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The talks are all happening at University of Southampton (some virtually, some in person), Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( s (dot) nishikawa (at) soton.ac.uk ) for a meeting link.
We also run a few seminars: Pure Lunchtime Seminar, Topology Seminar, and Pure Postgraduate Seminar (link soon).
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Abstract: The algebraic K-groups of an exact functor F: M \to N between exact categories are classically defined as the homotopy groups of the homotopy fiber of the induced map on the K-theory spaces of M and N. In 2016, Daniel Grayson produced a conjectural presentation of these K-groups using a certain category built of chain complexes. In this talk, I will present work proving that these descriptions agree.
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Abstract: One approach to studying infinite groups is through their actions on various classes of spaces. Recent research on affine actions of groups on Banach spaces has been particularly active, with significant applications in areas such as combinatorics, ergodic theory, smooth dynamics, operator algebras, and the Baum-Connes conjecture. In 1989, Pansu initiated the study of the $L^p$-cohomology of negatively curved groups acting on $L^p$ spaces. This work was later extended by Yu, Bourdon, Valette, Cornulier, Tessera, and Nica, who established general results concerning the existence of proper affine actions on $L^p$ spaces for large values of $p$, in relation to certain dimensional invariants. After providing a brief introduction to the subject and an overview of the existing research, I will present an original result in this direction.
Abstract: A group is said to be C*-simple if its reduced C*-algebra is simple. A celebrated result by Kalantar and Kennedy says that a group G is C*-simple if and only if the action of G on its Furstenberg boundary is free. I will present on recent work in which we extend this result, proving that even the maximal ideal structure of the reduced C*-algebra of a discrete group is governed by the action of G on its Furstenberg boundary: given a point x in the Furstenberg boundary of G, we prove that there is a bijection between maximal ideals of the reduced C*-algebra of G and maximal co-induced ideals of the C*-algebra of the stabiliser subgroup of x. Interestingly, our result reduces the problem of computing maximal ideals in reduced group C*-algebras to computing ideals in C*-algebras of amenable group. This is joint work with Kevin Aguyar Brix, Kang Li, and Eduardo Scarparo.
Abstract: The distribution of lattice points is a classical topic that lies at the intersection of number theory with harmonic analysis and also has connections to spectral theory and mathematical physics. A central problem in this area is the Gauss circle problem, which is to determine the number of integer lattice points that lie within a circle with a large radius. In this talk I will discuss the distribution of lattice points lying in circles with large radii and will also describe some recent results. The talk is aimed at a broad mathematical audience and no prior knowledge on the topic will be assumed.
Abstract: To a metric space X we associates a family of C*-algebras capable of coding some of the large scale geometry of X; these are called Roe like algebras, defined for the first time by John Roe for index theoretic purposes. We study the following question: if two Roe like algebras associated to X and Y are isomorphic, what can be said about the geometries of X and Y? We provide results about some of these C*-algebras and their quotients, where answers to the above question are shown to depend on the set theoretic ambient.
Abstract: The first part of the talk is very elementary; it is just about fractions! We go back to a problem raised by John Farey in 1815 which led to Farey sequences of rationals. This led to some theorems about these sequences proved by Cauchy fourteen years later. These theorems led me to construct the Farey map which turns out to be the universal triangular map.
If we consider Farey fractions mod n we get some finite triangular maps such as the icosahdreon and Klein's map on a surface of genus 3.
In Doha Kattan's Southampton Ph.D.thesis of 2019 she constructed universal q-gonal maps for q>3 and we then studied and drew a particular finite 4-gonal map associated to Bring's surface, which is a surface of genus 4 similar to Klein's surface.
Abstract: What kind of mathematical object is the category of representations of a group? The answer depends on what kind of group we're considering: finite/algebraic /topological/ etc...
In all cases, after having axiomatised the kind of categories that look like Rep(G), a surprisingly fruitful question is to ask whether there are other categories that satisfy all the same axioms, but are not of the form Rep(G).
Abstract: This talk will perhaps be a little different to the normal topics covered in the Pure Math Colloquium. The objective is to foster collaboration and new ideas. I'll set out the key mathematical ideas behind modern machine learning (specifically deep learning and deep neural networks) as I see it, and highlight some of the big open research questions with an emphasis on how a deeper mathematical insight could help. I'll focus on the two aspects that I work on with my machine learning colleagues and students: (a) the functional design of the learning machine; and (b) understanding what that machine has learned, particularly with an eye to to desired properties of generalisation and robustness. I'll cover some examples of our own work including the design of contractive recurrent networks that can learn algorithms on simple tasks to later solve harder problems by iterating for longer, and work towards building better tools for explaining what a network has learned.