Please note: the instructor and course information changes from semester to semester for this course number. Specifics for each semester below.
Diophantine Approximations and Transcendental Numbers
This course concerns two subjects which are closely related: approximations of special numbers by algebraic numbers and theory of transcendental numbers. The main results will be covered in details, in particular :
-Baker theory oflinear forms of logarithms
Among several applications I will give a solution of the Gauss Class Number One Problem for imaginary quadratic fields.
No advanced knowledge of number theory is required, but participant's curiosity in special numbers will make the course enjoyable.
J.W.S. Cassels, An Introduction to Diophantine Approximations, Alan Baker, Transcendental Number Theory
Spectral theory of automorphic forms
This will be a one semester course on automorphic forms from analytic point of view. The main topics are:
- trace formula
-sums of Kloosterman sums
-distribution of eigenvalues of the Laplace operator (Weyl’s law, exceptional eigenvalues)
-distribution of Hecke eigenvalues
-hyperbolic lattice point problems
-application to equidistribution of roots of congruences
Henryk Iwaniec, Spectral Methods of Automorphic Forms, AMS Grad.Stud. Vol.53, 2002
Good knowledge of functional analysis and complex function theory will be helpful