# TTT116: Online

Hosted by Liverpool

## Date

13 May 2022

## Speakers

### Thomas Huettemann (Queens' University, Belfast)

Having fun with K-theory

Having fun with K-theory

Abstract: In the first part of the talk, I will recall some applications of the algebraic K-groups K_0 and K_1 of a ring in algebra, topology and geometry, starting with rather elementary and entertaining linear algebra. In the second part, I will discuss a variation of the so-called fundamental theorem which (in essence) identifies K_0 and K_1 of a ring R as K_1 of the Laurent polynomial ring R[t, 1/t].

The variation involves embedding R as the degree-0 component of an arbitrary strongly Z-graded ring. The graded approach yields a more general splitting theorem for algebraic K-theory, applicable to twisted Laurent polynomial rings and Leavitt path algebras of finite graphs without sink.

### Matt Booth (Lancaster University)

Global Koszul duality

Global Koszul duality

Abstract: Koszul duality is the name given to various duality phenomena between differential graded algebras and coalgebras involving the bar and cobar constructions. For example, the above functors give a Quillen equivalence between augmented dgas and coaugmented conilpotent dgcs. There is a similar statement for modules and comodules, where the equivalence is given by twisting.

In this talk, I'll survey the above landscape before talking about to what extent one can remove the word 'conilpotent' from the above theorems; this is the 'global' setting of the title. This is a report on ongoing joint work with Andrey Lazarev.

### Ximena Fernandez (Durham University)

Morse theory for group presentations and the persistent fundamental group

Morse theory for group presentations and the persistent fundamental group

Abstract: Discrete Morse theory is a combinatorial tool to simplify the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will present a refinement of this theory that guarantees not only a homotopy equivalence with the Morse CW-complex, but also a Whitehead simple homotopy equivalence. Moreover, it provides an explicit description of the attaching maps of the critical cells in the simplified complex and bounds on the dimension of the complexes involved in the deformation.

This result provides the suitable theoretical framework for the study of different problems in combinatorial group theory and topological data analysis. I will show an application of this technique that allows to prove that some potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture. Moreover, the method can also be extended to filtrations of CW-complexes, providing an efficient algorithm for the computation of the persistent fundamental group of point clouds in terms of group presentations.

This is joint work with Kevin Piterman.

Fernandez, X. Morse theory for group presentations. arXiv:1912.00115.