For decades partition function differences have been studied. These include a famous problem of Henry Alder posed in the 1950's and solved only recently by Yee, Oliver et al. In 1978, Szekeres and Richmond partially solved a problem of this type concerning the Rogers-Ramanujan continued fraction. Unknown to them, the problem had essentially been solved by Ramanujan in the Lost Notebook. The late Leon Ehrenpreis asked in 1987 if one could prove that the number of partitions of n into parts congruent to 1 or 4 mod 5 is always at least as large as the number with parts congruent to 2 or 3 mod 5 WITHOUT using the Rogers- Ramanujan identities. Subsequently Baxter and I gave a "sort of" solution to the problem. Our work was more designed to give a motivated proof of the Rogers-Ramanujan identities. The first "real" solution to Ehrenpreis's problem was found by Kadell in the 1990's. Recently major methods have been developed by Berkovich, Garvan and Grizzell. In this talk, I will begin with the history of such problems. I will conclude with some observations on the anti-telescoping method for treating some such problems.

In his lost notebook, in one of three partial manuscripts on diophantine approximation, Ramanujan gives the best possible diophantine approximation to e2/a, where a is a nonzero integer. This best diophantine approximation was not proved in the literature until 1978 in a paper by C. S. Davis. We apply the Ramanujan-Davis diophantine approximation to a conjecture of Jonathan Sondow on the approximations of e by partial sums ¿n ¿ N1/n!. It was then realized that our methods could be used to attack similar problems on the diophantine approximation of hypergeometric functions and the exponential generating functions arising from Dirichlet L-series and L-series of elliptic curves. We briefly sketch our proof of Sondow's Conjecture, but similar conjectures about the other aforementioned exponential generating functions remain unresolved. Our proof of Sondow's Conjecture also arises from a careful study of the partial denominators of the simple continued fraction of e. We establish several new results about these partial denominators and about certain p-adic functions that naturally arise, but several open problems about the p-adic functions also emerge. Our talk is based on joint work with Sun Kim and Alexandru Zaharescu.

We study Sturm type of bounds on Jacobi forms and Siegel modular forms of genus 2. Congruences involving an analogue of Atkin¿s U(p)- operator applied to Siegel modular forms of genus 2 will be considered, and we discuss various examples. This is joint work with various people.

Let n > 1. For an (n-1)-tuple of integers, let Af(M) denote the Mth Fourier coefficient of a Maass form f for SL(n, Z). We are interested in a certain weighted average (over Maass forms f) of Af(M)¿\overline{Af(M')}, and conjecture that this weighted average is either 1 or 0 depending on whether M = M' or not. In joint work with Alex Kontorovich it is shown that this conjecture can be proved for GL(3). Applications to Katz-Sarnak symmetry types and Sato-Tate distribution problems will be given.

The discriminants of quadratic number fields for which the associated L-function is relatively small at the point s = 1 have exceptional power to grasp prime numbers. For example, a special nature of these discriminants (perhaps the first one in the literature, 1913) is revealed in the theorem of G. Rabinowitsch. We shall show (a joint work with John Friedlander) that the exceptional positive fundamental discriminants can be represented as the sum of a square and a prime number.

This will be a survey talk on results that have been obtained over the past ten years about sign changes of the Fourier coefficients of cusp forms, both in the integral as well as in the half-integral weight case.

We discuss the algebraic and arithmetic properties of vector-valued modular forms of level 6, a subject that finds its roots in the work of Fricke. We pay particular attention to the contributions made by Marvin Knopp to the modern theory of vector-valued modular forms.

We describe recent joint work with A. Ghosh and A. Reznikov on the nodal domains of Maass forms with large eigenvalue. In particular a proof, conditional on the Lindelof Hypothesis, that locally the number of such domains goes to infinity with the eigenvalue. This relies on some related sharp results on L2 restrictions of these eigenfunction to geodesics which are established unconditionally.

Marvin Knopp was not only a great researcher and teacher, but a very gifted expositor. I will give an exposition about his expository work.