Topology Seminar talks at Southampton

The talks are all happening at University of Southampton (in person), on Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( d (dot) kasprowski (at) soton.ac.uk ) if you have questions.

More Pure Maths talks at Southampton: we run Pure Maths Colloquia, Pure Lunchtime Seminar, and Pure Postgraduate Seminar (link soon).

Upcoming talks

• Mon 12 May 2025 14:00 (in 54 / 10031 (10C)): Robin Stoll (Cambridge): TBA
  No abstract yet.

Recent talks

• Mon 24 Mar 2025 14:00: Lukas Brantner (Oxford): Formal integration of derived foliations
  unroll abstract

  Frobenius' theorem in differential geometry asserts that every involutive subbundle E ⊂ TM of the tangent bundle of a manifold M integrates to a decomposition of M into smooth leaves. We prove an infinitesimal analogue for suitably nice schemes X. More precisely, we show that every partition Lie algebroid g → T_X integrates uniquely to a formal moduli stack X → S under S, where S is the formal leaf space and the fibres of X → S are the formal leaves. This talk is based on joint work with Magidson-Nuiten, and ties into work of Jiaqi Fu.

• Mon 17 Mar 2025 14:00: Sebastian Chenery (Bristol): Gyration Stability for Projective Planes
  unroll abstract

  Gyrations are operations on manifolds that first arose in geometric topology. A given manifold M may exhibit different gyrations depending on the chosen twisting, prompting the following natural question: do all gyrations of M share the same homotopy type regardless of which twisting we choose? Inspired by recent work of Duan, which demonstrated that the quaternionic projective plane is not gyration stable (but with respect to diffeomorphism) in this talk we will explore our question for projective planes in general, resulting in a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes up to homotopy. Moreover, we will also see that these results connect to several seemingly distinct contexts. This is joint work with Stephen Theriault.

• Mon 10 Mar 2025 14:00: Stephen Theriault (Southampton): Loop spaces of n-dimensional Poincare Duality complexes whose (n-1)-skeleton is a co-H-space
  unroll abstract

  Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincare Duality complexes of dimension n whose (n − 1)-skeleton is a co-H-space. This unifies many known decompositions obtained in different contexts and establishes many new families of examples. As consequences, we show that such a looped Poincare Duality complex retracts off the loops of its (n − 1)-skeleton and describe its homology as a one-relator algebra. This is joint work with Lewis Stanton.

• Mon 3 Mar 2025 14:00: Andrea Guidolin (Southampton): Koszul complexes and relative homological algebra of persistence modules
  unroll abstract

  In topological data analysis (TDA), functors from a poset category to vector spaces, called persistence modules, are a central object of investigation. Persistence modules can arise from data via several constructions, and often encode complex geometric information. Finding informative and computable descriptors of persistence modules is crucial to extract this information and use it in data analysis.

  A recent trend in TDA is to describe persistence modules over finite posets via relative homological invariants. Relative homological algebra extends constructions of standard homological algebra by redefining the notion of projective module, which depends on the choice of a family of basic modules. In this talk, we study relative resolutions of persistence modules. Under certain conditions, the multiplicities of the basic persistence modules in a relative resolution are unique, and define invariants called relative Betti diagrams. Using Koszul complexes, relative Betti diagrams can be computed in a simple and local way, avoiding the computation of the entire relative resolution.

  The talk is based on joint work with Wojciech Chachólski, Isaac Ren, Martina Scolamiero, and Francesca Tombari.

• Mon 17 Feb 2025 14:00: Daniel Galvin (MPIM Bonn): An obstruction theory for fillings of 3-manifolds
  unroll abstract

  It is a classical result, due to Milnor, that every spin 3-manifold spin bounds a 4-manifold. If we ask for this null-bordism to be of a prescribed normal 1-type, i.e. its stable normal bundle has a prescribed Postnikov-Moore approximation, then it may no longer exist. We fix this data on a given 3-manifold and describe a three stage geometric obstruction theory for the existence of a filling which extends this structure. Our main contribution is a new `tertiary' obstruction which is defined using Wall's equivariant self intersection number. This is joint work with Peter Teichner and Simona Veselá.

• Mon 10 Feb 2025 14:00: Jérôme Scherer (EPFL): Module spaces over homology theories and completion towers
  unroll abstract

  This is joint work with W. Chacholski and W. Pitsch. To any coaugmented functor X=>EX one can associate its modules, i.e. spaces X which are retracts of EX. When E is ordinary homology, this functor is the infinite symmetric product and modules are generalized Eilenberg-Mac Lane spaces (GEMs). Given two such functors E and F we introduce a new one, we denote by a bracket [E, F], as a pullback of two natural transformations E=>EF and F=>EF. We study modules over such brackets and construct two towers by bracketing systematically either on the left or the right. For ordinary homology we obtain two well-known towers: the Bousfield-Kan completion tower and the modified one as introduced by Dror Farjoun.

• Mon 3 Feb 2025 14:00: Jan Steinebrunner (Cambridge): Moduli spaces of 3-manifolds with boundary are finite
  unroll abstract

  In joint work with Rachael Boyd and Corey Bregman we study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. I will recall this B Diff(M), which is also called the moduli space of M, and explain how it parametrises smooth families of manifolds diffeomorphic to M.

  Using Milnor's prime decomposition and Thurston's geometrisation conjecture we can cut M into geometric pieces, for which we have a better understanding of B Diff. The purpose of this talk is to explain a technique for computing the moduli space B Diff(M) in terms of the moduli spaces of the pieces. We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex, as was conjectured by Kontsevich.

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