Classification of proper holomorphicmaps between bounded symmetric domains is deeply related to the study of locally symmetric spaces. In this talk, we consider rigidity problem of proper holomorphic maps between bounded symmetric domains and related problems in locally symmetric spaces. Then we give an introduction to differential geometric techniques on rigidity problems, based on the similar phenomenon for local CR maps between arbitrary boundary components of two bounded symmetric domains of Cartan type I.

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions and discuss the large time behavior of the solution. The proof that the solution orbits are relatively compact is based upon rearrangement theory. We also characterise the limit function and prove that it is given by a step function. (This is joint work with Hiroshi Matano, Thanh Nam Nguyen and Hendrik Weber.)

We generate ring class fields of imaginary quadratic fields in terms of the special values of certain eta-quotients, which are related to the relative norm of Siegel-Ramachandra invariants. These give us minimal polynomials with relatively small coefficients from which we are able to solve certain quadratic Diophantine equations concerning non-convenient numbers.

Let A and B be finite nonempty subsets of a multiplicative group G, and consider the product set AB = { ab | a in A and b in B }. When |G| is prime, a famous theorem of Cauchy and Davenport asserts that |AB| is at least the minimum of {|G|, |A| + |B| - 1}. This lower bound was refined by Vosper, who characterized all pairs (A,B) in such a group for which |AB| < |A| + |B|. Kneser generalized the Cauchy-Davenport theorem by providing a natural lower bound on |AB| which holds in every abelian group. Shortly afterward, Kemperman determined the structure of those pairs (A,B) with |AB| < |A| + |B| in abelian groups. Here we present a further generalization of these results to arbitrary groups. Namely we generalize Kneser’s Theorem, and we determine the structure of those pairs with |AB| < |A| + |B| in arbitrary groups.

Given a curve in a plane, we construct a factorization of a polynomial multiplied by an identity matrix into the product of two matrices, by counting certain polygons in a plane. Such correspondences between geometric objects (curves, polygons) and algebraic objects (matrix factorizations of a polynomial) are instances of homological mirror symmetry. We explain the generalization of the construction to higher dimensions, and its application to the proof of homological mirror symmetry conjecture for certain spaces.