Expander graphs, in general, and Ramanujan graphs, in particular, have been objects of intensive research in the last four decades. Many application came out, initially to computer science and combinatorics and more recently also to pure mathematics (number theory, geometry, group theory ). In recent years, there has been an interest in generalizing this theory to higher dimensional simplical complexes. We plan to survey first the classical theory and then describe the more recent developments.
Mirror symmetry is a deep conjectural relationship between complex and symplectic geometry. It was first noticed by string theorists. Mathematicians became interested in it when string theorists used it to predict counts of curves on the quintic three-fold (just as there are famously 27 lines on a cubic surface, there are 2875 lines on a quintic three-fold, 609250 conics, and so on). Kontsevich conjectured that mirror symmetry should reflect a deeper equivalence of categories: his celebrated 'Homological Mirror Symmetry' conjecture.
Mathematical Theories of Interaction with Oracles: Active Property Testing and New Models for Learning Boolean Functions
With the notion of interaction with oracles as a unifying theme of much of my dissertation work, I discuss novel models and results for property testing and computational learning, with the use of Fourier analytic and probabilistic methods.
I will describe some recent results and problems regarding influence of sets of variables on Boolean functions: In 1989 Benny Chor conjectured that a balanced Boolean function with n variables has a subset S of size 0.4n with influence (1-c^n) where c0 follows from a theorem by Kahn, Kalai and Linial (KKL).I will present a recent counterexample by Kahn and me showing that up to the identity of c, the KKL bound cannot be improved.
We will discuss recent work on wave evolutions for large data. Particular emphasis will be placed on concentration compactness ideas. Amongst others, we will describe a result for wave equations from R^3 minus the unit ball into the sphere S^3 where we can show that any solution approaches the unique harmonic map in its degree class.
Joint work with Cote, Kenig, Lawrie, Nakanishi -- in various combinations.