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).
Groups model symmetry. But that is not the end of the story. Related but distinct structures capture aspects of symmetry beyond the reach of groups. In this talk, I will overview the algebraic theory of racks and quandles and their homological and homotopical aspects. This theory has been rediscovered many times, most often in the context of knot theory, but its reach goes well beyond that, as I will show with a broad range of examples.
Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. At the end we will mention explicit computations for some arithmetic groups and mapping class groups in terms of class numbers and special values of zeta functions. This is all joint with Wolfgang Lück and Stefan Schwede.
Polyhedral products are a topological space formed by gluing together ingredient spaces in a manner governed by a simplicial complex. They appear in many areas of study, including toric topology, combinatorics, commutative algebra, complex geometry and geometric group theory. A fundamental problem is to determine how operations on simplicial complexes change the topology of the polyhedral product. In this talk, we consider the substitution complex operation. We obtain a description of the loop space associated with some substitution complexes, and use this to build a new family of simplicial complexes such that the homotopy type of the loop space of the moment angle complex is a product of spheres and loops on spheres.
Barwick's theorem of the heart asserts that the connective algebraic K-theory of certain stable infinity-categories, namely those admitting a bounded t-structure, can be described in terms of purely algebraic data. Antieau, Gepner and Heller proved a generalisation of Barwick's theorem and posed two conjectures about the general behaviour of non-connective algebraic K-theory on such stable infinity-categories. The first of these conjectures was subsequently disproved by Neeman, and recent joint work of myself with Ramzi and Sosnilo answers their second conjecture in the negative as well.
After recalling the description of algebraic K-theory in terms of its universal property and formulating Barwick's theorem, I will explain the input that is required to produce counterexamples to the conjectures of Antieau, Gepner and Heller.
I will describe some recent work on Topological Data Analysis and then focus on joint work with David Mendez on directed persistent homology. Namely, we develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach. Our directed persistent homology theory is motivated by homology with semiring coefficients.
I will explain some recent progress on understanding mapping class groups of 4-manifolds, in both the smooth and topological categories.
Tom Brown and I gave a construction of an uncountable family of groups of type FP using graphical small cancellation, but *not* using Bestvina-Brady Morse theory. I shall discuss our construction and a recent extension of it.