**Haus Bergkranz · Kleinwalsertal · Austria · March 18-23**

A “coworkspace” is a place where people come together to work together or simply side-by-side.
The goal of our “combinatorial coworkspace” is to give a group of motivated young and promising as well as established mathematicians such a place to explore new directions, applications, cooperations, and alliances within combinatorics and beyond.
A context is provided by a suitable number of tutorials and lectures and the beautiful location encourages to take mathematical thoughts outdoors.

*Participation is by invitation only.*

General introduction to a topic, potentially including exercises.

**Thomas Lam***Total positivity and the amplituhedron*

Total positivity is the study of real matrices all of whose minors are positive. These matrices were first systematically studied in the 1930s in the works of Gantmacher, Krein, and Schoenberg. Total positivity has attracted the attention of combinatorialists due to the connections with planar networks (Lindstrom and Gessel-Viennot), the relation with canonical bases and cluster algebras (Lusztig and Fomin-Zelevinsky), and the generalizations to Grassmannians and flag varieties (Lusztig and Postnikov). In my lectures I will focus on the totally positive Grassmannian, which has a potent analogy with the theory of convex polytopes. We will discuss the combinatorics and geometry of the amplituhedron, a Grassmannian analogue of a cyclic polytope that was defined by Arkani-Hamed and Trnka in the study of the physics of scattering amplitudes. I won't assume any knowledge of physics, but we should see some hints of the relation between particle scattering and the amplituhedron.

The tutorial will be accompanied by exercise sessions led by**Amanda Schwartz**

Overview of a research topic and its connections to geometric, topological and algebraic combinatorics.

**Jan Draisma***Algebra and well-quasi-orders*

Theorems about well-quasi-orders play an important role in Noetherianity proofs for large algebraic structures. Conversely, such Noetherianity proofs lead to new well-quasi-orders. I will discuss more classical material on this interplay, mostly following Sam-Snowden's ideas on Groebner categories (e.g. algebra over FI and the Artinian conjecture); some applications in algebraic statistics (e.g. Noetherianity for iterated toric fibre products); and some newer results (e.g. the tensor restriction theorem over finite fields).**Vincent Pilaud***Quotientopes*

This talk will survey the combinatorial and geometric theory of lattice quotients of the weak order, and if time permits, explore some further topics such as quotients of the s-weak order, semilattice quotients, and quotients of intervals of the weak order.**Irem Portakal***Combinatorics of equilibria in game theory*

Nash 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 nonlinear algebra and convex geometry. In particular, we examine the set of correlated equilibria, which is a convex polytope in the probability simplex. We study its combinatorial types for small games and highlight potential future research directions. This is a joint work with Benjamin Hollering and Marie-Charlotte Brandenburg.**Petra Schwer***CAT(0) groups and cube complexes*

This talk provides an overview on algebraic, combinatorial and geometric properties of cube complexes and the groups acting on them. We will introduce Gromov's link condition, talk about cubulation of half-space systems and boundary constructions and highlight some properties of CAT(0) groups.

Be prepared to work yourself (under supervision) on the topic of the hands-on session. Starting with an introduction, the goal for everyone is to use your own computer to play and experience the session's topic.

**Lukas Kühne***Hands on ... AI*

Methods based on artificial intelligence are ubiquitous in our lives today. It is therefore not surprising that**AI methods have been successfully applied to mathematical research problems**, see for example here, here and here. In this session, we will briefly cover the**basics of a neural network**before discussing its applications and limitations. In the second part,**everyone is invited to train a simple neural network**to learn, for example, the number of descents of a permutation. No previous knowledge is required. We will work with a Python notebook similar to this one that you can run online without installing anything.**Moritz Firsching***Hands on ... LEAN*

Lean is a language based on dependent type theory, which can be used to express formal proofs. Its most extensive mathematical library, mathlib, contains many theorems and definitions and theorems from various areas of math.**The plan for this hands-on session:**(1) Understand why it might be beneficial to use a theorem prover and how it works. (2) Demo the current capabilities of Lean/mathlib and its ecosystem. (3) Get your hands dirty by trying it out yourself: you'll have the opportunity to prove a lemma!**Prerequisites:**We don't assume any familiarity with Lean/mathlib.**However, you will need some way to run Lean on your computer.**There are different options: (1)**The best way**: Install lean on your computer. (2)**The fast way**: Try lean online. (3)**Another way**: Use gitpod. If you are new to Lean/mathlib, a fun way to begin is by playing the the natural number game!

Research talk, keeping in mind the diverse audience.

- Tobias Boege:
*The Ingleton inequality for random variables* - Marie Brandenburg:
*Separating points by piecewise linear functions -- The real tropical geometry of neural networks* - Cesar Ceballos:
*Generalized Heawood Graphs and Triangulations of Tori* - Nathan Chapelier:
*Étale elements in Coxeter groups* - Ansgar Freyer:
*Unimodular Valuations Beyond Ehrhart* - Constantin Ickstadt:
*Semidefinite Games* - Katharina Jochemko:
*Weighted Ehrhart theory* - Martina Juhnke:
*cd-indices of simplicial manifolds* - Mario Kummer:
*New results on combinatorics of stable polynomials* - Léo Mathis:
*Zonoid algebra and application* - Mateusz Michalek:
*Interactions of intersection theory and tensors - from permutohedral variety to complete colineations* - Leonie Mühlherr:
*A separator-based approach to generating the module of logarithmic derivations of graphical hyperplane arrangements and applications* - Sophie Rehberg:
*Acyclotopes* - Martin Winter:
*Wachspress Coordinates - a Bridge between Algebra, Geometry and Combinatorics*

WhoIsWho, EveningTalks, OpenMic

- Carlos Amendola
- Gennadiy Averkov
- Aenne Benjes
- Benjamin Biaggi
- Tobias Boege
- Marie Brandenburg
- Paul Breiding
- Justus Bruckamp
- Cesar Ceballos
- Nathan Chapelier
- Clement Cheneviere
- Michael Cuntz
- Jan Draisma
- Tarek Emmrich
- Moritz Firsching
- Ansgar Freyer
- Sofía Garzón
- Zoe Karoline Geiselmann
- Christian Haase
- Ben Hollering
- Elena Hoster
- Constantin Ickstadt
- Aryaman Jal
- Katharina Jochemko
- Martina Juhnke-Kubitzke
- Thomas Kahle
- Viktória Klász
- Andreas Kretschmer
- Mario Kummer
- Joris Köfler
- Lukas Kühne
- Thomas Lam
- Georg Loho
- Felix Lotter
- Dante Luber
- Leo Mathis
- Mateusz Michalek
- Leonie Mühlherr
- Elke Neuhaus
- Benjamin Nill
- Vincent Pilaud
- Irem Portakal
- Germain Poullot
- Sophie Rehberg
- Paco Santos
- Raman Sanyal
- Amanda Schwartz
- Petra Schwer
- Rainer Sinn
- Johanna Steinmeyer
- Jan Stricker
- Christian Stump
- Bernd Sturmfels
- Martin Winter
- Francesca Zaffalon