Pure Mathematics Colloquium talks at Southampton

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).

Upcoming talks

• Fri 2 May 2025 15:00 (in Building 54, 10037(10B)): Haluk Sengun (Sheffield): Old and new synergies between representation theory and operator algebras
  unroll abstract

  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).

• Fri 9 May 2025 15:00 (in Building 54, 10037(10B)): Ian Charlesworth (Cardiff): Title(TBA)
  unroll abstract

  Abstract: TBA

• Fri 16 May 2025 15:00 (in Building 54, 10037(10B)): Jack Button (Cambridge): Title(TBA)
  unroll abstract

  Abstract: TBA

Recent talks

• Fri 28 Mar 2025 15:00: Andrea Guidolin (Southampton): Decompositions, invariants, and metrics in persistent homology
  unroll abstract

  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.

• Fri 21 Mar 2025 15:00: Safoura Zadeh (Bristol): Dilation theory in operator algebra
  unroll abstract

  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.

• Fri 14 Mar 2025 15:00: Henry Wilton (Cambridge): Surface groups among cubulated groups
  unroll abstract

  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.

• Fri 7 Mar 2025 15:00: Pavel Zalesskii (University of Brasília): Prosoluble subgroups of the profinite completion of 3-manifold groups
  unroll abstract

  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.

• Fri 28 Feb 2025 15:00: Alice Pozzi (Bristol): From Complex to Real Multiplication: a p-adic story
  unroll abstract

  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.

• Fri 21 Feb 2025 15:00: Matthew Tointon (Bristol): Structure, expansion and probability in vertex-transitive graphs
  unroll abstract

  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.

• Fri 14 Feb 2025 15:00: Shaked Bader (Oxford): Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings
  unroll abstract

  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).

• Fri 7 Feb 2025 15:00: Helena Verrill (Warwick): Folding curves and Fractal dragons
  unroll abstract

  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.

• Fri 31 Jan 2025 15:00: Macarena Arenas (Cambridge): Taut smoothings and shortest geodesics
  unroll abstract

  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.

• Fri 10 Jan 2025 15:00: Runlian Xia (Glasgow): Hilbert transforms for groups acting on right-angled buildings
  unroll abstract

  Abstract: https://drive.google.com/file/d/108I_nL2BZE85NxFeT3_Qv5F0v2zFph20/view?usp=sharing

• Fri 6 Dec 2024 13:00: GGSE
  unroll abstract

  https://www.ucl.ac.uk/~ucahllo/ggse/

• Fri 29 Nov 2024 15:00: Jane Turner (Southampton): Combinatorial Relative Algebraic K-theory
  unroll abstract

  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.

• Fri 22 Nov 2024 15:00: Kostas Karagiannis (Manchester): CANCELED
  unroll abstract

  Abstract: TBA

• Fri 15 Nov 2024 15:00: Kevin Boucher (Southampton): On affine actions of negatively curved groups at the critical index
  unroll abstract

  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.

• Fri 8 Nov 2024 15:00: Chris Bruce (Newcastle): Maximal ideals of reduced group C*-algebras
  unroll abstract

  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.

• Fri 1 Nov 2024 15:00: Steve Lester (KCL): The distribution of lattice points
  unroll abstract

  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.

• Fri 25 Oct 2024 15:00: Alessandro Vignati (Université de Paris Cité): Roe type algebras, their isomorphisms, and (some) set theory
  unroll abstract

  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.

• Fri 18 Oct 2024 15:00: David Singerman (Southampton): From Farey fractions to Farey maps
  unroll abstract

  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.

• Fri 11 Oct 2024 15:00: André Henriques (Oxford): ''What kind of thing is... Rep(G)?''
  unroll abstract

  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).

• Fri 4 Oct 2024 16:00: Jonathon Hare (Southampton, ECS): How can mathematics help us design better learning machines and understand what they've learned?
  unroll abstract

  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.

Archive