Alexander Duncan: Automorphisms of del Pezzo surfaces in small characteristic
The plane Cremona group Cr(k) is the group of birational automorphims of the projective plane or, equivalently, the group of automorphisms of the purely transcendental field extension k(x,y)/k. Over an algebraically closed field k of characteristic 0, the finite subgroups of Cr(k) have (mostly) been classified. We concern ourselves with the case where k has positive characteristic. The most interesting groups in the classification arise from automorphisms of del Pezzo surfaces. We consider mainly the case of del Pezzo surfaces of degree 4 and 3.
These are closely related to quadratic and cubic forms. A del Pezzo surface of degree 4 is a smooth complete intersection of two quadrics in P^4. A del Pezzo surface of degree 3 is a smooth cubic surface in P^3. In characteristic 0, classical normal forms exist which facilitate the study of their automorphisms. These normal forms fail to exist in small characteristics; we identify new normal forms in these cases and indicate how they can be used to classify automorphism groups of del Pezzo surfaces.
Andrew G. Earnest: Classification of Quadratic Lattices by Representations
In this talk, the general question of the extent to which quadratic lattices are classified by the lattices of smaller rank that they represent will be discussed, primarily focusing on the case of positive definite lattices over Z. For the case of the representation of integers, known results will be surveyed and the status of several conjectures will be discussed. A general framework to facilitate the discussion for the representation of lattices of arbitrary rank will then be introduced and some new results on the representation of lattices of codimension one will be presented.
Jesse Kass: How to use quadratic form to count curves arithmetically
Ideas in motivic homotopy theory suggest that many classical results about counting algebraic curves can be enriched to results giving counts valued in the group of quadratic forms. These enriched counts are to contain arithmetic information. Recent advances in homotopy theory, especially work of Marc Levine, provide powerful new tools for realizing this idea. I will talk about joint work with Kirsten Wickelgren on this topic, emphasizing aspects involving the theory of quadratic forms. In particular, no prior knowledge of homotopy theory (motivic or otherwise) will be assumed.
Andrew Obus: Reduction of Dynatomic Modular Curves: The good, the bad, and the irreducible
The dynatomic modular curves parameterize one-parameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. These results are motivated by uniform boundedness conjectures in arithmetic dynamics, which will be explained.
James Ricci: Establishing Finiteness Results for Regular Quadratic Polynomials
In 1954, G.L. Watson showed that there are only finitely many equivalence classes of positive definite primitive regular ternary quadratic forms. In this talk, we will generalize Watson's method of descent to apply to cosets of quadratic lattices, show how this can be used to establish a finiteness result for primitive regular ternary quadratic polynomials (joint work with Wai Kiu Chan), and then discuss how similar techniques might be used in the binary case.
Huy Dang: Connectedness of the Moduli Space of Artin-Schreier curves with genus $g$
In this talk, the connectedness of the moduli space of Artin-Schreier curves with genus g over an algebraically closed field of positive characteristic will be discussed. We introduce a combinatorial description of how irreducible strata of the moduli space fit together. As an application, when the characteristic is equal to 3, we show that the moduli space is connected for all possible genus. When the characteristic is greater than 3, it turns out that the moduli space is connected when the genus is sufficiently large, and the bound depends on the characteristic.
Solly Parenti: Unitary PSL_2 CM Fields and the Colmez Conjecture
In 1993, Pierre Colmez conjectured a relation between the Faltings height of a CM abelian variety and certain log derivatives of L-functions associated to the CM type, generalizing the classical Chowla-Selberg formula. I will discuss how we can extend the known cases of the conjecture to a certain class of unitary CM fields using the recently proven average version of the conjecture.