DIAMANT Symposium Spring 2022 & NMC 2022

Sato-Tate Groups in Dimension Greater than 3

The focus of this talk is on Sato-Tate groups of abelian varieties — compact groups predicted to determine the limiting distributions of local zeta functions. In recent years, complete classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimensions. In this talk, I will describe my recent work on families of higher dimensional Jacobian varieties. This work is partly joint with Melissa Emory.

Every finite abelian group is the group of rational points of an
ordinary abelian variety over F2, F3 and F5

We show that every finite abelian group G occurs as the group of rational points of an ordinary abelian variety over F2, F3 and F5. We produce partial results for abelian varieties over a general finite field Fq. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over Fq when q is large.
This is joint work with Caleb Springer.

Explicit methods for hyperelliptic genus 4 Kummer varieties

An explicit construction of the Kummer variety of a hyperelliptic curve has only been done for genus up to 3. As part of my PhD thesis I do such an explicit construction in the case of g=4. In this talk I give an overview which methods have been made explicit and what functionality can be provided in the Magma language such as an specific embedding of the Kummer into (P)^15, duplication polynomials or an algorithm to compute the canonical height of points. I will also go into problems encountered in the research. This work was done under supervision of M. Stoll and S. Müller.

Division in modules and Kummer theory

In this talk I will present some of the results of my recent paper “Division in modules and Kummer Theory”. Motivated by number-theoretic applications I will introduce a generalization of the classical concept of injective modules which also makes it possible to extend the concept of p-divisible abelian group over any ring. Classical results such as Baer’s criterion and the existence of injective hulls also hold in this more general setting. Finally I will show how one can use these purely algebraic concepts to study Kummer-like extensions arising from commutative algebraic groups.

One-Shot Capacity of Networks with Restricted Adversaries

In this talk, we will concentrate on the one-shot capacity of communication networks with an adversary who can possibly corrupt only a proper subset of network edges. That is, we want to compute the maximum number of information symbols that can be sent in a single use of network no matter how the adversary acts. We show that combining information linearly at the intermediate vertices of the network does not suffice in general to achieve capacity, in contrast with the standard scenario where the adversary is not restricted. We also propose new techniques to obtain bounds for the one-shot capacity of networks using combinatorics.

On the Simplex method for 0/1 polytopes

The Simplex method is one of the most popular algorithms for
solving linear programs, but despite decades of study, it is still not
known whether there exists a pivot rule that guarantees it will always
reach an optimal solution with a polynomial number of steps.
In fact, a polynomial pivot rule is not even known for linear programs
over 0/1 polytopes (0/1-LPs), despite the fact that the diameter of a
0/1 polytope is bounded by its dimension.
This talk will focus on the behavior of the Simplex method for 0/1-LPs,
and discuss pivot rules that are guaranteed to require only a
polynomial number of non-degenerate pivots.

Joint work with: Alexander Black, Jesus De Loera, Sean Kafer.

Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

The combinatorial diameter diam(P) of a polytope P is the maximum shortest path distance between any pair of vertices. For example, the n-dimensional cube has combinatorial diameter n, while any simplex has combinatorial diameter 1. The combinatorial diameter is a combinatorial quantity closely related to questions surrounding the running time of the simplex method for solving linear programming problems. In this talk, we describe upper and lower bounds on the combinatorial diameter of a random “spherical” polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. Joint work with Gilles Bonnet, Daniel Dadush, Uri Grupel, and Galyna Livshyts.

Northcott numbers and applications

A set of algebraic numbers with bounded degree and bounded height is a finite set, by Northcott’s theorem. The set of roots of unity is of height zero, but is infinite. What about other sets of algebraic numbers? When is a set of bounded height still infinite? A way to approach this question is through the Northcott number of these sets. We will study some of their properties, discuss links to Julia Robinson’s work on undecidability, and explain other applications towards height controls in Bertini statements. The talk is based on joint work with Technau and Widmer.

Negative curves on rational surfaces

Two curves C ≠ D on an algebraic surface X meet in a finite nonnegative number of points, called the intersection number C ⋅ D ≥ 0. On the other hand, the “self-intersection” C ⋅ C of a curve C may also take negative values. I will explain what this means, and present progress on the Bounded Negativity Conjecture, a folklore conjecture going back at least to Enriques (1871–1946).