Applied Analysis

The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

Applied analysis is an important area of research in the Department of Mathematics.

Analysis of Partial Differential Equations Arising in Fluid Dynamics, Hyperbolic Conservation Laws, Etc.

Wang's research area is analysis of partial differential equations arising in fluid dynamics, hyperbolic conservation laws, elastodynamics, geometry, kinetic theory, biology, etc. He works on nonlinear PDE problems including transonic flows, free boundary problems, vortex sheets, turbulence, isometric embeddings,  boundary layer, regularity, and so on.

Free Boundary Problems in Mathematical Finance

Chadam's recent research efforts have been focused on the study of free boundary problems that arise in mathematical finance. With his colleague, Xinfu Chen, their students, and foreign collaborators, he has studied early exercise boundaries for American style options analytically and numerically. In addition to giving precise estimates for the location of these boundaries, the work provided the first rigorous proof of the existence and uniqueness of the solution to the mathematical problem for the American put in the nonlinear integral equation formulation as well as convexity of its early exercise boundary. These methods have been carried over in a unified manner to treat a wide range of similar problems, including inverse first crossing problems in credit default and optimal strategies for prepayment of mortgages. Present research in this area is directed toward the study of default contagion in the context of higher dimensional value-of-firm models and multiple boundaries in callable convertible bonds.

Chadam and his students also are interested in a variety of other problems, such as pricing and hedging equity-linked securities and calibrating jump-diffusion and stochastic volatility models to electricity prices and using them to price futures contracts and swing options.

Ming Chen (Associate Professor, PhD)

Chen’s main area of research is PDEs with a particular focus on fluid mechanics and interfacial dynamics. Some specific topics include:

  1. The steady and dynamic aspects of the water wave problem, as well as various nonlinear dispersive equations arising from/related to the full water wave system.
  2. Stability of traveling waves in abstract Hamiltonian system with symmetry.
  3. Free boundary problem in compressible elastodynamics.
  4. Energy balance in incompressible and compressible Navier-Stokes equations. 

Nonlinear Analysis, Partial Differential Equations and Calculus of Variations.

Lewicka's research areas are nonlinear analysis, partial differential equations and calculus of variations. She has obtained results on the well-posedness and stability of systems of conservation laws and reaction-diffusion equations. Currently, her research focuses on the mathematical theory of elasticity with connections to Riemannian geometry and with an eye on the applications in morphogenesis of growth.

Nonlinear Differential Equations

Chen studies a wide variety of problems on such topics such as non-linear partial differential equations of parabolic and elliptic type, ordinary differential equations and dynamical systems, free boundary problems and interfacial dynamics, singular perturbation and asymptotic expansions, and mathematical finance.

Nonlinear waves in spatially discrete systems

Prof. Vainchtein's current research efforts focus on the dynamics of transition fronts, solitary waves, breathers and other nonlinear waves in spatially discrete systems. Her research program aims to advance the fundamental understanding of the energy transfer phenomena associated with the wave propagation. It also seeks to provide insights into the effects of nonlinearity, nonconvexity, dissipation, heterogeneity, and nonlocality on the existence and stability of different types of nonlinear waves and the extent to which some of these effects can be captured by dispersive continuum approximations. For more information, please visit Prof. Vainchtein's website.

Transient and Multiple Timescale Dynamics

Rubin works to develop and apply dynamical systems techniques to study transient dynamics and to study complicated dynamics in multiple timescale and hybrid dynamical systems (i.e., that are smooth except for discrete jumps or switching manifolds), including canards, mixed-mode oscillations and bursting.