A description of the product structure of the cohomology of polyhedral products is given in terms of the stable splitting. Several applications will be discussed. This report is based on joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

This survey talk will be a review results obtained in collaboration with Mattias Franz and Nigel Ray. As singular toric varieties, weighted projective spaces have an action of a real torus. The equivariant cohomology with respect to this action is computed to be isomorphic to the ring of piecewise polynomials on the defining fan. The theory is seen to parallel that for smooth toric varieties with the role of the Stanley-Reisner ring replaced by the ring of piecewise polynomials. If time permits, an alternative presentation of weighted projective spaces as iterated Thom complexes will be discussed briefly. Further collaboration with Mattias Franz, Nigel Ray and Dietrich Notbohm yields in a complete topological classification of weighted projective spaces.

Certain natural subspaces of a product of CW complexes, called polyhedral products, play an important role in a variety of different fields including: toric varieties, toric manifolds/orbifolds, intersections of quadrics, homotopy theory, algebraic combinatorics complements of subspace arrangements, robotics and group theory. The talk will survey certain combinatorics based constructions on polyhedral products which allow for a description of the cohomology rings of certain families of toric manifolds in a particularly compact form. The new constructions will be related to a generalization of the basic Davis-Januszkiewicz construction of toric manifolds. This report is based on joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

In 1975, Y. Morita conjectured that if an abelian variety defi ned over a number fi eld has the Mumford-Tate group which does not have any non-trivial Q-rational unipotent element, then it has potential good reduction everywhere. In this talk, we explain a proof of this conjecture. The main ingredients of proof include some newly established cases of the conjecture due to Vasiu, a generalization of a criterion of Paugam on good reduction of abelian varieties, and the local-global principle of isotropy for Mumford-Tate groups of abelian varieties.

A local factor of a Dedekind zeta function can be btained as a partition function of an appropriate C*-dynamical system. For a Dedekind zeta function itself such a dynamical system is related to the Galois group of the given number field and is called a BC system. We will ee how to extend this idea in the case of Hecke L-functions.

Using Todd operator one can express Euler-McLaurin formula in simple form. For recent decades, along with development of toric geometry and theory of polytopes, Euler-McLaurin formula has been generalized to a category of lattice cones and polytopes by Brion-Vergne, Karshon-Sternberg-Weitsman, Garoufalidis-Pommersheim and several others. In particular, Garoufalidis-Pommersheim expressed special values of zeta function associated to Todd operator associated to a certain cone decomposition. We extend the category of cones by Grothendieck group construction of ordinary cones. This new category contains 'virtually decomposed cones' considering cones with negative weight and the appropriate form of Todd operator construction generalizes the Euler-Mclaurin formula on it. We apply this generalization to obtain an alternating sum expression of special values of (partial) zeta functions at nonpositive integers associated (virtual) cone decomposition. This expression enables us to read polynomial behavior of special values of zeta function at nonpositive integers in some family of real quadratic fields. It is joint work with Byungheup Jun.