Research interests and work in the year 2017:
Worked on the semiclassical focusing Nonlinear Schroedinger equation (NSF supported work) towards resolving the long-standing problem of the mechanism of the second break, making substantial progress. I believe that the problem will be solved in the coming year.
MATHEMATICAL BIOLOGY: DORSAL CLOSURE
Participant of the Dorsal Closure project of the Kiehart lab. In collaboration with the Kiehart group:
1) Review of of dorsal closure, the biological part published, the mathematical modeling part on preparation,
2) Advanced mathematical model of dorsal closure in progress.
ACOUSTIC AND ELECTROMAGNETIC SCATTERING: THEORETICAL AND COMPUTATIONAL
Collaborative work on the scattering from scatterers with spatially periodic geometry.
1) Paper published. For the first time, the paper overcomes theoretical and computational difficulties at certain frequencies associated with the well-known Wood anomalies (NSF supported work),
2) Paper, that utilizes the approach of the above work, to be submitted in the current or next week (NSF supported work).
This area of physics deals with the prediction of magnetic behaviors at sub-micrometer length scales. Although little mathematical work has been done in this field, its physical and technological importance is obvious. From wikipedia: “Apart from conventional magnetic domains and domain-walls, the theory also treats the statics and dynamics of topological line and point configurations, e.g. magnetic vortex
and antivortex states or even 3d-Bloch points,
where, for example, the magnetization leads radially into all directions from the origin, or into topologically equivalent configurations. Thus in space, and also in time, nano- (and even pico-)scales are used. The corresponding topological quantum numbers
are thought to be used as information carriers, to apply the most recent, and already studied, propositions in information technology
I entered this area, working collaboratively in June 2017. Two papers are in preparation. By analyzing the related Landau-Lifschits-Gilbert equation, both works prove for the first time the existence of traveling domain walls under different circumstances in each case. In one of the papers, the proof utilizes a topological argument.