In the talk, I will describe recent attempts to understand the mysterious and beautiful geometry of nodal lines of random spherical harmonics and of random plane waves. If time permits, I will also discuss asymptotic statistical topology of other natural polynomial-like ensembles of random functions. The talk is based on a joint work with Fedja Nazarov.
In this talk I will describe a real-variable method to extract long-time asymptotics for solutions of many nonlinear equations (including the Schrodinger and mKdV equations). The method has many resemblances to the classical stationary phase method in the theory of oscillatory integrals.
This will be an introduction to special value formulas for L-functions and especially the uses of modular forms in establishing some of them -- beginning with the values of the Riemann zeta function at negative integers and hopefully arriving at some more recent work on the Birch-Swinnerton-Dyer formula.
Let p be an odd prime number and let F be a totally real field. Let F_cyc be the cyclotomic extension of F generated by the roots of unity of order a power of p . From the maximal abelian extension of F_cyc which is unramified (resp. unramified outside auxiliary primes), we get exact sequences of Iwasawa modules. We will discuss how splitting of these exact sequences are linked to Leopoldt conjecture for F and p . (JW with C. Khare)