Mon, 18 May 2026
15:30
L5

The stable Andrews-Curtis conjecture for thickenable group presentations

Marc Lackenby
((Mathematical Institute University of Oxford))
Abstract

The stable Andrews-Curtis conjecture remains one of the most notorious unsolved problems in group theory. It proposes that every balanced presentation of the trivial group can reduced to the standard presentation (with one generator and one relation) using a sequence of simple moves. In my talk, I will focus on group presentations that are ‘thickenable’, which means that their associated 2-complex embeds in a 3-manifold. For such presentations, the stable Andrews-Curtis conjecture is known to hold. In my talk, I will explain how one can also get an explicit exponential-type upper bound on the number of stable Andrews-Curtis moves that are required. This is in sharp contrast to what is known about non-thickenable presentations.

 

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Mon, 04 May 2026
15:30
L5

The prime decomposition fibre sequence for moduli spaces of 3-manifolds

Jan Steinebrunner
(Cambridge)
Abstract
Milnor's prime decomposition theorem states that every oriented 3-manifold M is diffeomorphic can be written as a connected sum of "prime" manifolds in an essentially unique way: M == P_1 # ... # P_n # (S^1 x S^2)^{#g}. This reduces many questions about 3-manifolds to the prime case, but when studying 3-manifolds in families this reduction is not so straightforward. For example, a diffeomorphism of M need not respect the decomposition into prime factors.
I will explain recent joint work with Boyd and Bregman, in which we use a homotopical version of the prime decomposition theorem to describe the classifying space BDiff(M) (the "moduli space" of M) in terms of moduli spaces of the P_i. More precisely, we establish a "prime decomposition fibre sequence" that describes the moduli space in terms of BDiff(P_1 u ... u P_n) and a space of handle-attachments that is amenable to computations. To illustrate this, I will discuss our calculation of the rational cohomology ring of BDiff((S^1 x S^2)#(S^1 x S^2)).
Mon, 27 Apr 2026
15:30
L5

Nilpotent Deformation Theory

Sofia Marlasca Aparicio
((Mathematical Institute University of Oxford))
Abstract

Deformation theory studies how varieties and other algebro-geometric objects vary in families. A central part of the subject is formal deformation theory, where one deforms over an Artinian base; such deformation problems are governed by Lie algebraic models. 

We pose the question of deforming varieties over nilpotent but not necessarily Artinian bases. These turn out to be classified by the same Lie algebraic models plus some topological structure. More precisely, we will consider partition Lie algebras in the category of ultrasolid modules, a variation of the solid modules of Clausen and Scholze that give a well-behaved category akin to topological modules.

To approach this result, we decompose deformation problems into n-nilpotent layers. Each of these layers is individually easier to understand, and is classified by simpler variants of partition Lie algebras.


 

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