One hundred and sixteenth meeting of
Transpennine Topology Triangle
(link to be circulated via the mailing list)
hosted by the University of Liverpool
Friday 13th May 2022
The speakers will be Matt Booth (Lancaster University), Ximena Fernandez (Durham University) and Thomas Huettemann (Queens’ University, Belfast).
The talks will be in the afternoon, starting at 14:00 UK time (=BST).
If you'd like to attend, but are not on the mailing list, please
contact Sarah Whitehouse
s.whitehouse (\a\t) sheffield.ac.uk
14:00 Thomas Huettemann (Queens’ University, Belfast)
"Having fun with K-theory"
15:00 Matt Booth (Lancaster University)
"Global Koszul duality"
16:00 Tea break
16:30 Ximena Fernandez (Durham University)
"Morse theory for group presentations and the persistent fundamental
Speaker: Thomas Huettemann (Queens’ University, Belfast)
Title: 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.
Speaker: Matt Booth (Lancaster University)
Title: 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.
Speaker: Ximena Fernandez (Durham University)
Title: 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
Everyone who wishes to
participate is welcome, particularly
Last updated: 6 May 2022