Talks
Janko Boehm
Recent Advances in Open Source Computer Algebra
In this short lecture series, we will focus on some recent trends in the development of open source computer algebra software. One focus will be on the combination of the computer algebra system Singular for commutative algebra and algebraic geometry with the Petri net based workflow management system GPI-Space to create a convenient, easy to use framework for massively parallel computations in computer algebra. Another focus will be the integration of Singular in the new, interdisciplinary open source computer algebra system OSCAR. We will illustrate these developments through examples arising from recent research topics, and discuss practical aspects in tutorial sessions.
Martin Bohnert
The Fine interior of three-dimensional lattice polytopes with width 2 and half-integral polygons
We investigate three-dimensional lattice polytopes with width 2 and their half-integral "middle"-polygons with the help of the Fine interior. In particular we calculate the possible Chern numbers for minimal surfaces of general type obtained from nondegenerate toric hypersurfaces, whose Newton polytope is a three-dimensional lattice polytope with few interior lattice points and two-dimensional Fine interior.
Gavin Brown
Working with flops and Fanos
I plan to explain various computer experiments with 3-fold flops as motivation for discussing or explaining a selection of algorithms we use along the way. This includes analysis of very particular quasiprojective varieties described by affine charts, consequences of Groebner basis in associative algebras, and a type of formal noncommutative singularity theory. As well as solutions, the area has a lot of scope for wider projects and open problems, and we may discuss ideas for those too. The experiments are part of joint work with Michael Wemyss. If time permits, I will also talk about the current state and ambition of the Graded Ring Database, for anyone who does not already know it.
Ana García Pulido
On the non-existence of low dimensional sympathetic Lie algebras
Sympathetic Lie algebras are Lie algebras which share similar properties to semisimple Lie algebras. There are results describing some of the structural properties of these algebras. However, these results have proved insufficient when it comes to classifying, or even proving the existence, of these algebras in low dimensions. In this talk, I will begin by introducing sympathetic Lie algebras. I will then present our recent results and computational methods that allow us to address the classification of a large family of low dimensional sympathetic Lie algebras. This work is in progress and joint with Gil Salgado.
Elana Kalashnikov
Quantum cohomology, Schubert calculus, and mirror symmetry for type A flag varieties
In this series of talks, I’ll survey different approaches to mirror symmetry for type A Grassmannians and flag varieties. I’ll start by reviewing the toric picture, which will give us a guide for what we can hope for in the case of non-Abelian GIT quotients like Grasssmannians and flag varieties. I’ll then discuss the role of toric degenerations in producing conjectural mirrors, with a focus on the oldest construction: the Eguchi—Hori—Xiong mirror of the Grassmannian and its generalizations. I’ll then discuss two other approaches to mirror symmetry for the Grassmannian, which is the best understood case: the Abelian/non-Abelian correspondence and the Plucker coordinate mirror approach. Along the way, I’ll review quantum Schubert calculus in type A, and explain the role of the Abelian/non-Abelian correspondence in quantum cohomology. I’ll conclude the lectures by discussing the Plucker coordinate mirror of the flag variety.
Marta Panizzut
Computing in tropical geometry
Many exciting research topics lie at the interface between polyhedral and algebraic geometry creating fruitful grounds for new computational methods. Tropical geometry has recently fueled these interactions, providing a systematic framework to study degenerations of algebraic varieties. In the course, I will introduce main computational tools to tropicalize varieties defined over valued fields. Specific examples will focus on polyhedral computations at the core of the study of curves and surfaces through tropical lenses. Computational examples and exercises will be discussed using the recent implementation of tropical geometry in the computer algebra system Oscar.
Eric Pichon-Pharabod
Computing periods of hypersurfaces
A period of a complex algebraic surface is the integral of a holomorphic 2-form on a homological 2-cycle. Such numbers carry important information about the geometry of the hypersurface (see for instance Torelli-type theorems), and arise in theoretical physics (Feynman integrals). We provide a direct method to efficiently carry out computations of numerical approximations of these periods. More precisely, our method relies on finding an explicit description – which stems from Picard-Lefschetz theory – of the 2-cycles in a way that is suitable for integration. In this talk, I will describe some elements of this method and give a demonstration of my code. This work is in progress and joint with my advisors P. Lairez and P. Vanhove.
Karin Schaller
Nobodies are perfect, semigroups are not
NObodies are asymptotic limits of certain valuation semigroups. Their construction depends on a given flag of subvarieties. We investigate toric surfaces together with non-toric flags and determine when the associated valuation semigroups are finitely generated. This is a joint work with K. Altmann, C. Haase, A. Küronya, and L. Walter.
Emre Sertöz
Computing limit mixed Hodge structures
Consider a smooth family of varieties over a punctured disk that is extended to a flat family over the whole disk, e.g., consider a 1-parameter family of hypersurfaces with a central singular fiber. The Hodge structures (i.e. periods) of smooth fibers exhibit a divergent behavior as you approach the singular fiber. However, Schmid's nilpotent orbit theorem states that this divergence can be "regularized" to construct a limit mixed Hodge structure. This limit mixed Hodge structure contains detailed information about the geometry and arithmetic of the singular fiber. I will explain how one can compute such limit mixed Hodge structures in practice and give a demonstration of my code.
Sara Veneziale
Data analysis and machine learning in algebraic geometry
The use of machine learning and data analysis techniques in pure mathematics to help formulate conjectures has been a growing research area, with examples in knot theory and representation theory. In this talk, we go through two successful examples of such applications in algebraic geometry, where toric varieties are the central object of study. In the first, we study the quantum period of toric varieties, and see if we can learn their dimension. In the second, we construct a neural network that can distinguish between terminal and non-terminal toric varieties from their GIT data, which motivates a proposed mathematical solution to the problem.