Seminars archive

In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer.

This dissertation provides three results: (1) A Hasse principle for rational points of projective homogeneous spaces under unitary or special unitary groups associated to hermitian or skew hermitian spaces over function fields of p-adic curves; (2) A Springer-type theorem for isotropy of hermitian spaces over odd degree field extensions of function fields of p-adic curves; (3) Exact values of Hermitian u-invariants of quaternion or biquaternion algebras over function fields of p-adic curves.

In this talk I will give a tour of recent results and open problems about vector bundles of conformal blocks on the moduli space of curves. I will discuss how these results fit into the context of some of the open problems about the binational geometry of the moduli space.

Let E/Q be an elliptic curve and let Q(3^infty) denote the compositum of all cubic extensions of Q. While the group E(3^infty) is not finitely generated, one can show that its torsion subgroup is finite; this holds more generally for any Galois extension of Q that contains only finitely many roots of unity. I will describe joint work with Daniels, Lozano-Robledo, and Najman, in which we obtain a complete classification of the 20 torsion subgroups that can and do occur, along with an explicit description of the elliptic curves E/Q that realize each possibility (up to twists). This is achieved by determining the rational points on a corresponding set of modular curves and relies on several recent results related to the mod-n Galois representations attached to elliptic curves over Q.

Let A/F be a division algebra of degree 3, and X its Severi-Brauer variety which is a form of the projective plane. The linear system of cubic curves is defined on X, and so we can let C \subset X be one such. If C is a nonsingular such curve, then C is a genus one curve with Jacobian E, an elliptic curve. The question we address is the one asked by Asher Auel, namely, which E arise. We give an answer that depends on the structure of A.

Summary : Let G be a connected linear algebraic group over a field k. By definition, a groupic G-variety X over k is a smooth (left) G-variety with a dense open set isomorphic to G with its (left) action on itself. Let X be a groupic G-variety over a number field. Under a suitable noncompactness hypothesis for the simple factors of the semisimple part of G at the archimedean places, Cao and Xu show that the Brauer-Manin obstruction is the only obstruction to strong approximation for X off the archimedean places. The proof builds upon the case X=G (handled in earlier papers by Xu and the speaker, Harari, Demarche). The toric case (G is a torus) was already handled in a previous paper by Cao and Xu. For X projective, the statement is a weak approximation result and the theorem has been known for a long time (Sansuc).. The proof of the strong approximation result for an arbitrary groupic G-variety X involves novel arguments, both geometric and arithmetic.

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We describe several generalizations of the moduli space of marked rational curves, their combinatorial structure and construction methods. In particular, we report joint work with Noah Giansiracusa in which we revisit the smooth configuration space compactifying n distinct points in affine space up to translation and homothety. I will also report joint work with Kenny Ascher in understanding a good locus inside the moduli space of weighted line arrangements constructed by V. Alexeev.

I will discuss a matrix completion problem arising in combinatorial statistics and explain how we can use results in algebraic geometry to understand it better. The object linking the two different areas is the cone of sums of squares and its properties as a convex cone.