on a complete riemannian manifold with ricci curvature bounded below, yau proved a sharp gradient estimate for positive harmonic functions. one comparison space is the hyperbolic space (but it is not unique). this talk will report on the corresponding question for complete noncompact kaehler manifolds.

stillman's conjecture asserts the existence of a uniform bound
on the projective dimension of an ideal in a polynomial ring generated
by a fixed number of polynomials of fixed degrees. ananyan and hochster
gave a proof of stillman's conjecture by proving the existence of ``small
subalgebras''. i'll describe a simplification of their approach using
ultraproducts (and in particular, explain what ultraproducts are and all
of the terms mentioned above). this is based on joint work with daniel
erman and andrew snowden.

given a semiabelian variety over a number field and a closed
subvariety thereof, it is a theorem (of various people) that the torsion
points on the subvariety have zariski closure equal to some finite union
of torsion cosets of semiabelian subvarieties. we will focus on this
question for (split) tori, where it manifests as the more concrete
problem of finding solutions to a system of polynomial equations valued
in roots of unity. we compare two effective methods for solving this
problem: a combinatorial approach introduced by conway-jones, and a
commutative algebra approach introduced by beukers-smyth.

hironaka's theorem states that every algebraic variety over a field of characteristic zero admits a resolution, however the algorithm to do so is very complex. it was conjectured by nash that repeated nash blowups (which replace each singular point by limiting positions of tangent spaces at nonsingular points) might provide a simple way to resolve. i plan to investigate the case of toric varieties.

in his landmark paper ``modular forms and the eisenstein
ideal,'' mazur studied congruences modulo a prime p between the hecke
eigenvalues of an eisenstein series and the hecke eigenvalues of cusp
forms, assuming these modular forms have weight 2 and prime level n. he
asked about generalizations to squarefree levels n. i will present some
work on such generalizations, which is joint with preston wake and
catherine hsu.