# The GEARS seminar

The Glasgow Edinburgh Algebra Research Student seminar

The Glasgow Edinburgh Algebra Research Student seminar is an informal meeting between algebra (loosely interpreted!) PhD students and postdocs from Edinburgh, Heriot-Watt, and Glasgow universities. We meet roughly five times per year and give participants the opportunity to either speak about their research, or present an important paper in their area. These meetings normally take place in the late afternoon/evening and the location alternates between Glasgow and Edinburgh.

The GEARS seminar is currently organised by Tuan Pham (Edinburgh), Parth Shimpi (Glasgow), and Oli Jones (Heriot–Watt). For previous organisers, please see the relevant tabs.

We are grateful for the financial support from: the Glasgow Mathematical Journal Learning and Research Support Fund, the Edinburgh Researcher Development Fund, the Heriot-Watt Small Project Grant Scheme, and the EPSRC Programme Grant "Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And Deformations."

## Upcoming Meetings

GEARS Meeting 17th October 2024

Date: 17th October 2024

Location: Room 337, St Andrew's Building, 11 Eldon Street, Glasgow

Speakers:

13:30-14:20: Julia Bierent

Title: Irregular flat connections and Stokes structures

Abstract: Systems of solutions of irregular flat connections are in 1-1 correspondence with Stokes local systems. This is an irregular version of the regular Riemann-Hilbert correspondence between regular flat connections and local systems. In order to open it to irregular flat connections, we need to define Stokes structures. This will lead to the definition of Stokes filtered local systems, which are local systems on the punctured surface with filtrations that are locally constant when they don't cross some particular directions (the Stokes directions). The stack of Stokes local systems is then very interesting to study. We will motivate and explain the irregular Riemann-Hilbert correspondence and the Stokes structures it implies.

14:30-15:20: Gabriel Corrigan

Title: Realising virtual cohomological dimension of automorphism groups of RAAGs

Abstract: In 1986, Culler & Vogtmann introduced 'Outer space' - a complex upon which Out(F_n), the outer automorphism group of a free group, acts properly. This had many applications; one is that the dimension of the so-called 'spine' of Outer space is precisely the virtual cohomological dimension (VCD) of Out(F_n). More recently, Charney-Stambaugh-Vogtmann constructed an 'untwisted Outer space' - an analogous space used for studying the group of untwisted automorphisms of a right-angled Artin group. However, in a departure from the free group case, sometimes the dimension of this rather natural 'untwisted spine' is larger than the VCD of the corresponding group of outer automorphisms! In this talk I will present work examining this phenomenon. I present graph-theoretic conditions under which we can perform an equivariant deformation retraction of the untwisted spine to produce a new complex which geometrically realises the VCD of the group of untwisted automorphisms. As a corollary, this proves that the gap between the VCD of the untwisted subgroup and the dimension of the untwisted spine can be arbitrarily large.

16:00-16:50: Francesco Tesolin

Title: Generalizing the non-commutative Stone Duality to a wider class of morphisms.

Abstract: There are many examples of dualities between categories of topological spaces and partially ordered sets. Generalizing the original Stone representation theorem of Boolean algebras, Kudryavtseva and Lawson have shown that Boolean inverse monoids are equivalent to Boolean groupoids. In joint work with Lawson, based on that of Kudryavtseva, we generalize this duality to include a wider class of morphisms. These include the unusual coherent relational covering morphisms between Boolean groupoids.

I will include an introduction to semigroups, topological groupoids, and the classical Stone duality.

Photos from our meetings: Dec 2023 (Edinburgh) and September 2023 (Glasgow)