Experimental methods in algebra, geometry and number theory

Daniele Agostini: Groups, automorphisms and Ulrich bundles
To certain groups, or rather to their Lie algebras one can associate naturally a skew-symmetric matrix of linear forms, which encodes the Lie bracket. Geometrically, these matrices are naturally interpreted in terms of so-called Ulrich sheaves on projective hypersurfaces. We use this point of view to study geometrically the automorphism of the original groups. This talk is based on work in progress with Daniele Faenzi, Dragos Fratila, Josh Maglione and Mima Stanojkovski.

Marie-Charlotte Brandenburg: Correlated equilibrium polytopes -- from game theory to combinatorics to varieties and back
Nash equilibria and correlated equilibria are topics of extensive research in economics and game theory. In this talk, we explore these classical notions using the framework of combinatorics and algebraic geometry. In particular, we examine the polytope consisting of all correlated equilibria of a game. Attempting to classify the combinatorial types of these polytopes will lead us to the study of semialgberaic sets, algebraic varieties, and the development of computational proof methods. No prior knowledge of game theory will be assumed in this talk. This talk is based on joint work with Benjamin Hollering and Irem Portakal.

Bettina Eick: The finite $p$-groups of maximal class with 'large' automorphism group
A finite group $G$ of prime-power order $p^n$ has maximal class if its nilpotency class is $n-1$. The $p$-groups of maximal class have been investigated extensively in the literature. For primes $p=2$ and $p=3$ they are fully classified, for prime $p=5$ there is a detailed conjectural classification available, for all primes $p \geq 7$ the classification of $p$-groups of maximal class is wide open.
A finite $p$-group $G$ of maximal class has a 'large' automorphism group if it satisfies $(p-1) \mid |Aut(G)|$. We consider the graph ${\cal G}_p$ associated with such groups: the vertices of this graph correspond one-to-one to the isomorphism types of considered groups and there is an edge $G \rightarrow H$ if $H / \gamma(H) \cong G$, where $\gamma(H)$ is the last non-trivial term of the lower central series.
In this talk we exhibit the structure of ${\cal G}_p$. This structure translates to a broad classification of the $p$-groups of maximal class with 'large' automorphism group.
This is joint work with Heiko Dietrich.

Giles Gardam: Solving semidecidable problems in group theory
Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

Yang-Hui He: The AI Mathematician: From Physics, to Geometry, to Number Theory
We present a number of recent experiments on how various standard machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry, to representation theory, to combinatorics, and to number theory. In particular, we focus on recent AI-assisted discovery of the Murmuration Phenomenon in Arithmetic.

Steffen Müller: Quadratic Chabauty for modular curves
By Faltings' theorem, a curve of genus at least 2 over the rationals has only finitely many rational points. However, computing the set of rational points explicitly for a given curve can be quite difficult. The Quadratic Chabauty method, developed by Balakrishnan and Dogra, is an instance of Kim's nonabelian extension of Chabauty's method that can sometimes be used for this purpose. After introducing this technique, I will discuss the application to several modular curves of interest in the classification of Galois representations of elliptic curves; this is joint work with Balakrishnan, Dogra, Tuitman and Vonk and, more recently, with Balakrishnan, Betts, Hast and Jha.

Eamonn O’Brien: Conjugacy in finite groups
We report on the successful outcome of a project to provide complete theoretical and practical solutions to conjugacy problems in finite classical groups. In particular we can list all classes; construct centralisers; and decide constructively conjugacy between elements of all classical groups. We discuss the implications of this work for the identical problems in finite groups. This is joint work with Giovanni de Franchesci and Martin Liebeck.

Anna-Laura Sattelberger: Combinatorial Correlators
We consider hyperplane arrangements in affine space and study their behavior when moving each hyperplane individually. To such an arrangement, we associate the Mellin integral of the product of the lines, each raised to an individual power. These functions occur as correlator functions in cosmology. Our aim is to encode such functions as holonomic functions in the constant terms of the hyperplanes. To do so, we construct an ideal of annihilating differential operators purely combinatorially from the arrangement. This talk is based on an ongoing project with Anaëlle Pfister.

Emre Sertöz: Computing linear relations between $1$-periods.
I will sketch a modestly practical algorithm to compute all linear relations with algebraic coefficients between any given finite set of $1$-periods. As a special case, we can algorithmically decide transcendence of $1$-periods. This is based on the "qualitative description" of these relations by Huber and Wüstholz. We combine their result with the recent work on computing the endomorphism ring of abelian varieties. This is a work in progress with Jöel Ouaknine (MPI SWS) and James Worrell (Oxford).

Sara Veneziale: Machine learning detects terminal singularities
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.

Nelly Villamizar: An algebraic approach to geometrically continuous spline spaces
In this talk, we will discuss a recent algebraic approach developed to study geometrically continuous spline spaces. In contrast to parametric spline functions, geometrically continuous splines are piecewise polynomial functions defined on a collection of patches that do not necessarily form a partition of a domain but are stitched together through transition maps. These functions are called Gr-splines if, after composition with the transition maps, they are continuously differentiable functions up to order r on each pair of patches with stitched boundaries. This type of spline has been used in computer-aided design and approximation theory to represent smooth shapes with complex topology. To establish an algebraic approach to these functions, we define Gr-domains and establish an algebraic criterion to determine whether a piecewise function is Gr-continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the Gr-spline space. This complex generalizes the Billera-Schenck-Stillman homological complex used to study parametric splines. We show how previous constructions of Gr-splines fit into this new algebraic framework, illustrate how our approach works with concrete examples, and prove a dimension formula for the Gr-spline space in terms of invariants of the chain complex. We also present an algorithm, based on symbolic computation, to construct bases for Gr-spline spaces. This is a joint work with Angelos Mantzaflaris, Bernard Mourrain, and Beihui Yuan.

Christian Wuthrich: Computations of necklaces on elliptic curves
I wish to present a moduli interpretation (via what we call "necklaces" on elliptic curves) for one of the less well known modular curves $X_{nsp}^+(p)$ and explain how one can perform explicit calculations using necklaces.