Schedule for: 24w5229 - Computational Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Polyhedral homotopies were originally introduced by Huber and Sturmfels nearly 30 years ago, and have since become a staple strategy for solving polynomial systems. Main topic of the talk is a generalisation thereof. Building on ideas of Jensen, Leykin, and Yu, we will discuss two distinct types of tropical homotopies: First, we will discuss how to use tropical points to construct homotopies for solving systems of polynomial equations. Second, we will discuss how to compute tropical points using homotopies for intersecting systems of balanced polyhedral complexes. Centerpiece of the talk are systems of parametrized polynomial equations, and we will focus two main cases: Vertically parametrized polynomial systems are systems in which parameters are shared between equations but always bound to the same monomial. These are for example the steady state equations of chemical reaction networks or they arise in the computation of ED or ML degrees. Horizontally parametrized polynomial systems are systems in which parameters are shared between monomials but always bound to the same equation. These are prominently studied using the theory of Khovanskii bases and Newton Okounkov bodies. We conclude the talk with an OSCAR demo on tropical geometry.

In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The primary invariant for a family of curves is the combinatorial description of `bad' fibers. When the curves are elliptic, the classification of possible geometric configurations (`reduction types') is due to Kodaira and Neron, in genus 2 to Namikawa-Ueno, and in genus 3 to Ashikaga-Ishizaka. In this talk, I would like to describe a possible classification for curves of arbitrary genus, and give an overview of what are the classes of curves for which we can currently compute their reductions.

Discussion lead by Frank Sottile. Description: Ambitious computational exploration of phenomena in geometry requires serious resources: computing, software development and maintenance, data storage, and of course people and their time. During this week at BIRS devoted to building a community of computational mathematicians working in geometry, I'd like to engage the participants in a formal discussion about procuring the resources we need for our work. This includes sharing ideas and stories about how we obtained resources as well as looking ahead about how we may induce grant agencies to respond to these research needs. With such a diverse, influential, and international group of attendees there is much that we can learn from one another and accomplish.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will explain the GAP-aided proof of the statement in the title. After a brief mathematical background based on the “magic of K3-surfaces’’ and reducing the problem to finding certain sets of square 4 vectors in (any) one of the 24 Niemeier lattices, I will concentrate on the code optimization that made the computation feasible. The most striking case is that of the so-called Leech lattice, where I have managed to reduce the computation time from 2+ months on Max-Planck’s servers (still unfinished) to less than 2 days on my laptop.

The L-functions and Modular Forms Database (beta.lmfdb.org) is an online database, centered on the Langlands correspondence, that includes sections on groups and varieties in addition to L-functions and various kinds of modular forms. I will talk about two projects within the LMFDB: finite groups, which will soon be launched as part of the production website, and K3 surfaces, which are at an early stage of development. I hope to offer insights that will be useful for others considering their own database projects.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Discussion lead by Alessio Corti and John Voigt.

We describe some tools recently implemented in Sagemath for rigorous arbitrary precision computation of periods of algebraic Riemann surfaces, for Abel-Jacobi maps, and for theta functions and their derivatives, with characteristics. The particular applications we have in mind include numerical computation of automorphisms, endomorphism rings, as well as explicit reconstructions of algebraic curves using Torelli-type results. The work reported on includes contributions from Linden Disney-Hogg, Sohrab Ganjian, Wuqian Gao, Jeroen Sijsling, and Alexandre Zotine.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Due to the fact that groebner bases don't really profit from parallelization, computational algebraic geometry was believed not to profit from it either. But changing the perspective slightly, there is a lot of natural (non-trivial) parallelism already built into the structures of algebraic geometric data. In this talk, I will highlight 2 or 3 applications in this context using a transparent parallelization framework.

Oguiso and Yu have shown the existence of an Enriques Surface automorphism $f : Y \to Y$ with minimal entropy. However, their proof takes place purely on the lattice theoretic side, i.e. they give a period in the K3 lattice to describe the universal covering $X$ of $Y$, together with an orthogonal transformation $\phi$ of the lattice for the lifting of $f$ to $X$ and another one $\iota$ for the covering involution. By virtue of the Torelli theorems for K3 surfaces, this describes $f$ uniquely. I will exhibit how we use the newly developed functionality for algebraic schemes in OSCAR to explicitly construct all the object involved and find the automorphism described by Oguiso an Yu.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Galois group of a polynomial system is a group of symmetries of the zeros of the system that reflects its intrinsic structure. These groups were initially studied algebraically by Jordan, and much later Harris interpreted them as geometric monodromy groups. This geometric view allows one to study Galois groups through analytic means and to approximate them via numerical methods. It has been shown that knowledge of the Galois group may be used to reduce computation in solving polynomial systems, and this has been used in computer vision, for example. We will consider Galois groups of sparse polynomial systems, systems whose coefficients are general and whose monomial support is fixed. There are two special structures that occur in sparse systems: lacunary systems are those that have been precomposed with a non-invertible monomial map, and triangular systems are those that contain a proper subsystem. Galois groups of lacunary systems and triangular systems act imprimitively on the zeros of the system and are expected to be equal to a certain wreath product. However, a classification of these Galois groups remains open. We determine the Galois group of a pure lacunary polynomial system, which is a sparse polynomial system that is lacunary and not triangular. We use analytic methods akin to those of Harris to show that the Galois group of a pure lacunary system is determined by the automorphism group of a certain variety. Further, using a resultant product formula of D'Andrea, this variety is defined by binomial equations and its automorphism group is a group of roots of unity acting by coordinate-wise multiplication. We use this to compute the Galois group of some pure lacunary systems and demonstrate that the Galois group may be strictly smaller than the expected wreath product in many cases.

The Schubert calculus of enumerative geometry is a rich and well-understood family of structured problems in enumerative geometry. With many millions of computable problems, it is a laboratory for investigating new phenomena in enumerative geometry. I have been using it as such for the past 25 years in projects to study arithmetical (reality and Galois groups) in enumerative geometry and more generally in systems of equations. These projects use different computational resources and well over 15 TeraHertz-years of computing. This talk will sketch some of the mathematics and some of the lessons learned from these investigations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The bottleneck for numerically solving a polynomial system can lie in the memory required to store the solutions. We propose an alternative way to represent solutions to such a system via how they are found during a monodromy solve algorithm. Such a representation is essentially an iterator for the solution set, whose decreased memory requirements come at the cost of an increased time requirement to produce the representation. We propose several uses for this space/time complexity trade-off, including data compression, certification, and verification.

Chair: Yue Ren. Submissions of ideas via Google form: https://forms.gle/7QPLKEg6PYcsSDJ17

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will focus on computing kernels of polynomial maps. This core problem in symbolic computation is known as implicitization. While there are extremely effective Groebner basis methods to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the map is multigraded, we consider an alternative approach. First, we will demonstrate how to find a maximal rank matrix for which the polynomial map is homogeneous, and then we will show how to compute a homogeneous minimal generating set up to a desired degree. We have implemented our techniques in Macaulay2 and show some examples where our methods can produce many generators of low degree where Groebner methods have failed.

Computational methods, and the generation and analysis of classifications of mathematical data have proved increasingly useful for research in recent years. The database polyDB (polyDB.org) for research data in discrete geometry and related fields aims to support this. With its growing set of collections, mostly in the area of discrete geometry, it intends to make datasets and classifications accessible to researchers independent of a specific software or computer algebra system. The database relies on MongoDB and its drivers for a stable and flexible access to the data. It comes with sevaral options to access the data, e.g. from polymake and OSCAR (via Polymake.jl), and APIs for convenient access in python, Julia, and via a REST interface. In the talk I will introduce polyDB, its design and data model, and some of the interfaces. In a second part I will show small use cases of polyDB in various systems, in particular in polymake and OSCAR.