Titles and abstracts for the session “Algebraic geometry through numerical computation”

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Daniele Agostini, Humboldt-Universität zu Berlin
Computing Theta Functions with Julia
I present a new package Theta.jl for computing with the Riemann theta function. It is implemented in Julia and offers accurate numerical evaluation of theta functions with characteristics and their derivatives of arbitrary order. The package is optimized for multiple evaluations of theta functions for the same Riemann matrix, in small dimensions. As an application, I will report on experimental approaches to the Schottky problem in genus five. This is joint work with Lynn Chua.

Francesca Bianchi, University of Groningen
Computational aspects of quadratic Chabauty over number fields
The quadratic Chabauty method is a p-adic approach to finding integral or rational points on certain types of curves over the rationals. In recent joint work with Balakrishnan, Besser and Müller, we generalised the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields. In this talk, I will discuss computational steps and challenges that arise when we want to put the theoretical strategy into practice.

Michael Burr, Clemson University
Correct computation with inexact calculations in algebraic geometry
In computations, it seems that one often has to choose between correctness and efficiency. For instance, in algebraic geometry, correct symbolic-based algorithms are often slow whereas efficient numerical-based algorithms may produce incorrect results. In this talk, I will discuss recently developed algorithms that combine the positive features of these two approaches. These algorithms use inexact floating point-based calculations, but guarantee the correctness of the final output. I will first provide a high-level description of the types of tools, philosophies, and approaches that are used in the development of these algorithms. Then, I will then provide details on how these ideas are incorporated into real-world algorithms such as those for efficient certified numerical homotopy continuation.

Netan Dogra, Jesus College, Oxford
New algorithms for Frobenius structures and applications
The computation of Frobenius structures on connections has applications to a diverse range of topics in arithmetic and geometry. In this talk I will review some applications to Diophantine geometry, and some recent work with Jan Vonk extending the scope of the algorithms currently available in this area.

Jonathan Hauenstein, University of Notre Dame
Computing sparse polynomials via witness sets
Sparse polynomials that vanish on algebraic sets are preferred in many computations since they are easy to evaluate and often arise from underlying structure. For example, a monomial vanishes on an algebraic set if and only if the algebraic set is contained in the union of the coordinate hyperplanes. This talk will consider joint work with Laura Matusevich, Chris Peterson, and Samantha Sherman which aims to compute sparse polynomials that vanish on an algebraic set which is represented by a witness set. Our approach relies on using numerical homotopy methods to sample points on the algebraic set along with incorporating multiplicity information using Macaulay dual spaces. If the algebraic set is defined by polynomials with rational coefficients, exactness recovery such as lattice based methods can be used to find exact representations of the sparse polynomials and Grobner basis techniques can be used to certify correctness. Several examples will be presented demonstrating the approach.

Nicolas Mascot, American University of Beirut
Hensel-lifiting torsion points on Jacobians, and computation of Galois representations from higher étale cohomology
We will describe an algorithm to p-adically lift torsion points on Jacobians of curves. As an application, we will show how to compute explicitly mod l Galois representations occurring in the Jacobians of curves, and also more generally in higher étale cohomology spaces of higher-dimensional varieties. This has many applications, ranging from exploring Langlands’s program to point counting in cryptography.
Although the topic may sound abstract, this talk will be very introductory, and will include several concrete examples; for instance, we will apply these techniques to a Galois representation found in the H2 étale of a surface and attached to a modular form of exotic type.

Margaret Regan, University of Notre Dame
Evaluating and differentiating a polynomial using a pseudo-witness set
For polynomials that arise via elimination, they could be difficult to compute explicitly. By using a pseudo-witness set, we develop an algorithm for evaluating and computing directional derivatives for the polynomial at any point. Several examples are used to demonstrate this new algorithm including an example for computing the critical points of the discriminant locus of a parameterized polynomial system.

Jose Rodriguez, University of Wisconsin–Madison
A numerical approach for computing Euler characteristics of affine varieties
One of the most fundamental topological invariants is the Euler characteristic.
For example, Euler’s polyhedron formula is derived from the Euler characteristic of a sphere.  Furthermore, a series of works have recently appeared that relate the complexity of an algebraic optimization problem to the Euler characteristic of a complex algebraic variety. With this in mind, we are motivated to compute this invariant using the tools of numerical algebraic geometry.
While there are several existing approaches to compute the Euler characteristic of a complex algebraic variety, we develop a new recursive method, which has the following advantages. First, our method directly computes the Euler characteristic of an affine variety without involving a compactification. This is useful because the projective closure of a smooth affine variety can have bad singularities along infinity. Second, our method does not rely on the inclusion-exclusion principle. Instead, we use the additive property of Euler characteristics to stratify our variety into locally closed smooth subvarieties. In this talk, we will explain how our method leads to a practical numerical algorithm that is tailored to take advantage of the recursive nature. This is joint work with Xiaxin Li and Botong Wang.

Frank Sottile, Texas A&M University
General witness sets for numerical algebraic geometry
Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory. Under appropriate assumptions, general witness sets enable numerical algorithms such as sampling and membership. These assumptions hold for products of flag manifolds. We introduce Schubert witness sets, which provide general
witness sets for Grassmannians and flag manifolds.

Simon Telen, KU Leuven
Solving Polynomial Systems using Cox Rings
The Cox ring of a compact toric variety is a generalization of the homogeneous coordinate ring of projective space. In this talk we present numerical algorithms for computing a subscheme of such a toric variety defined by the homogenization to the Cox ring of relations on its torus. We introduce generalizations of classical eigenvalue methods and homotopy path tracking algorithms for computing homogeneous coordinates of the solutions. The approach allows to understand the number of solutions in the torus of certain families of systems, as well as to deal with solutions `at infinity’ in a robust way.