Soton Pure Maths Colloq

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 ( c (dot) y (dot) mak (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 26 Apr 2024 15:00 (in 54/7033): Markus Steenbock (Vienna): Subgroups and diversity of left-orderable small cancellation groups

We discuss left-orderability of groups and its relation to other group properties, such as the unique product property. We pass on to left-orders on the free group and discuss combination theorems that help construct negatively curved groups that are left-orderable. We will argue that there are continuum many such small cancellation groups up to quasiisometry that are not locally indicable. Further, every finitely generated group is a quotient of a left-orderable small cancellation group by a finitely generated subgroup (Rips construction)

Fri 3 May 2024 15:00 (in 54/7033): Anna Felikson (Durham): Polytopal realizations of non-crystallographic associahedra

An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildrim.

In the first half of the talk, I will sketch the history of the associahedron and introduce generalised associahedra, then in the second half we will discuss how to produce the associahedra in the non-crystallographic case. The talk will not require any special background.

Fri 10 May 2024 15:00 (in 54/7033): Francesco Fournier Facio (Cambridge)
No abstract yet.

here

Recent talks

Fri 22 Mar 2024 15:00: Noah Porcelli (Imperial): Floer homotopy and smooth structures

One large class of symplectic manifolds of interest is cotangent bundles: for any closed manifold Q, T*Q is naturally a symplectic manifold. If Q and Q' are diffeomorphic, their cotangent bundles are isomorphic symplectic manifolds. A classical question is to ask whether the converse is true: if T*Q and T*Q' are isomorphic, are Q and Q' diffeomorphic? I'll discuss some progress in this direction (also obtained independently via different methods by Abouzaid--Alvarez-Gavela--Courte-Kragh) as well as some generalisations outside the case of cotangent bundles.
I won't assume any symplectic background in this talk. This is based on joint work with Ivan Smith.

Fri 15 Mar 2024 15:00: Yuhan Sun (Imperial): Symplectic cohomology and Lagrangian torus fibrations

We will start with a gentle introduction of symplectic cohomology, reviewing its importance in dynamics, symplectic topology and mirror symmetry. Then we discuss a covering formula to compute it when our symplectic manifold admits a Lagrangian torus fibration. The main examples are in real dimension 4 with nodal fibers, called symplectic cluster manifolds. This is joint with U.Varolgunes.

Fri 8 Mar 2024 15:00: Arman Darbinyan (Southampton): Quantification of residually finite groups

A method of quantifying residually finite groups is via residually finiteness growth (RFG), which was introduced by Bou Rabee and henceforth generated much interest within group theory. While much research is done towards understanding RFG, many natural questions still remain open. My talk will address some of those questions. In particular, in my talk, I will discuss a construction of a family of residually finite groups with a very diverse and refined set of RFG functions. In particular, I will discuss the first examples of residually finite groups with intermediate RFG.

Additionally, I will discuss how to show that RFG is not a quasi-isometry invariant and, as a by product, how to derive answers to some open questions of related nature.

Fri 1 Mar 2024 15:00: Sam Fisher (Oxford): Kaplansky's zero divisor conjecture via embeddings into division rings

A famous conjecture often attributed to Higman or Kaplansky predicts that group algebras of torsion-free groups have no zero divisors. In this talk, we will survey what is known about this conjecture, with an emphasis on group algebras that embed into division rings. We will also present new results, which confirm the zero divisor conjecture for all (twisted) group algebras of torsion-free 3-manifold groups and virtually compact special groups. Joint with Pablo Sánchez-Peralta.

Fri 23 Feb 2024 15:00: Emanuele Dotto (Warwick): Characteristic polynomials of self-adjoint endomorphisms

The algebraic properties of the characteristic polynomial of a matrix can be efficiently packaged by expressing the characteristic polynomial as a ring homomorphism from the cyclic K-theory ring to the ring of Witt vectors. This homomorphism can moreover be interpreted as the effect in pi_0 of a homotopy theoretic trace map. The talk will introduce these ideas and explain how, by extending them to an equivariant context, they give rise to a refined version of the characteristic polynomial for self-adjoint endomorphisms.

Fri 16 Feb 2024 15:00: Ed Segal (UCL): The McKay correspondence and GIT

Kleinian singularities are very classical objects in algebraic geometry, they are the surface singularities that appear when we quotient the plane by a finite subgroup of SL_2(\C). Each one has a minimal resolution by a smooth algebraic surface. It was famously observed by McKay, over forty years ago, that the geometry of this resolution is encoded in the representation theory of the finite group.

A modern viewpoint on McKay's correspondence is that the singularity has a second minimal resolution which is an orbifold, and the two resolutions are birational. I'll explain some of the traditional and modern perspectives on this story, and also a new approach to constructing the birational equivalence using Geometric Invariant Theory (which is joint work with Tarig Abdelgadir).

Fri 9 Feb 2024 15:00: Abigail Ward (Cambridge): Volume-preserving birational transformations of P^2 and symplectomorphisms of open symplectic four-manifolds

Within the Cremona group of all birational maps from P^2 to itself, there exists the subgroup G consisting of birational maps which preserve a standard meromorphic volume form. This group is interesting in its own right; in 2010, Jeremy Blanc exhibited a particularly nice set of generators for G and used these generators to give a simple presentation for the underlying discrete group \pi_0(G). We will discuss how one can use Blanc's work along with previous work of Hacking and Keating in homological mirror symmetry to find interesting new symplectomorphisms of certain open symplectic four manifolds. We will also speculate how such results might be used to find a presentation for G itself. This is joint work in progress with Keating.

Fri 2 Feb 2024 15:00: Charles Cox (Bristol): Spread and infinite groups

The notion of the spread of a finite group was introduced by Brenner and Wiegold in 1975. A group G has non-zero spread if every non-trivial g in G is part of a generating pair. If we focus on groups which have non-zero spread, then this restricts our attention to 2-generated groups for which every proper quotient is cyclic. Surprisingly, this condition on quotients is sufficient for a finite group to have non-zero spread. This is a result of Burness-Guralnick-Harper from 2021 which has now appeared in Annals of Math. The world of 2-generated infinite groups is much more wild, e.g. there exist simple groups requiring k generators for any natural number k (it is currently open, however, whether all finitely presented simple groups are 2-generated). After covering some of the history, we will see that the result of Burness-Guralnick-Harper does not generalise to 2-generated infinite groups. We will do this by motivating a concrete example which acts naturally on the integers (and is a subgroup of the second Houghton group H_2).

Fri 26 Jan 2024 16:00: Delaram Kahrobaei (CUNY): Post-quantum Group-based Cryptography

Group-based cryptography is a relatively new family in post-quantum cryptography.

In the first part of my talk, I speak about the first dedicated security analysis of one of the central problems in group-based cryptography, so-called Semidirect Discrete Logarithm Problem (SDLP). In particular, we provide a connection between SDLP and group actions, a context in which quantum subexponential algorithms are known to apply. We are therefore able to construct a subexponential quantum algorithm for solving SDLP, thereby classifying the complexity of SDLP and its relation to known computational problems.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

Fri 8 Dec 2023 16:00: GGSE meeting: no colloquium this week
No abstract available.

Fri 1 Dec 2023 16:00: Baris Kartal (Edinburgh): A Morse-Bott approach to the equivariant stable homotopy type

Morse theory provides an effective way to calculate the homology of smooth manifolds, in terms of critical points of a function and its gradient flow. Floer applied this idea in the infinite dimensional setting to produce new invariants in symplectic and low dimensional topology. Motivated by this, Cohen, Jones and Segal has shown how to obtain finer information about the topology of a smooth manifold from the Morse theory, thus providing a framework for refining Floer's invariant too. However, neither Morse theory nor the framework of Cohen-Jones-Segal are compatible with the compact group actions on the underlying manifold. In this talk, I will explain how to define a new framework for Morse-Bott function in order to extract information about the equivariant homotopy type. If time remains, I will discuss applications to equivariant Floer theory. Joint work with Laurent Cote.

Fri 24 Nov 2023 10:00: Jose Antonio Vilches (Universidad de Sevilla) [note the unusual time and location]: Lusternik-Schnirelmann category and Topological Complexity under a combinatorial point of view.

The goal of this talk is presenting a general overview of our (nowdays) main research line: establishing and developing L-S category and Topological Complexity in a discrete setting (the simplicial, poset, finite topological spaces contexts are good examples). In order to carry it out, no only the theoretical development will be presented, but also some motivational examples and applications shall be introduced. Joint work with D. Fernández-Ternero (Universidad de Sevilla) and E. Macías-Virgós (Universidade de Santiago de Compostela).

Fri 17 Nov 2023 16:00: Georg Gruetzner (Luxembourg): Hilbert spaces attached to Möbius spaces

In 2001, Y. Shalom asked if every hyperbolic group admits a uniformly bounded representation with a proper cocycle.
In this talk we construct Sobolev spaces H(d) associated with a Möbius structure M. A Möbius structure is a collection of metrics {d} sharing the same cross-ratios. We adapt Muckenhoupt's theory to uniform Ahlfors-David regular Möbius spaces. With this we show that the norms on H(d) for different metrics d in the Möbius structure are comparable for a large class of functions. This is a partial result of a program to construct and study uniformly bounded representations for all hyperbolic groups.

Fri 10 Nov 2023 16:00: Dmitri Panov (KCL) - cancelled, no colloquium this week
No abstract available.

Fri 3 Nov 2023 16:00: Sarah Whitehouse (Sheffield): Obstruction theory and equivariance

I aim to give a friendly overview of obstruction theories for higheralgebraic structures in homotopy theory, especially E-infinity structures.In the non-equivariant setting,this is an old story, involving work ofBasterra, Goerss-Hopkins, Robinson and myself, among others. Morerecently there has been a great deal of interest in the equivariant analoguesof these structures which come in many different flavours. I'll surveysome of this landscape and mention some joint work in progress withDan Graves.

Fri 27 Oct 2023 16:00: Nóra Szakács (Manchester): Inverse semigroups as metric spaces, and their uniform Roe algebras

Since Gromov's work in the 90s, studying groups as metric spaces has become a main trend in group theory. In turns out that there is a strong link between some of the algebraic properties of the group sand some of the large scale geometric properties of the metric associated to it, and in some cases, this links to properties of a C*-algebra called the uniform Roe algebra associated to the metric space. In the past few years, some of these results have been extended to so-called inverse semigroups by Y.C. Chung, R. Gray, D. Martínez, P. Silva, and the speaker. Inverse semigroups are a generalization of groups which encapsulate partial symmetries similarly as to how groups encapsulate symmetries. During the talk, we will give an exposition of these results.

Fri 20 Oct 2023 16:00: Ramon Flores (Universidad de Sevilla): From right-angled Artin groups to graph properties via cohomology

In the last years, thorough research has been conducted in order to understand graph properties in terms of group properties of the associated right-angled Artin group (RAAG). These properties should be intrinsic, in the sense that they should not depend on a concrete system of generators of the group. In this talk we will show how to approach some graph properties (as for example planarity or outerplanarity) using as input different bases of the cohomology of the RAAG and the good behaviour of the Colin de Verdière invariant with respect to minors.

Fri 13 Oct 2023 16:00: Simon Smith (Lincoln): The theory of local action diagrams

In this talk, which concerns groups acting on trees, I will describe joint work with Colin Reid in which we develop a 'local action' complement to classical Bass-Serre theory. We call this the theory of local action diagrams. This theory addresses a limitation of Bass-Serre theory that frequently occurs when one tries to construct novel examples of large groups acting on trees. This limitation has been particularly troublesome in the rapidly-developing theory of totally disconnected and locally compact groups.

To highlight the significance of our theory, I will describe how we can use it
to obtain a complete classification of all group actions on trees that have Tits'
Independence Property (P).

Fri 6 Oct 2023 16:00: Shintaro Nishikawa (Southampton): NCG meets harmonic analysis on symmetric spaces

The representation theory on reductive symmetric spaces G/H was studied in depth in the 1990s by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, among many others. In our collaborative efforts with A. Afgoustidis, N. Higson, P. Hochs and Y. Song, we delve into this subject from the perspective of noncommutative geometry. I will describe this exciting new development, with a particular emphasis on providing an elementary overview of harmonic analysis on reductive symmetric spaces.

Fri 19 May 2023 10:00: Operator Algebras in the South meeting

No pure math colloquium

Thu 18 May 2023 15:00: Ibrahim Trifa (École Normale Supérieure): Non simplicity of some groups of area-preserving homeomorphisms in higher genus surfaces

In recent years, significant progress has been made in understanding the algebraic structure of the groups of area-preserving homeomorphisms of surfaces. In particular, in 2021 and 2022, Cristofaro-Gardiner, Humilière, Mak, Seyfaddini and Smith have used Heegaard Floer Homology to introduce new spectral invariants and prove the non-simplicity of several groups in the case of the disc and the sphere. In this talk, after a short introduction to symplectic geometry, I will present a generalization of their results in higher genus. This is joint work with Cheuk Yu Mak.

Fri 12 May 2023 15:00: Lawk Mineh (Southampton): Combining quasiconvex subgroups of relatively hyperbolic groups

In a (relatively) hyperbolic group, a very natural class of subgroups to consider are the 'quasiconvex' subgroups, whose embedding as a subgroup respects the geometry of the ambient group. The intersection of two quasiconvex subgroups Q and R is always quasiconvex but the subgroupthey generate may not be. We will see how the algebraic property of separability can be used, however, to deduce the existence of finite index subgroups of Q and R that do generate a quasiconvex subgroup. This is joint work with Ashot.

Fri 5 May 2023 15:00: Daniel Drimbe (KU Leuven): Rigidity theory for von Neumann algebras

In the early 1940s, Murray and von Neumann found a natural way to associate a von Neumann algebra L(G) to any countable group G. The classification of von Neumann algebras has since been a central theme in operator algebras driven by the following question: what aspects of the group G are remembered by L(G)? The goal of this talk is to present major breakthroughs in the theory of von Neumann algebras and to survey some of the recent progress which is due to Popa's deformation/rigidity theory. We emphasize results which provide classes of groups that can be completely recovered from their von Neumann algebras.

Fri 28 Apr 2023 15:00: Aleksander Doan (UCL): Symplectic invariants of Calabi-Yau manifolds

One of the central problems of geometry is to classify manifolds which admit special geometric structures. Calabi-Yau manifolds provide a particularly interesting class of examples. Since their discovery forty years ago they have stimulated extraordinary research activity, leading to new, beautiful mathematics and surprising connections with physics. A distinctive feature of Calabi-Yau manifolds is that they lie at the intersection of three branches of geometry: algebraic, differential, and symplectic. As a result, there is an abundance of examples of these manifolds as well as tools to understand them. The talk will focus on invariants of Calabi-Yau manifolds of complex dimension three, which are defined by counting holomorphic curves. In particular, I will discuss a recent proof, joint with E. Ionel and T. Walpuski of the Gopakumar-Vafa finiteness conjecture, which concerns the algebraic structure of these invariants.

Fri 24 Mar 2023 15:00: Tyler Kelly (Birmingham): Mirror Symmetry between Landau-Ginzburg models

A Landau-Ginzburg model is a triplet of data: a quasi-affine variety X with a (reductive or finite) group G acting on it, along with a G-invariant regular function W : X → C. Here, the geometry of a Landau-Ginzburg model is encoded in a mix of the geometry of the quotient stack [X/G] and the singularity theory of W. While they may seem synthetic, they are often found as deformations of some of our favourite projective varieties. We will discuss the state of the art of their mirror symmetry along with some new resultsin enumerative geometry for Landau-Ginzburg models found in joint work with Mark Gross (Cambridge) and Ran Tessler (Weizmann).

Fri 17 Mar 2023 15:00: Strike day (cancelled)
No abstract available.

Fri 10 Mar 2023 15:00: Christian Voigt (Glasgow): Quantum symmetries

It is well-known that symmetries play central role in Mathematics, and that the study of symmetries is at the origin of group theory. In recent years it has emerged that, in certain situations, it can be beneficial to look more generally for so-called quantum symmetries as well. In fact, quantum symmetries feature naturally in the study of free probability, subfactors and in quantum information. In this talk I will start out by explaining what quantum symmetries are, and motivate why they are interesting. I will focus in particular on the case of graphs, and how this links with problems in quantum information. In the case of infinite graphs I will present a general scheme which allows one to define and study quantum symmetries. This will be illustrated with a number of examples, and - time permitting - I will also highlight some intriguing open problems in the area.

Fri 3 Mar 2023 15:00: GGSE meeting

No colloquium this week

Fri 24 Feb 2023 15:00: Robert Gray (U of East Anglia): The geometry of Schutzenberger graphs and subgroups of special inverse monoids

In the 1960's Higman was able to characterize the finitely generated subgroups of finitely presented groups. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain the motivation for studying this class of inverse monoids, and also indicate how we established these results by developing new geometric approaches to maximal subgroups, exploiting the fact that they are isomorphic to automorphism groups of Schutzenberger graphs.

Fri 17 Feb 2023 15:00: Roman Sauer (Karlsruhe): Higher property T of arithmetic lattices

The talk is based on joint work with Uri Bader and with Uri Bader, Alex Lubotzky and Shmuel Weinberger.

A fundamental property of the unitary representations of a group is Kazhdan's property T. This classical notion will be explained in the talk.In recent years there were various attempts to define and study higher analoga.

With Uri Bader we prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and geometric group theory. We describe applications to the cohomology and stability of arithmetic groups. In the former case we reprove results of Borel on the cohomology of lattices.

Fri 10 Feb 2023 15:00: David Méndez (Champalimaud Centre): An introduction to Algebraic Machine Learning

Model Theory is the field of mathematics that studies the relationships between algebraic structures and the syntactic expressions that they make valid. As such, this field emphasizes the embedding of knowledge into models.

In this talk we will introduce a meta-mathematical approach to Machine Intelligence using Model Theory. In particular, we will show how we can introduce knowledge and data into semilattices and how we can later manipulate it, by adding to these structures special elements called atoms that both encode the structure of the semilattice and can be operated on to enforce syntactic expressions. We will see how, using such ideas, we are guaranteed to find a simple rule in our data, if it exists. By doing so, we are able to find generalising models.

We will also discuss some relevant properties of our approach, like its mathematical transparency, the fact that we are only using set-theoretical operations that do not rely on optimisation, the possibility of combining data with formal knowledge and the reduced dependency on statistics and hyperparameters.

Fri 3 Feb 2023 15:00: Cheuk Yu Mak (Southampton): Quantum cohomology and loop group action

In this talk, we will first explain the nature and properties of some important invariants of symplectic manifolds, including quantum cohomology, Floer cohomology and Fukaya category. We will give a biased overview of why they are interesting and what we can do with them. Finally we will present a new algebraic structure on G-equivariant quantum cohomology when a compact Lie group G acts Hamiltonianly on a symplectic manifold. This is a joint work with Eduardo Gonzalez and Dan Pomerleano.

Fri 13 Jan 2023 15:00: Frank Neumann (Leicester): Hochschild cohomology of differential graded categories and applications to geometry and topology

Abstract: The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret this map as an edge homomorphism in a spectral sequence, which allows to study the characteristic homomorphism systematically in many interesting examples from algebra, geometry, topology and physics. To illustrate this, we will discuss several concrete examples related to coherent sheaves on algebraic curves and cochains of classifying spaces of Lie groups. If time permits, I will also indicate a new extension of this framework to $A_\infty$-categories. Some of this is joint work with M. Szymik (Sheffield) other with A. Phimister (Leicester).

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