# Lecture series on modified traces in algebra and topology

What?

Monoidal categories with duals have a notion of the trace of a morphism, generalizing the classical traces of matrices. These categorical traces are a well-studied key tool in low-dimensional topology, representation theory and other fields. However, it is often the case that categories of interest are not semisimple and the categorical traces vanish, becoming a trivial concept. The the past 10 years it became clear there exist non-trivial replacements for traces on non-semisimple categories called modified traces. The study of these traces leads to new, interesting quantum invariants of links and 3-manifolds as well as applications in the study of representation theory.

The aim of these two lecture series is to give a reasonably self-contained introduction to modified traces, with an eye on its origin in representation theory and low-dimensional topology.

**Speakers, titles and abstracts**

Jonathan Kujawa (University of Oklahoma) - An introduction to modified traces

The trace of a map and the dimension of a representation are fundamental invariants in representation theory. They are useful both for proving results in representation theory and for applications in other areas (e.g., low-dimensional topology). Unfortunately, in the non-semi-simple setting these invariants are often trivial. For example, this happens in modular representation theory, graded representation theory, and representations of quantum groups at roots of unity. Over the last decade, it became clear that many of these settings admit replacement invariants which can serve much the same role. However, it is still mysterious when exactly they exist.

In this lecture series I will give a friendly introduction to the theory of these invariants with an emphasis on examples, basic properties, and some applications within representation theory.

Marco de Renzi (Universität Zürich) – TQFTs from non-semisimple modular categories and modified traces

Topological Quantum Field Theories (TQFTs for short) provide very sophisticated tools for the study of topology in dimension 2 and 3: they contain invariants of 3-manifolds that can be computed by cut-and-paste methods, and their extremely rich structure induces representations of mapping class groups of surfaces. In recent years, so-called non-semisimple constructions have deeply generalized the standard semisimple approach of Reshetikhin and Turaev to the theory, producing powerful topological invariants and representations of mapping class groups with remarkable new properties. In this series of talks, we will explain how to construct a TQFT starting from an algebraic ingredient called (non-semisimple) modular category, and using the theory of modified traces. We will also describe some features of the invariants and representations issued by this construction.

Based on joint works with A. Gainutdinov, N. Geer, B. Patureau, and I. Runkel.

**Schedule**

All lectures will take place 19:00 local time (that is 5pm UTC), and always Mondays, Tuesdays and Thursdays; the first part of this mini lecture series is scheduled for 12.Oct.2020, 13.Oct.2020 and 15.Oct.2020, the second for 19.Oct. 2020, 20.Oct.2020 and 22.Oct.2020.

Talks are given online and zoom links will be sent to the registered participants in due time. Recordings of the talks will be posted online.

Each talk will be 25 + 5 (break) + 25 minutes plus time for questions.

**Registration**

If you are interested in attending the lecture series please register online here.

**Questions?**

If you have any questions, please feel free to contact any of the organizers:

Nicolle Gonzales, Amit Hazi, Louise Sutton, Daniel Tubbenhauer, Paul Wedrich, Jieru Zhu

**Slides/Videos**

**Jonathan Kujawa (University of Oklahoma) - An introduction to modified traces**

Lecture I

Lecture II

Lecture III

**Marco de Renzi (Universität Zürich) – TQFTs from non-semisimple modular categories and modified traces**

Lecture I

Lecture II

Lecture III