Cluster Structures
Emine Yıldırım, University of Leeds
Cluster algebras are in the nexus of many mathematical and physics phenomena such as total positivity, configurations spaces, algebraic varieties, representation theory of algebras, integrable systems in physics and many more. Briefly, cluster algebras are a class of commutative algebras defined recursively forming clusters and certain exchange relations. Thus, they are very combinatorial in nature. We will present the definition of cluster algebras and investigate some of their connections to other fields in the series of talks.
Combinatorics of Markov Numbers
Ezgi Kantarcı Oğuz, Galatasaray University
We will start by introducing the Markov Diophantine equation. We will see how to generate new solutions from known ones via Vieta jumps and show all solution triples occur in the Markov tree exactly once. We will introduce Frobenius’s Uniqueness conjecture and discuss partial process. Then we will move on to different combinatorial models: Christoffel words, Cohn matrices, snake graphs, lattice paths and posets. We will also look at connections to cluster algebras.
Representation theory of quivers
Francesca Fedele, University of Leeds
Any finite dimensional algebra over an algebraically closed field k corresponds to some directed multigraph, called a quiver, with relations. Conversely, starting with a quiver, one can construct an associative (unitary) k-algebra, that is finite dimensional under certain assumptions. This approach to studying the representation theory of algebras is very powerful as it gives ways to visualise algebraic structures using combinatorial methods.
In this series, we will explore this connection and the link between the representations of quivers and the modules of the corresponding algebras. We will also relate these constructions to some of the topics discussed in the other series of talks, such as mutation of quivers in cluster algebras.
Combinatorics of q-deformations
Kağan Kurşungöz, Sabancı University
A q-deformation of a formula is a way to introduce a new variable to the formula where we can recover the original by setting q to a predetermined value. In this series of lectures, we will start with introducing the classical q-deformation for integers. We will discuss their properties and applications in counting. We will also look at cases where the power of q denotes a particular statistic, such as the area under a lattice path.
Quantum Groups
Léa Bittmann, Université de Strasbourg
Quantum groups were introduced in the 70s to build non trivial solutions to the Yang-Baxter equation (also called the braid relation - the relation between consecutive permutations in the symmetric group). These objects have since then grown into their own subject of research, with many applications in different area of mathematics. One of these applications is directly linked to the original construction: because representations of quantum groups satisfy the braid relation, they can be represented using tangle diagrams. As a further application, they can be used to compute knot invariants, and so help answer the question: can this knot be unknotted?
Frieze Patterns
Matthew Pressland, Université de Caen‐Normandie
Frieze patterns were introduced and developed by Coxeter and Conway in the 80s, who presented them as a kind of combinatorial game, whereby the player attempts to fill a grid with positive integers obeying certain rules. It turns out, however, that frieze patterns appear naturally in a range of mathematical problems from a number of different areas. For example, in geometry, a frieze pattern represents a positive integer valued point in a decorated Teichmüller space, or in the totally positive Grassmannian. In representation theory, the entries in a frieze pattern count submodules of quiver representations. In this series of lectures, we will look at some of these different interpretations of frieze patterns and the connections between them.