**Title:** Topology of continuous cluster categories

**Speaker:** Kiyoshi Igusa

**Speaker Info:** Brandeis

**Brief Description:**

**Special Note**:

**Abstract:**

Cluster categories are triangulated additive categories which ``categorify'' cluster algebras. The simplest examples are of type $A_n$ and $D_n$. Continuous cluster categories are topological triangulated categories which are basically given by taking the limit as $n$ goes to infinity and completing. For type A the space of isomorphism classes of indecomposable objects forms an open Moebius strip. However, the definition of continuous triangulation implies that there are an even number of objects in every isomorphism class. As an exercise in topology, I will classify all possible topological triangulated categories which are equivalent to the continuous cluster category of type A and have only finitely many objects in every isomorphism class.In this talk, I will review basic concepts in representation theory of quivers and the algebraic theory of cluster categories, in particular how the triangulated structure of the category is used in cluster mutation. Then I will explain why embedding these discrete categories into topological categories can be useful. I will concentrate on type A, but I will also say a few words about type D and cluster categories of surface type.

This talk is based on joint work with Gordana Todorov, some of which is on arXiv.

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