**Tom Brady** (Dublin City University, Ireland)

*Title:* Triangulating the permutahedron [1]

*Abstract:* For an Artin group A(W) of finite type W, we construct a homotopy equivalence
from the A(W) classifying space of Salvetti to the one defined by noncrossing partitions.
The construction involves the type-W asssociahedron.
This is joint work with Emanuele Delucchi and Colum Watt.

**Frédéric Chapoton** (University of Strasbourg, France)

*Title:* Representation-theoretic aspects of cluster theory [1-2] [3]

*Abstract:* Since the introduction of cluster theory by Fomin and Zelevinsky, it is clear that the combinatorics of associativity naturally fits in this general framework as the case of type A in the classification of finite Dynkin diagrams. Some time after its introduction, some aspects of the theory of cluster algebras were better understood using representations of quivers. The lectures will introduce this point of view on cluster algebras, starting from basics of quiver representations. Then we will describe the cluster categories and the cluster characters that allows to describe individual cluster variables using Euler characteristics of Grassmannians of submodules. The last part of the lectures will be devoted to results about the representations of the Tamari lattices, which is another story intertwining associativity and representation theory.

**Patrick Dehornoy** (University of Caen, France)

*Title:* Associativity, Thompson's group *F*, and the Tamari lattices [1] [2] [3]

*Abstract:* Essentially following the original approach of Richard Thompson in the 1960s, we shall explain the connection between the associativity law and Thompson's group *F*: for every algebraic law (or family of algebraic laws), there exists an associated "geometry monoid" that describes a partial action of the law(s) on abstract terms; in good cases (such as the case of associativity), the geometry monoid projects without loss to a group, which is *F* in the case of associativity, and *V* in the case of associativity plus commutativity. In the case of *F*, the orbit of the partial action of *F* on a size *n* term is the 1-skeleton of the *n*th Stasheff associahedron.

This approach naturally leads to a presentation of the geometry monoid/group, consisting of trivial (quasi)commutations plus specific relations reflecting the geometry of the considered law(s): as can be expected, the Mac Lane-Stasheff pentagons (resp. pentagons and hexagons) appear in the case of *F* and *V*, and proving that the involved relations make a presentation amounts to establishing a coherence result and can be done by internalizing the action on terms.

It turns out that, in the case of *F*, the above geometrical presentation consists of positive relations only, so it leads to also considering the associated presented monoid. One checks that *F* is a group of fractions for this monoid, and that the latter admits least common multiples and greatest common divisors: thus, its Cayley graph is a lattice, namely the union of all Tamari lattices.

For each of the above points, comparing associativity with other algebraic laws, like the self-distributivity law *x(yz)=(xy)(xz)*, whose geometry monoid is connected with Artin's braid groups, will help to understand what is general and what is specific.

**Stefan Forcey** (University of Akron, USA)

*Title:* Clades and tubes: facets of graph associahedra and phylogenetic polytopes. [1] [2] [3]

*Abstract:* The faces of graph associahedra correspond to certain topological bases called graph tubings. This fact inspired us to look for analogous faces of the balanced minimal evolution polytope. We begin by defining the concepts of tubes, clades, phylogenetic trees; and our two families of polytopes. One sort of polytope is related to well known graphical zonotopes -- but there is an open question about its connection to brick polytopes. The other is related to NP-hard problems.

Next we will describe the faces and facets of the balanced minimal evolution polytope that were recently discovered. Similarly to graph associahedra, there are faces which have the form of smaller dimensional examples from the same family -- an operad-like structure. There are also faces which correspond to certain phylogenetic networks. We will define split-networks and show that some of these do correspond to facets of the BME polytope. The open problem is to finish this list.

Finally we will look at two sorts of algebra. First there is linear algebra--actual algorithms for linear programming over the polytopes and their relaxations. Then there is abstract algebra: some (Hopf) algebraic interpretations of combining tubings to make new larger ones.

**Mikhail Kapranov** (Kavli IPMU, Japan)

*Title: * Associativity and higher Segal structures [1-3]

*Abstract: * The concept of 2-Segal simplicial objects, built around the
combinatorics of triangulations of coonvex polygon, is intimately
related to associativity issues. The mini-course will cover the following topics:

- Segal and 2-Segal conditions, associativity and triangulations. Hall algebras and their modules. Waldhausen groupoids and their 2-Segal property.
- Waldhausen spaces of abelian and pre-triangulated dg-categories. Their 2-Segal nature.
- Cyclic and paracyclic 2-Segal objects. Their relation to geometry of surfaces. Paracyclic structure on the Waldhausen spaces of pre-triangulated categories.
- The relative Waldhausen space of a functor. Its 2-Segal nature and relation to semi-orthogonal decompositions in triangulated categories.
- Spherical functors, perverse sheaves and perverse schobers on surfaces.

**Yuri I. Manin** (Max Planck Institute for Mathematics, Germany)

*Title:* Arithmetic and geometry in characteristic 1. [1] [2]

*Abstract:* In the first part of mini-course, I will give a survey
of various constructions related to the "arithmetic in characteristic 1"
stressing those constructions that did not yet find their way
into a systematic theory.

In the second part of the mini-course, I will explain
in some detail constructions of geometric categories
(schemes, pro-schemes etc.), based principally upon
recent works by Sh. Haran and A. Connes with C. Consani.

**Martin Markl** (Mathematics Institute of the Czech Academy of Sciences, Czech Republic)

*Title: *Distributive laws between the Three Graces (with Murray Bremner)

*Abstract:* Motto: All algebras are equal, but some algebras
are more equal than others.

Experience teaches us that the most common classes of
algebras are the Three Graces - associative, commutative
associative, and Lie - together with other that combine these in
a specific way. The most prominent example of a combined
structure are Poisson algebras which are combinations of Lie and
commutative associative algebras by means of a quadratic
homogeneous distributive law.

Our aim was to investigate whether the commonly known
combinations of the Three Graces are the only possible ones via
such a distributive law. The answer turned out, rather
surprisingly, no.

In my talk I will present results of our on-going work
including examples of "exotic" structures combining the Three
Graces.

Our research was facilitated by advances in computer-assisted
mathematics, and in particular the computer algebra system Maple
worksheets written by the first author expressly for this project
extended hand calculations of the second author dating from some
20 years ago.

**Vincent Pilaud** (École Polytechnique, France)

*Title: * Three new friends of the associahedron [1-3]

*Abstract:*
Originating from early works of Dov Tamari and Jim Stasheff, the associahedron is a fascinating mathematical object that can be considered under several perspectives, in particular from the point of view of lattice theory and polyhedral geometry. It has motivated several relevant generalizations, including secondary polytopes, generalized associahedra (in connection to finite type cluster algebras), graph associahedra, brick polytopes, etc. These lectures will explore combinatorial and geometric aspects of three recent generalizations of the associahedron:

- Permutreehedra. The permutrees are combinatorial objects in between permutations, binary trees and binary words. They give rise to relevant lattices generalizing the weak order, the Tamari lattice and the boolean lattice, to polytopes that interpolate between the permutahedron, the associahedron and the cube, and to Hopf algebras containing the Malvenuto-Reutenauer algebra, the Loday-Ronco algebra, and the descent algebra.
- Quotientopes. The work of Nathan Reading showed that lattice quotients of the weak order have strong combinatorial and geometric structures. In particular, he provided a very convenient combinatorial model for lattice quotients of the type A weak order in terms of non-crossing arc diagrams and he proved that any lattice quotient of the weak order is realized by coarsening the Coxeter fan according to the congruence classes of the quotient. In turn, these properties can be used to construct polytopal realizations for all lattice quotients of the type A weak order.
- Non-kissing associahedra. Recently introduced on the grid by Thomas McConville, non-kissing complexes provide combinatorial models for support-tau tilting complexes on arbitrary gentle quivers. Generalizing the associahedron, they have rich lattice theoretic properties and nice polyhedral realizations.

**Jim Stasheff** (University of North Carolina at Chapel Hill, USA)

*Title: * Associahedra to infinity and beyond [1]

*Abstract: * I will indulge in recollections of some of the history (and pre-history) of the associahedra, as I lived it . I will mention as well some of its progeny, its generalizations and applications.