The nonlinear wave equations, that arise in specific problems in areas like general relativity, elasticity or field theory, are quite well understood from physical point of view. However, to claim a rigorous mathematical picture, there is still work to be done, and this is why solving problems connected to these equations could lead to new ideas and new interpretations in their area of origin. This project plans to investigate issues like local well-posedness, lifespan of low regularity solutions, and blowup phenomena. Also, the investigator will focus on similar problems for stochastic nonlinear wave equations. It is likely that recent advances for deterministic hyperbolic equations (e.g. vectorfield method, wave packet parametrices) could improve the current state of the art for these problems, which have applications in mathematical finance.
Problem solving is widely considered as one of the focal points in the instruction process of mathematics. The investigator will work on an educational plan integrated into the above research program, which has both an undergraduate component and a K-12 outreach one. The former includes the continuous development of the current Problem Solving Seminar, together with the creation of new courses both at undergraduate and graduate level, which will address the subject of partial differential equations. The K-12 component focuses on the creation of a Rochester Area Math Circle, aimed at middle-school and high-school students with an interest in solving challenging math problems. This activity will feature weekly lectures, problem sessions and math olympiads, in which a wide range of math educators and students will be involved.
