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: Given a group G and an automorphism phi of G, one can use phi to define a relation on G which is akin to conjugacy. We then say that G has property R-infinity if for every automorphism of G, the determined relation partitions G into infinitely many equivalence classes. In this talk, we will use a result of Levitt--Lustig to show that any finitely presented group with infinitely many ends has property R-infinity. This is joint work with Armando Martino, Wagner Sgobbi, and Peter Wong.
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Abstract: In this talk I will explore the rich interplay between algebraic and analytic methods in the mathematically rigorous formulation of quantum field theory (QFT). Starting from the foundational framework of algebraic quantum field theory (AQFT) (local algebras of observables described using functional analysis and operator algebraic techniques), I will then shift focus to perturbative aspects (perturbative AQFT) where homological algebra and description in terms of formal power series play central role. In both cases, microlocal analysis and propagation of singularities allow one to construct models on a large class of Lorentzian manifolds. I will also illustrate the general mathematical framework with examples from physics.
Abstract: One easy way of representing a knot is via a knot diagram. However, inferring properties of the knot from its diagram and deciding when two diagrams represent the same knot are quite difficult problems. Surprisingly, when the diagram is sufficiently twisty then some structure starts to emerge. I will discuss two results of this nature: hyperbolicity of highly twisted knot diagrams and uniqueness of highly twisted plat diagrams.
Based on joint works with Yoav Moriah, Tali Pinsky and Jessica Purcell. All relevant notions will be explained in the talk.
Abstract: In 1965, Thompson introduced a group V, with subgroups F ≤ T ≤ V, to provide (counter-)examples of finitely presented groups for certain conjectures. Today, they form a very important class of groups with many interesting and uncommon properties. One such property, shown by Brown and Geoghegan in 1982, shows that Fis the first known group of type FP_∞whose cohomology vanishes with coefficients in the group ring ℤF.
In this talk, we will focus on the totally disconnected locally compact analogue of Thompson's F, known asalmost automorphism groups,and building from the work of Sauer and Thurmann, we will show that these almost automorphism groups satisfy similar cohomological properties. This is joint work with Laura Bonn, Bianca Marchionna and Lewis Molyneux.
Abstract: The talk is based on the papers [1,2,3] extending
Topological Data Analysis (TDA) to a wider area of Geometric Data
Science, which aims to continuously parametrise moduli spaces of real
objects under practical equivalences. The key example is a cloud A of
unordered points under isometry in R^n. Standard filtrations
(Vietoris-Rips, Cech, Delaunay) of complexes on A are invariant under
isometry (any distance-preserving transformation). Hence, persistent
homology of these filtrations can be considered a partial solution to
the following geo-mapping problem: design an invariant I of clouds of
m unordered points satisfying the conditions below. (a) Completeness:
any clouds A,B in R^n are related by rigid motion if and only if
I(A)=I(B); (b) Realisability: the invariant space {I(A) for all clouds
A in R^n} is explicitly parameterised so that any sampled value I(A)
can be realised by a cloud A, uniquely under motion in R^n; (c)
Bi-continuity: the bijection from the space of clouds to the space of
complete invariants is Lipschitz continuous in both directions in a
suitable metric d on the invariant space; (d) Computability: the
invariant I, the metric d, and a reconstruction of A in R^n from I(A)
can be obtained in polynomial time in the size of A, for a fixed
dimension n. The talk will outline a full solution to this problem,
which remains open for other data (embedded graphs, meshes, or
complexes) and relations (dilation, affine, or projective maps).
[1] V.Kurlin. Complete and continuous invariants of 1-periodic
sequences in polynomial time. SIAM Journal of Mathematics of Data
Science, 2025.
[2] P.Smith, V.Kurlin. Generic families of finite metric spaces with
identical or trivial 1-dimensional persistence. J Applied and
Computational Topology, v.8, p.839-855 (2024).
[3] D.Widdowson, V.Kurlin. Recognizing rigid patterns of unlabeled
point clouds by complete and continuous isometry invariants with no
false negatives and no false positives. Proceedings of CVPR 2023,
p.1275-1284.
Abstract: The Cuntz semigroup of a C*-algebra A consists of equivalence classes of positive elements, where equivalence means roughly that two positive elements have the same rank relative to A. It can be thought of as a generalization of the Murray von Neumann semigroup to positive elements and is an incredibly sensitive invariant. We give a brief introduction to this object and its relevance to the classification theory of separable nuclear C*-algebras. We then present a calculation of the homotopy groups of these Cuntz classes as topological subspaces of A when A is classifiable in the sense of Elliott.
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