My research is at the crossroads of analysis, differential geometry, dynamical systems and mathematical physics. I develop and exploit analytic and geometric tools to study nonlinear partial differential equations (PDE) from mathematical physics in connection with differential geometry.
Often the most interesting nonlinear PDE arise naturally in mathematical physics as approximate models for a diverse range of physical phenomena. Examples include PDE from gas dynamics, hydrodynamics, elasticity, gravitation, relativity, ecology and thermodynamics. A particularly interesting example from hydrodynamics is the Navier-Stokes equation where the nonlinearity comes from inertial effects. In general, when attacking a particular nonlinear PDE there are two fundamental questions to focus on: (i) how to establish existence and uniqueness of solutions for the associated boundary value problem, and (ii) how to ''solve'' the equation in an appropriate sense.
One of the challenging aspects of nonlinear PDE is that very few general techniques have been developed that work for all such PDE. Indeed it is almost a rule that each particular equation requires study as a separate problem in order to successfully address questions (i) and (ii) above. On the other hand nonlinear PDE that arise from integrable systems can often be completely solved in an appropriate sense. Perhaps the best known example is the Korteweg-de Vries equation. However there is no general criteria known for verifying integrability of infinite dimensional systems modeled by nonlinear PDE, and often special tools must be developed for each particular system.
Most of my research has been focused on attacking the following two fundamental questions for nonlinear PDE of one space dimension: (i) establishing existence, uniqueness and continuous dependence on initial data for the associated Cauchy problem, and (ii) developing tools for ''integrating'' the equation.
Ph.D. Mathematics, August 2004, University of Notre Dame, Notre Dame, Indiana.
M.S., Electrical & Electronics Engineering, May 1999, Bo˘gazi¸ci University, Istanbul, Turkey.
B.S. Electrical & Electronics Engineering, May 1999, Bo˘gazi¸ci University, Istanbul, Turkey.
Area of Expertise
Partial differential equations