Numerical adjunction formulas for weighted projective planes and lattice point counting

Mirror Symmetry for Weighted Projective Planes and Their Noncommutative Deformations

Special Point Sets in Finite Projective Planes

We consider the following three problems: 1. Let U be a q-subset of GF(q 2) with the properties 0; 1 2 U and u ? v is a square for all u; v 2 U. Does it follow that U consists of the elements of the subbeld GF(q)? Here q is odd. 2. Let f : GF(q) ! GF(q) be any function, and let be the set of diierence quotients (directions, slopes). What are the possibilities for jD f j? 3. Let B be a subset of...

An Adjunction for Modules over Projective Schemes

For modules over a ring there is an adjunction between the associated sheaf functor AMod −→ Mod(SpecA) and the global sections functor Mod(SpecA) −→ AMod. In this note we develop the graded version of this result. All of this material is taken from EGA.

Curves, dynamical systems, and weighted point counting.

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobi...

Lattice point counting and harmonic analysis

We explain the application of harmonic analysis to count lattice points in large regions. We also present some of our recent results in the three-dimensional case.