Abstract: TBA
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).
Abstract: TBA
Abstract: One notion indicating vastness for a given finitely
presented group G is whether or not G contains a non-abelian
free subgroup. Another, stronger, notion is that of being SQ
universal: every countable group is a Subgroup of a Quotient
of G. We investigate which finitely presented groups are known
to be SQ universal.
Abstract: The essentially-bounded random variables on a probability space form an example of a tracial von Neumann algebra, as $L^\infty(\Omega, \mathbb{P})$ can be viewed as bounded operators acting on $L^2(\Omega, \mathbb{P})$ by pointwise multiplication. This leads one to consider arbitrary tracial von Neumann algebras as essentially-bounded non-commutative random variables on a formal non-commutative probability space, and the field of free probability leverages this point of view to great effect. One common question is of how much information can be learned about a von Neumann algebra from the joint law of a generating tuple: for example, when can a family of freely independent variables generate an algebra with a finite-dimensional direct summand? How does the distribution of a sum of freely independent variables depend on their individual distributions? When can one rule out algebraic relations between a (not necessarily freely independent) set of variables?
In this talk I will give a survey of the field -- including basic definitions and examples -- and some of the tools used to tackle the above questions. I will mention connections to random matrices and more traditional approaches to von Neumann algebra theory. I will conclude with some discussion of recent and ongoing work with David Jekel investigating finite dimensional summands in algebras arising through mixtures of classical and free independence.
Abstract: The representation theory of locally compact groups and operator algebra theory had important interactions very early in their infant years but they began to move in separate directions soon afterwards. Recent years have seen efforts to bring the two theories back together in the fundamental case of reductive groups.
In this expository talk, we will touch upon some of these synergies; the interactions in the theory of unitary induction dating back to the 1970s and the recently discovered interactions in the theory of theta correspondence (aka Howe duality). This is based on joint works with Bram Mesland (Leiden) and with Magnus Goffeng (Lund).
Abstract: Persistent homology is one of the main methods of Topological Data Analysis, a field that studies principled ways of extracting geometric and topological information from data. A fundamental ingredient in the theory of persistent homology are decompositions of certain algebraic objects, which are used to define and compute invariants associated with the data. While based on classical results, decompositions and invariants in Topological Data Analysis are studied from a new metric perspective, with a focus on their stability properties. In this talk, I will discuss the interactions between decompositions, invariants,and metrics in the context of persistent homology, showing some applications to data analysis.
Abstract: Dilation theory provides a framework for understanding operators by realizing them as compressions of larger, better behaved operators. Classical results, such as those of Sz.-Nagy, Stinespring, Akcoglu, and Rota, illustrate how this approach leads to deeper structural insights. In this talk, I will introduce some of the fundamental dilation results and discuss some applications. I will also highlight an obstruction to extending Akcoglu's dilation theorem to the noncommutative setting.
Abstract : Free groups and surface groups are often thought of as cousins. One way to make this precise is through their subgroup structure: they share the property that every subgroup of infinite index is free. In fact, it turns out that this property characterises free and surface groups among cubulated hyperbolic groups, and also among other natural classes including one-relator groups. I will discuss these results and their proofs. The main new tool is the Whitehead complex associated to a convex subcomplex of a CAT(0) cube complex. The Whitehead complex generalises the classical Whitehead graph used to study automorphisms of free groups.
Abstract: In recent years there has been a great deal of interest in detecting
properties of the fundamental group $\pi_1M$ of a $3$-manifold via its
finite quotients, or more conceptually by its profinite completion.
This motivates the study of the profinite completion $\widehat
{\pi_1M}$ of the fundamental group of a $3$-manifold. I shall discuss
a description of the finitely generated prosoluble subgroups of the
profinite completions of all 3-manifold groups and of related groups
of geometric nature.
Abstract: Elliptic Curves are algebraic plane curves defined by certain cubic equations. An elliptic curve is said to have complex multiplication if it has extra symmetries, that is, if its endomorphism ring is larger than the integers. The theory of elliptic curves with complex multiplication has yielded some striking applications, ranging from questions in transcendence theory to Diophantine geometry. This has shed light on many questions related to the arithmetic of imaginary quadratic fields.
For real quadratic fields, similar questions remain much more mysterious. Can one formulate a theory of real multiplication? In this talk, I will discuss a recent (and largely conjectural) p-adic approach via rigid cocycles, certain group cocycles for the action of p-arithmetic groups on p-adic symmetric spaces.
Abstract:Celebrated theorems of Gromov, Trofimov and Coulhon--Saloff-Coste combine to give a remarkable dichotomy for vertex-transitive graphs: such graphs must either resemble highly structured Cayley graphs, or must exhibit expansion in a certain sense. This in turn has had a number of striking applications, particularly to probability, such as Varopoulos's famous characterisation of those transitive graphs on which the random walk is recurrent (i.e. eventually returns to its starting point with probability 1). I will describe a number recent quantitative, finitary refinements of these results that allow us to give meaningful extensions of results like Varopoulos's to finite transitive graphs and finite regions of infinite transitive graphs.
Abstract: In 1996 Gersten proved that if G is a word hyperbolic group of cohomological dimension 2 and H is a finitely presented subgroup (and actually it's enough to assume the algebraic condition FP_2), then H is hyperbolic as well. The definitions of cohomological dimension and FP_2 can be generalised for arbitrary rings. In this talk, I will present a joint work with Robert Kropholler and Vlad Vankov generalising Gersten's result to show that the same is true if G is only assumed to have cohomological dimension 2 over some arbitrary ring R and H is of type FP_2(R).
Abstract: John Heighway invented the fractal dragon curve in 1966 by repeatedly folding a strip of paper. This is a special case of folding curves, further developed by many other authors. I will discuss the L-systems of space filling folding curves and their boundaries, and other aspects of these curves, and different ways they can be constructed.
Abstract: In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between asmoothing, which is a type of surgery on a curve that resolves a self-intersection, andk-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay.
Abstract: https://drive.google.com/file/d/108I_nL2BZE85NxFeT3_Qv5F0v2zFph20/view?usp=sharing
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.
Abstract: TBA
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.