Mathematical Sciences: Classical Analysis, Number Theory andSupercomputers

Work on problems at the interface of complex analysis and analytic number theory will be the central theme of this mathematical research. To obtain additional insight, supercomputer experimentation will also play a prominent role. From the number-theoretical point of view, the objective of the work is in understanding properties of roots of various kinds of transcendental functions: zeta functions. A particular instance is the Epstein zeta function, a complex function depending on a class of parameters. As the parameters vary, the roots traverse certain paths. This work seeks to understand this motion from the standpoint of a statistical/dynamical analysis. Similar (statistical) analyses have been found useful in studying many other types of zeta functions - though here the emphasis will shift to studying the roots on the critical line where they are believed to reside. Work on complex analysis centers on Schwarzian differential equations on compact Riemann surfaces. Studies of the monodromy group and the conformal mapping properties of the solution as the forcing term of the differential equation - the quadratic differential - diminishes will be carried out. Particular interest focuses on the case where the differential passes through the Bers boundary of Teichmuller space.