Bootstrap percolation models have been applied to many problems involving interacting particle systems and cellular automaton growth processes. Balogh has studied the behavior of this process on lattice-like graphs. Recently he determined the critical probability (giving a sharp transition) of bootstrap percolation in dimension on the d-dimensional grid, solving a long-standing open problem, and he continues to investigate the process on other underlying structures.
Research interests include applied topology, singularity theory, stochastic geometry, random processes and their applications in combinatorics, robotics, networking and social choice.
Using a mix of differential geometry and algebraic geometry, Bradlow studies questions whose origins go back to electromagnetism and other gauge theories in physics, but which now concern moduli spaces of vector bundles and related objects. One major current interest is the relation between Higgs bundles and geometric structures on surfaces.
Differential equations and stochastic processes, with concrete applications in science and engineering. Particularly focused on systems defined on networks. Limit theorems and critical phenomena. Applications include synchronization of neuronal and other networks, designing fast multiscale algorithms for particle methods in the geosciences, and the consequences of symmetry on networked dynamical systems.
Works in topology and geometry with applications to computer science. In particular, studies algorithms for finding least-complexity objects, for instance least area surfaces (like those of a soap bubble), with prescribed topology (e.g. bound a loop in 3-space or missing voids in a material).
Works on geometrical visualization problems in wholly immersive virtual environments (CAVE, Cube). His particular interest are problems which are amenable only to real-time interactive computer animation, such as regular homotopies of surfaces (sphere eversions), 3D non-Euclidean geometries (Thurston) and quasicrystals (Penrose), and 4D phenomena and effects (quaternions).
Works in actuarial science and financial mathematics, especially the modeling of stochastic economic, financial, and insurance processes. Specific areas of research application involve dynamic financial analysis, enterprise risk management, and financial pricing techniques. Interest in multidisciplinary applications such as behavioral economics and complexity science.
Hur studies nonlinear partial differential equations that arise in physical contexts. Particular interests are in surface water waves and related interfacial fluids flows.
Katz's research is at the interface of geometry and physics, particularly algebraic geometry and string theory. Applications include the use of algebraic geometry to build topological string theories, topological quantum field theories, and "geometrically engineered" string models for realistic grand unified theories.
Uses representation-theoretical and combinatorial methods applied to physical systems such as: Exactly solvable models in statistical mechanics; topological states in condensed matter systems such as the quantum Hall effect; integrable quantum field theories and related discrete integrable systems. Key mathematical concepts include representations of quantum groups and affine algebras and cluster algebras.
Works in statistical mechanics, nonlinear partial differential equations, and dynamical systems, using a variety of methods from probability, functional analysis, harmonic analysis, and number theory. Applications include condensed matter physics, biophysics of polymers, and genotype pedigrees.
Works in differential equations and mathematical physics, especially on the effects of nonlinear couplings. Applications include: bifurcations and stability of nonlinear waves, and long time behavior of nonlinear dispersive systems.
Works in differential equations and mathematical physics, especially on spectral problems with a geometric flavor. Applications include: stability properties of steady states, and estimates of energy levels for classical and quantum systems.
Lerman uses category theory to understand continuous time, discrete time and hybrid dynamics on networks, and would like to use differential geometric stacks as a basis for a fresh approach to equivariant bifurcation theory. He is also interested in the connection between stochastic Petri nets and toric dynamical systems, and by a possible connection between continuous time Markov chains, weighted directed graphs and schemes over Z.
Julian Palmore's research is on dynamical systems, including discrete dynamics and computation, and spaceflight. He teaches a course in spaceflight for the Campus Honors Program.
Rapti's research interests lie in the areas of differential equations and dynamical systems with applications to biology and physics. Specifically, she studies the stability of solutions to nonlinear partial and ordinary differential equations that model systems such as DNA or interacting populations.
Schenck's research is in algebraic geometry and commutative algebra, especially problems with applied and computational aspects. Current applied work includes algebraic coding theory, rational surface modelling and implicitization, and approximation theory (splines).
Sowers currently works in modelling complex financial systems and interactions. This involves stochastic analysis of various events and interactions.
Nikolaos Tzirakis My research area is Harmonic Analysis and Partial Differential Equations (PDE). More precisely I am interested in the short/long time behavior of the Cauchy problem for dispersive nonlinear PDE . The problems that I am investigating include, local and global-in-time existence and uniqueness of solutions, existence of scattering states and soliton solutions, smoothing effects of the associated solution maps.
Research interests in Dynamical Systems and their applications in Physics and Engineering. In particular, he studies finite and infinite dimensional Hamiltonian systems, their dynamics and stability. More specifically, mathematical billiards and related systems with impacts and search problems.