The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the minimal model program, we introduce very general stringy invariants, associated to "almost all" singularities, more precisely, to all singularities that are not strictly log canonical. They specialize to the invariants of Batyrev when the singularity is log terminal. For example, the simplest form of our stringy zeta function is, in general, a rational function in one variable, but it is just a constant (Batyrev's stringy Euler number) in the log terminal case.

We study a class of stationary processes indexed by ℤ^d that are defined via minors of d-dimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong K-property, a particular strengthening of the usual K (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to independent identically distributed (i.i.d.) processes in the sense of ergodic theory). We obtain estimates of their entropies, and we relate these processes via stochastic domination to product measures.

Consider the Fulton-MacPherson configuration space of n points on ℙ¹, which is isomorphic to a certain moduli space of stable maps to ℙ¹. We compute the cone of effective $\mathfrak{S}$_n-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of SL₂ in the space of binary forms of degree n≥3

Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental forms and Seashi's work on linear differential equations of finite type, we reduce the proof to the vanishing of certain Spencer cohomology groups. The vanishing is checked by Kostant's harmonic theory for Lie algebra cohomology.

We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures posed by T. Shioda twenty-five years ago.