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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.
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.
Ohio State University | Math 1151 Calculus I, spring 2017 |
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Math 1148 College Algebra, autumn 2016 | |
Math 1075 Precollege Mathematics II, spring 2015 | |
Math 1151 Calculus I, spring 2015 | |
Math 1135 Number and Operations for Teachers, autumn 2014 | |
Math 1075 Precollege Mathematics II, Ohio State University, spring 2014 | |
Math 2177 Mathematical Topics in Engineering, autumn 2013 | |
Math 4512 Partial Differential Equations for Science and Engineering, autumn 2013 | |
Purdue University | |
Math 36600 Ordinary Differential Equations, spring 2013 | |
Math 34100 Foundations of Analysis, fall 2012 | |
Math 26100M Multivariate Calculus, fall 2012 | |
University of Western Ontario | |
Math 1225B Methods of Calculus, spring 2012 | |
Math 0110A Introductory Calculus, fall 2011 | |
University of New Orleans | |
Math 2108 Calculus II, single variable, summer 2008 | |
Math 2108 Calculus II, single variable, spring 2008 | |
Math 2221 Elementary Differential Equations, spring 2008 | |
Math 6490 Topics in Analysis-Introduction to Hilbert Spaces, fall 2007 | |
Math 2107 Calculus I, single variable, fall 2007 | |
Math 2221 Elementary Differential Equations, summer 2007 | |
Math 3221 Methods in Differential Equations, spring 2007 | |
Math 2108 Calculus II, single variable, spring 2007 | |
Math 4221 Intermediate Ordinary Differential Equations, fall 2006 | |
Math 2108 Calculus II, single variable, fall 2006 | |
Math 2107 Calculus I, single variable, summer 2006 | |
Math 2108 Calculus II, single variable, summer 2006 | |
Math 2107 Calculus I, single variable, spring 2006 | |
Math 1125 Precalculus Algebra, spring 2006 | |
Math 2107 Calculus I, single variable, fall 2005 | |
Math 2221 Elementary Differential Equations, fall 2005 | |
University of Notre Dame | |
Math 20550 Calculus III for Science and Engineering, Spring 2003 | |
Math 10560 Calculus II for Science and Engineering, Fall 2003 | |
Math 10250 Elementary Calculus in Basic Science, Spring 2002 |