# Combinatorics Day

## Abstracts for the talks

**Alex Fink ***Queen Mary University of London***Universal Tutte characters via combinatorial coalgebras**

The Tutte polynomial is a favourite invariant of matroids and graphs. So when one is working in a generalisation of these settings, for example arithmetic matroids or ribbon graphs, it is a tempting question to ﬁnd a counterpart of the Tutte polynomial; answers have been given in many cases. Our work uniﬁes these answers, providing general machinery to turn a combinatorial object with ”minor” operations into a coalgebra, and from that coalgebra extract the most general possible Tutte-like invariant. We build on earlier work by Krajewski, Moﬀatt, and Tanasa, who used Hopf algebras for this purpose.

Joint with Clément Dupont and Luca Moci.

**Christian Haase ***Freie Universität Berlin***The Kingman Coalescent as a Density on a Space of Trees**

Randomly pick n individuals from a population and look at their genealogy. The Kingman n-coalescent is a probabilistic model for the tree one obtains this way. It can be described by a probability density function on the space of equidistant trees with its fine fan structure, given by the Bergman fan of the complete graph.

I will describe this density and report on work in future with Lena Walter about relations of population genetics and algebraic geometry via the tropical connection.

**Nils Haug ***Medizinische Universität Wien***Asymptotics and scaling properties of two-dimensional lattice paths and polygons**

We consider the generating functions of different models of two-dimensional lattice polygons, weighted with respect to their area and perimeter. Many of them are expected to show similar asymptotic features which do not depend on the exact details of the model, such as the existence of a scaling limit, which is described by the Airy function. For rooted self-avoiding polygons, this scaling behaviour has been conjectured by Guttmann, Richard and Jensen. However, a proof of it seems to be currently out of reach. One can make progress by considering models which are analytically more tractable, such as area-weighted Dyck paths. In this talk I will show how a uniform asymptotic expression for their area-perimeter generating function can be obtained by using the method of steepest descents, generalised to the case of two coalescing saddle points. Afterwards I will explain how a new type of asymptotic behaviour can be observed by adding a perturbation to the area-perimeter generating function of Dyck paths, leading to a model which we call deformed Dyck paths.

**Christian Stump ***Technische Universität Berlin***Science fiction for Catalan and Shi arrangements**

Catalan and Shi arrangements for finite Weyl groups and their discrete-geometric structure have been considered in many contexts in Algebraic Combinatorics. Unfortunately, they have resisted from being generalized to more general finite reflection groups so far. I will start this talk with recalling in detail this discrete-geometric structure, and will then discuss surprising numerical observations in the module of derivations of a hyperplane arrangement that indicate that we still miss the "correct" combinatorics to study Catalan and Shi arrangements for general reflection groups. This talk can be seen as a combinatorial guidance of joint work with Torsten Hoge, Toshiyuki Mano and Gerhard Roehrle.

**Günter Ziegler ***Freie Universität Berlin***New Proofs from THE BOOK**

Two new proofs for very old theorems, prepared for the sixth edition of "Proofs from THE BOOK"

-- Fermat's Two Squares (after Heath-Brown and Spivak)

-- Euler's Series (after Borwein & Borwein)

[Joint Work with Martin Aigner]

## Date and Location

**May 23, 2018**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Amanda Cameron**

MPI für Mathematik in den Naturwissenschaften**Xue Liu**

MPI für Mathematik in den Naturwissenschaften

## Administrative Contact

**Saskia Gutzschebauch**

MPI für Mathematik in den Naturwissenschaften

Contact by Email