to understand the behavior, singularity formation, and stability of solutions to the nonlinear partial differential equations governing these flows. These equations have applications to the geometry and topology of manifolds, as well as to problems originating in materials science and physics.

renormalization in dynamical systems, computer-assisted proofs in dynamical systems and PDEs, phase transitions in statistical mechanics.

A major focus in my research is a study of hyperbolic 3-manifolds and Kleinian groups and attempting to understand their geometries. I am also interested in various related areas in the study of topology of 3-manifolds and geometry of surfaces and Teichmuller theory.

I work on geometric problems which are influenced by high energy physics, more specifically supersymmetric field theory and string theory. For the last several years my main focus has been on using N=2 supersymmetry to understand hyperkahler geometry. This has led to surprising connections between wall-crossing phenomena for Donaldson-Thomas invariants (and their generalizations) and new constructions of hyperkahler metrics.

Rational points on twists of modular curves through Atkin-Lehner involutions over local and global fields, Galois covers of curves over characteristc p

My research interests are focused on PDEs that arise in fluid dynamics and on nonlinear dispersive equations. More precisely, I have been investigating problems related to well-posedness and regularity of fluid equations and nonlinear dispersive equations. Also, recently I extended research interests in a new direction, to include topics on derivation of the nonlinear dispersive equations as the mean field limits of interacting Boson gases.

My interests are in symplectic topology, low-dimensional topology (with an emphasis on 4-manifolds) and interactions between these areas. My preferred techniques are those of pseudo-holomorphic curves, gauge theory and Floer cohomology theories.

I use geometric features of dense packings, of spheres and other bodies, in the modeling of equilibrium and nonequilibrium materials. Probability and ergodic theory often play a role.

My interests are in hyperbolic manifolds, discrete groups and low-dimensional topology with a particular interest in connections with number theory.

My research focuses on 1) mathematical and numerical analysis of inverse problems of partial differential equations, with applications in imaging sciences, and 2) numerical simulation of the propagation of waves and particles in random media.

I study the topology and dynamics of tiling spaces, mostly (but not exclusively) tilings of Euclidean space. These are multi-dimensional generalizations of symbolic dynamics, with geometry playing a substantial role. I am also interested in the interplay of differential geometry, statistical mechanics, and quantum mechanics.

My research area is Mathematical Finance and Stochastic Control. I am mainly interested in pricing/hedging and optimal investment in incomplete markets and markets with frictions and the stochastic control methods associated to these models.

I work on the design, implementation, and analysis of large-scale improvement initiatives in mathematics and science education both in higher education and in urban school districts. I am actively involved in the formulation and analysis of public policies that affect education--especially efforts to address systemic inequality in access to high-quality education. programming.

I am interested in the partial differential equations which arise in mathematical physics. The most famous of these are Einstein equations of relativity and Maxwell's equation for electrodynamics. My best known work has been on the Yang-Mills equations. However, there are other integrable systems which are connected to quantum topology, as well as newer version of gauge theoretic equations. My most recent students have worked in analytical aspects of these gauge theoretic equations.

I am working on Partial Differential equations. I am particularly interested in the PDE's used in Fluid mechanics.

Arithmetic of function fields. Diophantine geometry over function fields. Geometry of algebraic curves. Algebraic varieties over finite fields. Modular forms, elliptic curves and abelian varieties. Finite fields and applications to coding theory and cryptography.

My main research area is mathematical physics, and I am particularly interested in applying renormalization group ideas to study relevant problems arising in physics and probability.

My interests include probability theory and stochastic analysis and their applications in mathematical finance.