You may have noticed we are running a series of short films on the maths behind popular games (card, board, digital, nothing is off limits). So we want contributors for the following games plus any ideas you have of your own.
Poker, Blackjack, Roulette, Chess, Go, Bridge, Monopoly, Tsuro, Carcasonne, Cathedral, Minecraft, Catan, Ticket to Ride, Saboteur, Projective Noughts and Crosses, Projective Set, Splendour, Minesweeper, Backgammon, etc.
Mon, 04 May 2026
15:30
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
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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