*Fourth annual conference of the SFB-TRR 195*took place

**virtually**on

**Sept. 22–24, 2020.**The goal of the conference was to bring together people from different areas in algebra who apply computational methods in their research. The main organizer is

**Ulric**

**h**

**Thiel**(TU Kaiserslautern), see contact details below.

**Slides**and

**recordings**can be found in the list of talks below.

### Impressions

### Conference Portal

Rooms are accessible via the **conference portal**. Access code was sent to registered participants.

### Speakers

#### From outside the SFB-TRR

**Philippe Biane,** Université Paris-Est

**Tim Dokchitser,** University of Bristol

**Olivier Dudas,** Université Paris Diderot

**Gavril Farkas,** Humboldt-Universität zu Berlin

**Lars Thorge Jensen**, University of Clermont Auvergne

**Eric Katz**, The Ohio State University

**Michael Stillman,** Cornell University

**Nicolas Thiéry,** Université Paris Sud

#### From within the SFB-TRR

**Simon Brandhorst,** Universität des Saarlandes

**Claus Fieker**, TU Kaiserslautern

**Anne Frühbis-Krüger**, Universität Oldenburg

**Pascal Schweitzer**, TU Kaiserslautern

**Andrea Thevis**, Universität des Saarlandes

**Gabriela Weitze-Schmithüsen**, Universität des Saarlandes

### Schedule

You can find the schedule **here** (PDF).

### Talks

**Philippe Biane,** Université Paris-Est

**Free Lévy processes**

There is a notion of processes with free increments in free probability theory which parallels that of processes with independent increments in classical probability. When one tries to characterize those which are time homogeneous (like Lévy processes in classical

probability theory) one finds that there are two natural classes: those with homogeneous increments and those with homogeneous transition probabilities. I will explain how these classes are parameterized by convex sets of analytic functions on the upper half-plane.

**Tim Dokchitser,** University of Bristol

**Exact p-adic computation [Slides]**

I will review available packages for p-adic numbers and their extensions in computer algebra systems, focussing on the recent one in Magma by Christopher Doris. It provides lazy exact arithmetic, and is the first native implementation for provable p-adic computations.

**Olivier Dudas,** Université Paris Diderot

**Computing decomposition numbers for finite unitary groups **

(work in progress with R. Rouquier) In this talk I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. I will first explain how one can produce a “natural” self-equivalence in the case of $\mathrm{GL}_n(q)$ coming from the topology of the Hilbert scheme of $\mathbb{C}^2$. The combinatorial part of this equivalence is related to Macdonald’s theory of symmetric functions and gives $(q,t)$-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence.

**Gavril Farkas,** Humboldt-Universität zu Berlin

**Green’s Conjecture via Koszul modules**

Using ideas from geometric group theory we provide a novel approach to Green’s Conjecture on syzygies of canonical curves. Via a strong vanishing result for Koszul modules we deduce that a general canonical curve of genus g satisfies Green’s Conjecture when the characteristic is zero or at least (g+2)/2. Our results are new in positive characteristic (and answer positively a conjecture of Eisenbud and Schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Joint work with Aprodu, Papadima, Raicu and Weyman.

**Lars Thorge Jensen**, University of Clermont Auvergne

**How to teach a computer the Elias-Williamson graphical calculus? [Slides]**

Based on recent work of Achar-Makisumi-Riche-Williamson, one can calculate tilting characters of a reductive algebraic group

in positive characteristic p using the p-canonical/p-Kazhdan-Lusztig basis of the anti-spherical module. To calculate the p-Kazhdan-Lusztig basis one needs to calculate ranks of certain Hom-pairings (called intersection forms) in the diagrammatic Hecke category.

Unfortunately, string diagrams are well suited for calculations by hand, but how does one approach this problem using a computer?

I will explain an algorithm to calculate the p-canonical basis of the anti-spherical module (joint work with Geordie Williamson),

which allows to replicate the 10 months calculations Williamson performed for his Billiards conjecture in a couple of days.

**Eric Katz**, The Ohio State University

**p-adic integration on bad reduction hyperelliptic curves [Slides]**

Coleman’s theory of p-adic integration is important for finding rational and torsion points on curves. An algorithm for computing Coleman integrals on good reduction hyperelliptic curves was introduced in work of Balakrishnan, Bradshaw, and Kedlaya and was refined by many others. We discuss work with Enis Kaya on extending this algorithm to bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich–Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. We discuss how to compute Berkovich–Coleman integrals and then related them to abelian integrals by using tropical geometric techniques.

**Yue Ren**, Swansea University

**Tropical varieties of neural networks [Slides]**

In this talk, we introduce tropical varieties arising from neural networks with piecewise linear activations, and discuss how their geometry affects their expressivity. In particular, we will use Weibel’s f-Vector Theorem to derive optimal bounds for single-layered maxout networks, and Speyer’s f-Vector theorem to analyse networks with heavily restricted weights. We conclude with an initializing strategy for maxout networks based on our results.

**Nicolas Thiéry,** Université Paris Sud

**Categories, axioms, constructions in SageMath: Modeling mathematics for fun and profit**

The SageMath systems provides thousands of mathematical objects and tens of thousands of operations to compute with them. A system of this scale requires some infrastructure for writing and structuring generic code, documentation, and tests that apply uniformly on all objects within certain realms. In this talk, we describe the infrastructure implemented in SageMath. It is based on the standard object oriented features of Python, together with mechanisms to scale (dynamic classes, mixins, …) thanks to the rich available semantic (categories, axioms, constructions). We relate the approach taken with that in other systems (e.g. GAP), and discuss open problems.

This is meant as a basis for discussions: how are the equivalent challenges tackled in Oscar? Is there ground for crossfertilization!

**Michael Stillman,** Cornell University

**Computational algebraic geometry, applications in string theory, and Macaulay2**

In this talk we describe some open problems and interesting questions in computational algebraic geometry motivated by investigations in string theory, whose solutions would be useful to researchers in string theory. We describe recent work done in collaboration with Liam McAllister, Cody Long, Andreas Braun and others in this domain, and we keep it concrete by giving examples using Macaulay2.

**Simon Brandhorst,** Universität des Saarlandes

**Equations for the K3-Lehmer map [Slides]**

The dynamical complexity of an automorphism of a complex surface is measured by its topological entropy. The entropy is the logarithm of a Salem number, that is, a real algebraic integer $\lambda>1$ which is conjugate to $1/\lambda$ and all whose other conjugates lie on the unit circle. Conjecturally the smallest Salem number is Lehmer’s number $\lambda_{10}$. Lehmer’s conjecture is true for entropies: $\log(\lambda_{10})$ is the minimum among entropies of automorphisms of complex surfaces.

In a series of papers McMullen proved the existence of such an automorphism on a complex *projective* K3 surface. His strategy combines ideas from integer programming with the theory of lattices, number fields and reflection groups. The final step of the proof relies on the Torelli-type Theorem for K3 surfaces which is non constructive. In this talk I present equations for this automorphism. To find it we used Kneser’s neighbor method, elliptic fibrations and their linear systems, finite non-symplectic automorphisms and p-adic lifting.

This is joint work in progress with Noam D. Elkies.

**Claus Fieker**, TU Kaiserslautern

**OSCAR: current, plans and dreams [Slides]**

OSCAR is the next generation computer algebra system developed in the central software project of the SPP/ TRR 195. Written in the (rather young) Julia language, it combines the features and capabilities of Singular, Gap and Polymake with the newly created number theory package Hecke.

In this talk I will showcase some achievements, give an overview over the current status and indicate our current (immediate) plans and projects. I will conclude with some dreams about the longterm prospects.

**Johannes Flake**, RWTH Aachen

**PBW deformations of smash products and computer algebra**

PBW deformations of smash products form large classes of algebra deformations which include graded affine Hecke algebras, rational Cherednik algebras, symplectic reflection algebras, current Lie algebras, and many more of your favorite algebraic objects as special cases. I will explain how they can be studied from a general point of view, why some people find them interesting, why I find them interesting, and how I think computer algebra can help everybody to understand them better.

**Anne Frühbis-Krüger**, Universität Oldenburg

**Zeta-functions, p-adic integrals and simultaneous monomialization**

In this talk, we will discuss an approach to a class of order-zeta-functions through computation of p-adic integrals, whose domain of integration is rather far from being monomial. To render these integrals accessible to explicit computation, they need to be split up into a sum of simpler integrals. In our case the case distinction and simplification rely on a specific variant of resolution of singularities to obtain monomial condition on the domains of integration.

**Laura Maaßen**, RWTH Aachen

**Interpolating Partition Categories [Slides]**

In this talk we introduce tensor categories which interpolate the representation categories of partition quantum groups, which we view as subcategories of Deligne’s interpolation categories $\underline{\mathrm{Rep}(S_t)}$ for the symmetric groups. We compute the set of interpolation parameters yielding semisimple interpolation categories for all group-theoretical quantum groups, an uncountable family containing all but countably many partition quantum groups. A crucial ingredient is an abstract analysis of certain subobject lattices developed by Knop, which we adapt to categories of partitions.

We go on to present a parametrisation of the indecomposable objects for non-zero interpolation parameter using the representation theory for partition quantum groups developed by Freslon and Weber. This yields a description of the associated graded rings of the Grothendieck rings.

This is joint work with Johannes Flake.

**Pascal Schweitzer**, TU Kaiserslautern

**Computing Symmetries: Isomorphism, Automorphism and Canonization of** **Graphs versus other Combinatorial Objects**

The graph isomorphism problem, which asks for the existence of an isomorphism between two given finite input graphs, is known to be equivalent to the problem of computing automorphisms of graphs. One commonly applied method to solve this problem is via canonization.

In this talk I will relate the isomorphism problem of graphs to that of computing symmetries of other combinatorial objects. In fact there are general techniques to reinterpret finite combinatorial objects as graphs, while preserving their symmetry structure. This makes the graph isomorphism problem universal for isomorphism and automorphism problems of explicitly given combinatorial objects.

However, for implicitly given objects, such as those given by generating sets, the matter is different. Describing joint work with Daniel

Wiebking, the talks explains that a unified view of implicit combinatorial objects gives improved algorithms. This is in particular

the case for canonization algorithms and this view has subsequently also found applications in the computation of normalizers for permutation groups.

**Andrea Thevis**, Universität des Saarlandes

**$p$-Origamis: Strata, Veech Groups and Sums of Lyapunov Exponents [Slides]**

In this talk, we study a certain class of translation surfaces called $p$-origamis. These surfaces arise as normal covers of the torus with $p$-groups as deck transformation group. We classify the types of singularities of p-origamis and show that these depend in most cases only on the isomorphism class of the deck transformation group. For this, we use the rich theory of $p$-groups.

Veech groups of $p$-origamis are finite index subgroups of SL(2,Z) and capture a lot of information about the respective surfaces. We describe first results regarding Veech groups of $p$-origamis. Using these results, we compute the sum of Lyapunov exponents for certain example series of $p$-origamis.

This is partially joint work with Johannes Flake.

**Gabriela Weitze-Schmithüsen**, Universität des Saarlandes

**Systoles on Origami Translation Surfaces [Slides]**

A finite translation surface is a closed surface X together with the choice of a holomorphic differential. The moduli space of translation surfaces of genus g is stratified by the orders of the zeroes of the differentials.

Although translation surfaces have been intensively studied since the 1980’s, there are natural questions which are still wildly open. One of these questions is: Does there exist in each stratum a translation surface with a maximal systolic ratio, i.e. with a maximal shortest curve relative to the area. We study this question in the stratum H_2(1,1) of genus 2 surfaces with two zeroes of order 1. This is joint work with Columbus, Herrlich and Mützel.

### Confirmed Participants

**Total number: 107**

Firoozeh Aga, Saarland University

Aslam Ali, TU Kaiserslautern

George Balla, RWTH Aachen

Mohamed Barakat, University of Siegen

Reimer Behrends, TU Kaiserslautern

Marc Bellon, CNRS et Sorbonne Université

Dominik Bernhardt, RWTH Aachen

Philippe Biane, CNRS, LIGM Université Paris Est

Janko Böhm, TU Kaiserslautern

Jendrik Brachter, TU Kaiserslautern

Simon Brandhorst, Saarland University

Jens Brandt, RWTH Aachen

Sofia Brenner, Friedrich-Schiller-Universität Jena

Thomas Breuer, RWTH Aachen University

Eirini Chavli, Universität Stuttgart

Michael Cuntz, Leibniz Universität Hannover

Wolfram Decker, TU Kaiserslautern

Tim Dokchitser, University of Bristol

Gérard Duchamp, Université Paris 13

Olivier Dudas, CNRS and Université de Paris

Holger Eble, TU Berlin

Kurusch Ebrahimi-Fard, Norwegian University of Science and Technology

Gavril Farkas, Humboldt-Universität zu Berlin

Claus Fieker, TU Kaiserslautern

Johannes Flake, RWTH Aachen

Anne Frühbis-Krüger, Universität Oldenburg

Nan Gao, Shanghai University

Sabrina Gaube, UOL / LUH

Meinolf Geck, University of Stuttgart

Christoph Goldner, Tübingen University

Daniel Gromada, Universität des Saarlandes

Pierre Guillot, Université de Strasbourg

Melanie Harms, RWTH Aachen University

William Hart, TU Kaiserslautern

Jonas Hetz, University of Stuttgart

Tommy Hofmann, Universität des Saarlandes

Johannes Hoffmann, Universität des Saarlandes

Max Horn, TU Kaiserslautern

Jens Hubrich, TU Kaiserslautern

Lars Thorge Jensen, Université Clermont Auvergne

Birte Johansson, TU Kaiserslautern

Pooja Joshi, IISER Bhopal

Kunda Kambaso, RWTH Aachen

Lars Kastner, TU Berlin

Eric Katz, The Ohio State University

Enis Kaya, University of Groningen

Hanieh Keneshlou, IMPAN

Markus Kirschmer, Universität Paderborn

Matthias Klupsch, RWTH Aachen

Michael Kunte, TUK

Caroline Lassueur, TU Kaiserslautern

Vladimir Lazić, Universität des Saarlandes

Felix Leid, Saarland University

Viktor Levandovskyy, RWTH Aachen

Benjamin Lorenz, TU Berlin

Frank Lübeck, RWTH Aachen

Laura Maaßen, RWTH Aachen University

Antonio Macchia, Freie Universität Berlin

Verity Mackscheidt, RWTH Aachen

Gunter Malle, TU Kaiserslautern

Dario Mathiä, University of Kaiserslautern

Aleksander Morgan, RWTH Aachen

Gabriele Nebe, RWTH Aachen University

Alice Niemeyer, RWTH Aachen

Emily Norton, TU Kaiserslautern

Gyan Datt Panday, PRSU, Prayagraj

Pierre-Guy Plamondon, Université Paris-Saclay

Sebastian Posur, RWTH Aachen

Parisa Pourghobadian, Teheran University

Ludwig Rahm, Norwegian University of Science and Technology NTNU

Iryna Raievska, Institute of Mathematics of National Academy of Sciences of Ukraine

Maryna Raievska, Institute of Mathematics of National Academy of Sciences of Ukraine

Yue Ren, Swansea University

Lukas Ristau, TU Kaiserslautern

Liam Rogel, TU Kaiserslautern

Emil Rotilio, TUK

Mahsa Sayyary Namin, MPI Leipzig

Johannes Schmitt, TU Kaiserslautern

Leonard Schmitz, RWTH Aachen

Hans Schönemann, TU Kaiserslautern

Mathias Schulze, TU Kaiserslautern

Pascal Schweitzer, TU Kaiserslautern

Farideh Shafiei, Institute for Research in Fundamental Sciences (IPM)

Vishal Shankhaval, Ganpat University

Farrokh Shirjian, Tarbiat Modares University

Carlo Sircana, TU Kaiserslautern

Roland Speicher, Saarland University

Mima Stanojkovski, Max-Planck-Institut fuer Mathematik in den Naturwissenschaften

Michael Stillman, Cornell University

Bernd Sturmfels, MPI Leipzig

Jayantha Suranimalee, TU Kaiserslautern

Melis Tekin Akcin, Hacettepe University

Nicolas Thiéry, Université Paris Sud

Ayush Kumar Tewari, TU Berlin

Andrea Thevis, Universität des Saarlandes

Ulrich Thiel, TU Kaiserslautern

Paul Vater, MPI MiS Leipzig

Claude Viallet, Centre National de la Recherche Scietntifique / Sorbonne Université

Laura Voggesberger, TU Kaiserslautern

Maria Walch, DFKI/IAV/TUK

Moritz Weber, Saarland University

Yvonne Weber, Technische Universität Kaiserslautern

Gabriela Weitze-Schmithüsen, Universität des Saarlandes

Oguzhan Yürük, TU Braunschweig

Eva Zerz, RWTH Aachen

Yinan Zhang, Australian National University