# Sessions

Abstracts are arranged alphabetically by Speaker Surname.

Details of the Session, Venue and Day are provided at the end of each abstract.

Cristina Ana Maria Anghel (Oxford)

Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of link invariants. In this context, the quantum group Uq(sl(2)) leads to the sequence of coloured Jones polynomials, which contains the original Jones polynomial. Dually, the quantum group at roots of unity gives the sequence of coloured Alexander polynomials. On the topological side, Lawrence defined representations of braid groups on the homology of coverings of configurations spaces. Then, Bigelow and Lawrence gave a topological model for the original Jones polynomial.

We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum (defined over 3 variables) of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.

BMC03 Topology, Clyde 1, Thursday

Scott Balchin (MPIM Bonn)

Equivariant homotopy commutativity and the Catalan numbers

Studying the concept of commutativity up to homotopy is already a difficult problem, however, with the introduction of a group structure the problem is drastically more complicated. Luckily, we shall see that in the case of a finite group, the possible options for equivariant homotopy commutativity can be encoded using simple combinatorics via objects called indexing systems.

In particular, we show that for cyclic groups of prime power order, this combinatorial problem retrieves the construction of the Catalan numbers. We furthermore see a relationship to the associahedra when equipping these indexing systems with a natural order.

[This is joint work with David Barnes and Constanze Roitzheim].

BMC03 Topology, Clyde 1, Wednesday

Agnese Barbensi (Oxford)

A topological selection of knot folding pathways from native states

A small percentage of catalogued proteins is known to form open ended knots. Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of their projections using topological objects called knotoids. We define a topologically inspired distance between the knotoid distributions of knotted proteins, and we use it to detect and distinguish specific geometrical features for proteins sharing the same dominant topology. Our method allows us to reveal different folding pathways for proteins forming open ended trefoil knots by directly looking at the geometry and topology of their native states. This is joint work with N.Yerolemou, O.Vipond, B.Mahler and D.Goundaroulis.

BMC03 Topology, Clyde 1, Thursday

Christian Bönicke (Glasgow)

Dynamic asymptotic dimension and groupoid homology

Dynamic asymptotic dimension is a dimension theory for group actions and more generally for étale groupoids developed by Guentner, Willett, and Yu, which generalizes Gromov’s theory of asymptotic dimension. Having finite asymptotic dimension is known to have important implications for the structure of the involved C*-algebras. In this talk I will report on recent joint work with Dell’Aiera, Gabe, and Willett in which we prove a homology vanishing result for groupoids with finite dynamic asymptotic dimension. Our result allows us to present the first abstract class of groupoids satisfying Matui’s HK conjecture, which claims that the K-theory of a groupoid C*-algebra is completely encoded in the homology of the groupoid.

BMC04 Operator Algebras, Clyde 2, Wednesday

Matteo Casati (Ningbo)

Discrete Poisson Cohomology

I will present an unified formalism to describe functional Poisson bivectors in the differential and differential-difference case.

This allows to define their associated Poisson-Lichnerowicz cohomology, carrying information about the Casimir functionals, the symmetries and the admissible deformations of the corresponding Hamiltonian operator. The notion has been widely investigated in the differential case: here I will present the results for (-1,1) order scalar difference operators. An extension of the notion to the noncommutative cases will be briefly discussed to. (partly based on Casati, Wang, "A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators", Commun. Math. Phys. (2019)

BMC05 Math. Physics, Clyde 2, Thursday

Daniele Celoria (Oxford)

A discrete Morse perspective on knot projections

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse matching (dMm) on the 2-sphere, extending a construction due to Cohen. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMms, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of simple moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations. This is joint work with Naya Yerolemou.

BMC03 Topology, Clyde 1, Thursday

Robert Chamberlain (Warwick)

Minimal Permutation Representations of Finite Groups

Permutation representations are commonly used to represent finite groups on a computer. The degree of such representations can significantly effect the speed of computations involving the group. It is therefore ideal to use a faithful permutation representation of least degree. Computing this in general is not currently feasible, but this talk provides some of the more useful basic results and some new results, including the minimal degree of a faithful permutation representation of $2.A_n$, the double cover of the alternating group $A_n$, for each $n$.

Yemon Choi (Lancaster)

Completely almost periodic elements of group von Neumann algebras

On a locally compact group $G$, one may consider those $h\in L^\infty(G)$ whose left (or right) translates form a relatively compact subset of $L^\infty(G)$ in the norm topology: such functions are said to be (Bochner-)almost periodic, and they form a unital $C^{\ast}$-subalgebra ${\rm AP}(G)\subseteq L^\infty(G)$, whose Gelfand spectrum coincides with the Bohr compactification of $G$. In particular, if $G$ is compact then ${\rm AP}(G)=C(G)$.

There is a natural analogue of this construction where $L^\infty(G)$ is replaced by ${\rm VN}(\Gamma)$ for a locally compact group $\Gamma$, and the action of the algebra $L^1(G)$ on $L^\infty(G)$ is replaced by the action of the Fourier algebra ${\rm A}(\Gamma)$ on ${\rm VN}(\Gamma)$. Operator space considerations suggest that we should replace the usual notion of compactness by one which takes into account matricial structure, and the resulting space ${\rm CAP}(\widehat{\Gamma})$ is the subject of this talk. We will sketch a proof that ${\rm CAP}(\widehat{\Gamma})$ is always a unital $C^{\ast}$-subalgebra of ${\rm VN}(\Gamma)$, significantly extending previous results of Runde who established this under amenability/injectivity assumptions, and we will indicate how the question is ${\rm CAP}(\widehat{\Gamma})$ equal to $C^{\ast}_r(\Gamma)$ whenever $\Gamma$ is discrete?'' is equivalent to an open problem concerning the uniform Roe algebra.

BMC04 Operator Algebras, Clyde 2, Tuesday

Nirvana Coppola (Bristol)

Wild Galois representations of hyperelliptic curves

In this talk we will investigate the Galois action on a certain family of hyperelliptic curves defined over local fields. In particular we will look at curves with potentially good reduction, which acquire good reduction over a wildly ramified "large" extension. We will first clarify what large means and then show how to determine the Galois representation in the case considered.

BMC02 Number Theory, Bute 2, Tuesday

Well-Quasi-Orderability on Graphs

A class of graphs is well-quasi-ordered with respect to a containment relation if it contains no infinite set of incomparable elements (an anti-chain) and no infinite decreasing sequence. Being well-quasi-ordered is a highly desirable property, which has frequently been discovered in assorted areas of combinatorics and theoretical computer science. As an example, the Robertson-Seymour Theorem states that the class of all finite graphs is well-quasi-ordered with respect to the minor relation.

We can also consider the question of well-quasi-orderability of graphs with respect to other containment relations. If we consider the induced subgraph relation, the class of cycles forms an infinite anti-chain, so the class of all graphs is not well-quasi-ordered. It is therefore natural to ask what classes of graphs have the property of being well-quasi-ordered with respect to this relation.

This talk will describe some of the recent results in this area and include an introduction to the proof techniques that are available. No prior knowledge of graph classes or well-quasi-orders is needed to understand this talk.

Joint work with Vadim Lozin and Daniël Paulusma.

Ben Davison (Edinburgh)

Refined invariants of flopping curves

Associated to a flopping curve in a 3-fold X is a finite-dimensional algebra, the contraction algebra of Donovan and Wemyss, that represents the noncommutative deformations of the structure sheaf of that curve in the category of coherent sheaves on X. A conjecture of Brown and Wemyss states (roughly) that all finite-dimensional Jacobi algebras arise this way. As I'll explain, this conjecture would imply the positivity of all BPS invariants for finite-dimensional Jacobi algebras - these are invariants defined in terms of noncommutative Donaldson-Thomas theory. This is a surprising claim since it is easy to cook up negative DT invariants for infinite-dimensional Jacobi algebras. Finally, I'll explain that this positivity does indeed hold, providing evidence for the conjecture.

BMC07 Algebraic Geometry, Bute 2, Thursday

Matt Daws (UCLAN)

An introduction to (quantum) symmetries of (quantum) graphs

We will give a survey about quantum automorphism groups, concentrating upon automorphisms of graphs. We plan to quickly introduce the notion of a compact quantum group, describe how quantum groups can (co)act on spaces and algebras, and then describe a universal construction due to Wang which leads to the idea of quantum automorphisms of a finite set. Viewing a finite (simple) graph as a finite set of vertices with a relation describing the edges allowed Banica to define the compact quantum group of automorphisms of a graph. Surprisingly, this construction has recently appeared, repeatedly, in quantum information theory, and we will give a brief indication of how this is. Time allowing, we will also discuss "quantum graphs", a non-commutative generalisation of a graph, and their quantum automorphisms. The talk will concentrate upon setting the scene and describing some of the technical machinery which occurs.

BMC04 Operator Algebras, Clyde 2, Tuesday

Anastasia Doikou (Heriot-Watt)

Set theoretic Yang-Baxter equation, braces and quantum groups

We examine novel links between the theory of braces (nil potent rings) and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify quantum groups and admissible Drinfeld twists associated to set-theoretic solutions coming from braces. We also derive new classes of symmetries for the corresponding periodic transfer matrices.

BMC05 Math. Physics, Clyde 2, Thursday

Jonathan Eckhardt (Loughborough)

The peakon resolution conjecture for the conservative Camassa-Holm flow

The Camassa-Holm equation is an integrable nonlinear partial differential equation that models unidirectional wave propagation on shallow water. I will show how it can be proved that global conservative weak solutions form into a train of solitons (peakons) in the long-time limit.

BMC05 Math. Physics, Hillhead 1, Tuesday

Jonny Evans (Lancaster)

Contact and symplectic geometry of compound Du Val singularities

Joint work with YankI Lekili. The link of a terminal 3-fold hypersurface singularity (or compound Du Val singularity) is a dynamically convex contact 5-manifold. We compute a contact invariant called the positive symplectic cohomology (SH^+) of the link for a range of cDV singularities including some Brieskorn-Pham examples and for the base of the Laufer flop. Based on this we formulate a conjecture: if the singularity admits a small resolution whose exceptional set has p components, then SH^+ has rank p in every negative degree. Despite the resulting insensitivity of this invariant, we are still able to distinguish many links as contact manifolds by equipping SH^+ with a bigrading.

BMC03 Topology, Clyde 1, Tuesday

Alex Evetts (ESI)

Growth and equations in virtually abelian groups

Growth and equations in virtually abelian groups In 1983, Max Benson introduced a framework to study the structure of finitely generated virtually abelian groups, and used it to prove that the growth series of such a group is always a rational function. I will explain this framework (consisting of patterned words and polyhedral sets) and show how it can be extended and modified to study other forms of growth including conjugacy growth, and relative growth. In particular, I will discuss the solution sets to systems of equations in virtually abelian groups (i.e. those group elements which satisfy equations whose constants lie in the group) and demonstrate that their associated relative growth series are rational. Partially based on joint work with Alex Levine.

Enrico Fatighenti (IMT Toulouse)

Fano varieties from homogeneous vector bundles

The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.

BMC07 Algebraic Geometry, Bute 2, Thursday

Free boundary minimal surfaces with connected boundary and arbitrary genus in the unit ball

A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e. the unit sphere).

It is natural to ask if there are FBMS in the unit ball of any given genus g and number of boundary components b. Several different examples have already been discovered, but the answer was so far still unknown already in the simple case g=b=1.

In this talk, we will present the construction of a family of FBMS with connected boundary and any given genus. This answers affirmatively to the aforementioned open question.

This is joint work with Alessandro Carlotto and Mario Schulz.

Magnus Goffeng (Gothenburg)

Exotic examples in Fell algebras

As a partial result towards classifying the C*-algebras that arises from Smale spaces, Robin Deeley and Allan Yashinski studied a Fell algebra arising from Smale spaces with totally disconnected stable sets. The Fell algebra in question has compact spectrum and trivial Dixmier-Duoady class. Using work of Robin Deeley and Karen Strung, the existence of a projection in this Fell algebra was the missing piece needed to finish their classification and a projection was later on (tautologically enough) constructed using classification results. Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant is just a Hausdorff assumption away from being a unital commutative C*-algebra so one might naively think that general approaches from noncommutative geometry will let us put Fell algebras with smooth spectrum on equal footing with manifolds. For instance, one might suspect that all Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant are stably unital. In this talk I will discuss some examples where this fails. Based on joint work with Robin Deeley and Allan Yashinski.

BMC04 Operator Algebras, Clyde 2, Wednesday

Non-uniquely ergodic arational trees in the boundary of Outer space

The mapping class group of a surface is associated to its

Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arational trees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called non-uniquely ergodic'. I will first talk about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

Rod Halburd (UCL)

Finding exact special solutions to non-integrable equations

We will describe methods to find all meromorphic solutions to different classes of (generally non-integrable) equations and how these methods can be extended to other types of functional equations. The analysis here is much more subtle than Kowalevskaya-Painlevé analysis as global methods are needed to track the value distribution of individual solutions. We will discuss how these methods can be modified to allow for branching at fixed singularities.

BMC05 Math. Physics, Hillhead 1, Tuesday

Martin Hallnäs (Gothenburg)

Soliton scattering in the hyperbolic relativistic Calogero-Moser system

Integrable N-particle systems of relativistic Calogero-Moser type were first introduced by Ruijsenaars and Schneider (1986) in the classical- and Ruijsenaars (1987) in the quantum case. In the hyperbolic regime they are closely related to several soliton equations, in particular the sine-Gordon equation.

In this talk, I will focus on the quantum case and discuss a proof of the long-standing conjecture that the particles in the relativistic Calogero-Moser system of hyperbolic type exhibit soliton scattering, i.e. conservation of momenta and factorization of scattering amplitudes.

The talk is based on joint work with Simon Ruijsenaars.

BMC05 Math. Physics, Hillhead 1, Tuesday

Florian Hanisch (Potsdam)

Relative Traces in Obstacle Scattering

In obstacle scattering, one is interested in properties of the Laplacian $\Delta$ on the complement of a compact set $\mathcal{O} \subset \mathbb{R}^d$ (the obstacles) with suitable boundary conditions. It may be compared with the free Laplacian $\Delta_0$, defined on all of $\mathbb{R}^d$. For functions $f$ satisfying restrictive assumptions, it is known that differences $f(\Delta) - f(\Delta_0)$ are trace class operators and traces are given by integrals of the Krein spectral shift function associated with $\mathcal{O}$.

We will discuss a relative version of this result. Assuming that $\mathcal{O}$ has two connected components, we look at the setting where both obstacles are present relative to the situation, where one of them has been removed. The former is described by the operator $\Delta$; let $\Delta_1$ and $\Delta_2$ denote the Laplacians after removal of an obstacle. We show that the operator $f(\Delta) - f(\Delta_1) - f(\Delta_2) + f(\Delta_0)$ is now trace class for a much larger class of functions $f$. This is important for physical applications when the choice $f(x) = \sqrt{x}$ corresponds to relative (Casimir) energy densities.

Joint work with A. Strohmaier and A. Waters.

Fritz Hiesmayr (University College London)

Asymptotics of two-valued minimal graphs

A two-valued function is a function defined on a subset $\mathbb{R}^n$ and taking values in the set of unordered pairs of real numbers. We consider its graph as a subset of $\mathbb{R}^{n+1}$, with two (possibly equal) points lying above every point in the domain. This is called minimal if it is a critical point for the area functional, and the function satisfies a quasilinear elliptic PDE. Two-valued minimal graphs are of interest as they provide perhaps the simplest non-trivial setting where the challenging issue of branch point singularities arises. We present recent work on entire two-valued minimal graphs, including results concerning their rigidity in low dimensions.

Andrew Hone (Kent)

Heron triangles with two rational medians and Somos-5 sequences

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. In fact Schubert made the erroneous assertion that even two rational medians was impossible, but Buchholz and Rathbun later showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve $E(\Q)$ with Mordell-Weil group $\Z/2\Z\oplus\Z$, and they observed an apparent connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths and the area in this infinite family of Heron triangles with two rational medians.

BMC05 Math. Physics, Clyde 2, Thursday

Nayab Khalid (St Andrews)

An infinite geometric presentation for Thompson's group $F$

In this talk, I will present the algorithms I developed during the course of my PhD to find an infinite presentation for Thompson's group $F$ which reflects the geometric structure of the unit interval. Time permitting, I will also discuss how this presentation could help us solve the rotation distance problem.

Rachel Kirsch (ISU)

Universal partial words

Chen, Kitaev, Mütze, and Sun recently introduced the notion of universal partial words, a generalization of universal words and de Bruijn sequences. Universal partial words allow for a wild-card character $\diamond$, which is a placeholder for any letter in the alphabet. For non-binary alphabets, we show that universal partial words have periodic $\diamond$ structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for an infinite family of universal partial words over non-binary alphabets. Finally, we give a preliminary report on ongoing work.

Based on joint work with Bennet Goeckner, Corbin Groothuis, Cyrus Hettle, Brian Kell, Pamela Kirkpatrick, and Ryan Solava.

Theodoros Kouloukas (Kent)

Cluster maps associated with discrete KdV equation

In the context of cluster algebras, nonlinear recurrences are generated from cluster mutation-periodic quivers, i.e. quivers with certain periodicity property under sequences of mutations. In this talk we focus on a particular class of cluster recurrences associated with the discrete KdV equation which arise as plane wave reductions of the Hirota-Miwa (discrete KP) equation. We study the integrability of the corresponding discrete systems using the properties of the underlying cluster algebra structure.

BMC05 Math. Physics, Clyde 2, Thursday

Marius Leonhardt (Heidelberg)

Plectic Galois action on Hilbert modular varieties

The modular curve has had various applications to number theory, in particular to the theory of elliptic curves. But what about applications of other Shimura varieties to higher dimensional abelian varieties? In this talk, we will focus on the Hilbert modular variety and explain one attempt to achieve such analogous applications, via the so-called "plectic conjecture" of Nekovar--Scholl. It involves a mysterious plectic Galois group and actions of this group on various quantities associated to the Hilbert modular variety. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible.

BMC02 Number Theory, Bute 1, Thursday

Abigail Linton (NTNU)

Massey products in moment-angle complexes

Massey products are higher cohomology operations that are notoriously difficult to compute. Coming from Toric Topology, moment-angle complexes are spaces with many applications in commutative algebra, complex geometry and combinatorics. The topology of a moment-angle complex is often encoded combinatorially in its corresponding simplicial complex. I will demonstrate this by presenting systematic combinatorial constructions of non-trivial Massey products in moment-angle complexes. These constructions produce infinitely many families of manifolds with non-trivial Massey products and generalise all existing examples of Massey products in Toric Topology. This is joint work with Jelena Grbić (Southampton).

BMC03 Topology, Clyde 1, Wednesday

Florian Litzinger (Queen Mary University of London)

Optimal regularity for Pfaffian systems and the fundamental theorem of surface theory

The fundamental theorem of surface theory asserts the existence of a surface immersion with prescribed first and second fundamental forms that satisfy the Gauss–Codazzi–Mainardi equations. Its proof is based on the solution of a Pfaffian system and an application of the Poincaré lemma. Consequently, the regularity of the resulting immersion crucially depends on the regularity of the solution of the corresponding Pfaffian system. This talk shall briefly review both the classical smooth case and the existing regularity theory and then introduce a recent extension to the optimal regularity.

Alan Logan (Heriot-Watt University)

Equalisers of free group homomorphisms and Post's correspondence problem

For two free group homomorphisms, we investigate the set of points where the two maps agree, called the equaliser of the maps, and we focus on two questions of Stallings from the 1980s about the bases of these equalisers.

There are two worlds from which to draw ideas. Firstly, fixed subgroups of free group automorphisms, which are particular instances of equalisers and which have generated a lot of literature since the 1980s. Secondly, equalisers of free monoid homomorphisms, which have been studied in computer science for over 70 years, starting with Post's correspondence problem, the (undecidable) decision problem asking is the equaliser of two free monoid homomorphisms trivial?'.

Working somewhere between these worlds, we prove strong, positive results when both maps are immersions' of free groups, and for the free group of rank two. Joint work with Laura Ciobanu.

Sara Lombardo (Loughborough)

Integrability and plane wave instabilities: an algebraic-geometric approach

In this talk, I will review the approach to linear stability via integrability proposed in collaboration with Toni Degasperis and Matteo Sommacal (see Journal of Nonlinear Science, 28, pages 1251–1291, 2018). I will then apply it to study the linear stability of plane wave solutions of a novel long wave-short wave system which contains both the Yajima-Oikawa and Newell models for a particular choices of the coefficients. The stability spectra and the associated eigenfrequencies are explicitly computed, leading to a relation between the topology of the spectra and the gain function of the system.

Preliminary analysis indicates that, similarly to the case of vector Non-Linear Schroedinger (VNLS), the classification of the stability spectra allows one to predict regions of existence of rogue wave type solutions.

This work has been done in collaboration with Toni Degasperis, Matteo Sommacal and Marcos Caso Huerta.

BMC05 Math. Physics, Hillhead 1, Tuesday

Daniel Loughran (Bath)

Probabilistic Arithmetic Geometry

A theorem of Erdos-Kac states that the number of prime divisors of an integer behaves like a normal distribution (once suitably renormalised). In this talk I shall explain a version of this result for integer points on varieties. This is joint work with Efthymios Sofos and Daniel El-Baz.

BMC02 Number Theory, Bute 1, Thursday

Local-global principles for norms

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions.

In this talk, I will present work developing explicit methods to study this principle for non-Galois extensions. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for products of norms as well as consequences in the statistics of these local-global principles.

BMC02 Number Theory, Bute 2, Tuesday

Diane Maclagan (Warwick)

Toric Bertini Theorems and Higher connectivity of Tropical Varieties

The classical Bertini theorem is a basic and fundamental tool in algebraic geometry. Recent work of Fuchs, Mantova, and Zannier extended this to a version where a hyperplane is replaced by a subtorus. I will discuss this result, and an application in tropical geometry (joint with Josephine Yu), where we use this to show that tropical varieties are highly connected.

BMC07 Algebraic Geometry, Bute 2, Wednesday

The Morse Index of Willmore Spheres

Robert Bryant showed that any closed immersed Willmore sphere in Euclidean three-space is the inversion of a complete minimal sphere with embedded planar ends. We proved that the Willmore Morse Index of the closed surface can be computed by using unbounded Area-Jacobi fields of the related minimal surface. As a consequence, we get that all immersed Willmore spheres are unstable except for the round sphere. This talk is based on work with Jonas Hirsch and Rob Kusner.

Ciprian Manolescu (Stanford)

Relative genus bounds in indefinite four-manifolds

Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X - B^4, with boundary a knot K. We give several methods to produce bounds on the genus of such surfaces in a fixed homology class. Our techniques include relative adjunction inequalities and the 10/8 + 4 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold. We give an example showing that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. Further, we give examples of knots that are topologically but not smoothly H-slice in some indefinite 4-manifolds. This is joint work with Marco Marengon and Lisa Piccirillo.

BMC03 Topology, Clyde 1, Tuesday

Andrew McLeod (Oxford)

Pyramid Ricci Flow

In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).

Anthony Nixon (Lancaster)

Graph rigidity and flexible circuits

A framework is a geometric realisation of a graph in Euclidean d-space. Edges of the graph correspond to bars of the framework and vertices correspond to joints with full rotational freedom. The framework is rigid if every edge-length preserving continuous deformation of the vertices arises from isometries of d-space. Generically, rigidity is a rank condition on an associated rigidity matrix and hence is a property of the graph which can be described by the corresponding row matroid. Characterising which graphs are rigid is solved in dimension 1 and 2, but open in dimension at least 3. One fundamental problem in higher dimensions is the existence of flexible circuits; these are non-rigid graphs that are circuits in this matroid.

I will give a survey of graph rigidity including recent joint work with Georg Grasegger, Hakan Guler and Bill Jackson where we analyse flexible circuits in dimension d.

Nils Prigge (ETH)

Embedding calculus and automorphisms of manifolds

Considering the space of diffeomorphisms of a closed manifold as the space of self-embeddings, we can study it using the homotopy theoretic approximations from embedding calculus. I will discuss this approach as well as some recent advances, and I will focus on how we might detect the difference between the approximation and the space of diffeomorphisms using classical invariants of fibre bundles.

BMC03 Topology, Clyde 1, Wednesday

Jacqui Ramagge (Durham)

What can an algebraist bring to Operator Algebras?

Having asked some colleagues about what they would like to hear about, it appears that it would be useful to give my perspective on self-similarity and algebraic techniques in Operator Algebras. Most of this will not be new although some of it is not yet published. Hopefully all of it will be comprehensible.

BMC04 Operator Algebras, Clyde 2, Wednesday

Dhruv Ranganathan (Cambridge)

4/2 ways of counting curves in a pair

I will discuss the geometry surrounding a simple question: how does one understand the space of rational plane curves with tangency conditions along two distinct lines? There are (probably more than) two very sensible sounding ways of studying this problem: via orbifolds and via logarithmic structure. We’ll see why these give rise to different answers. I’ll then try to explain how an amazing formula of Aluffi, the intersection theory canon of Fulton-Macpherson, and a bit of tropical geometry tell us how to think about this and much more. This is joint work with Navid Nabijou (Cambridge).

BMC07 Algebraic Geometry, Bute 2, Wednesday

Alice Rizzardo (Liverpool)

New examples of non-Fourier-Mukai functors

Functors between the derived categories of two smooth projective varieties are a fundamental object of study. Almost all known such functors are so-called Fourier-Mukai: roughly speaking, they are well-behaved with respect to the geometry of the varieties. They admit a lift to a functor between the enhancements of the two derived categories. The first example of a non-Fourier-Mukai functor was given by myself and Van den Bergh in 2015. I will show that this is not a pathological example by providing a way to construct a non-Fourier-Mukai functor from the derived category of any smooth projective variety of dimension greater or equal to 3 admitting a tilting bundle. This is joint work with Theo Raedschelders and Michel Van den Bergh.

BMC07 Algebraic Geometry, Bute 2, Thursday

Simon Schmidt (Saarbruecken)

On the quantum symmetry of distance-transitive graphs

To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowicz's compact matrix quantum groups. An important task is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative. We will see that a graph has quantum symmetry if its automorphism group contains a certain pair of automorphisms. Then, focussing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute and deduce that several families of distance-transitive graphs have no quantum symmetry.

BMC04 Operator Algebras, Clyde 2, Tuesday

Dirk Schuetz (Durham)

A Scanning Algorithm for Odd Khovanov Homology

We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. The main difficulty comes from the sign assignments in the cochain complex, which are not local in the odd theory. To deal with this we use a mapping cone construction instead of a tensor product to handle the gluings of tangles.

The algorithm has been implemented in a computer program, and we can also use it to make efficient calculations for a concordance invariant that was recently introduced by Sarkar-Scaduto-Stoffregen.

BMC03 Topology, Clyde 1, Thursday

Sue Sierra (Edinburgh)

Poisson geometry of the Virasoro algebra

Let W be the Witt Lie algebra of algebraic vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W, with central generator z. We describe the geometry associated to prime Poisson ideals in Sym(Vir).

We focus first on understanding Poisson primitive ideals: Poisson cores of maximal ideals in Sym(Vir). We classify maximal ideals which have nontrivial Poisson cores and calculate their Poisson cores; using this we show that if \lambda \neq 0 then Sym(Vir)/(z-\lambda) is Poisson simple.

There is a notion of coadjoint orbit of a point in Vir^* = MSpec(Sym(Vir)). Although Vir^* is infinite-dimensional, these coadjoint orbits are all finite-dimensional. We describe them explicitly and also discuss general prime Poisson ideals and implications for the ideal structure of the universal enveloping algebra of Vir and for the representation theory of Vir.

This is joint work with Alexey Petukhov.

BMC01 Algebra and Representation Theory, Bute 1, Tuesday

Samir Siksek (Warwick)

Efficient resolution of Thue–Mahler equations

A Thue–Mahler equation has the form F(X,Y)=p_1^{m_1} \cdots p_r^{m_r} where F is an irreducible homogeneous binary form of degree at least 3 with integer coefficients, and p_1,..,p_r are primes. A standard algorithm due to Tzanakis and de Weger solves Thue–Mahler equations when the degree and the number of primes is small. We give lattice-based sieving techniques that are capable of handling large Thue–Mahler equations. This is joint work with Adela Gherga, Rafael von Känel and Benjamin Matschke.

BMC02 Number Theory, Bute 1, Thursday

Steven Sivek (Imperial)

Framed instanton homology and Dehn surgery

Framed instanton homology is a 3-manifold invariant which is usually very difficult to compute. However, it turns out that for any knot K in S^3, the homologies of all Dehn surgeries on K are determined by a single pair of integers, one of which is even a smooth concordance invariant. In this talk we’ll discuss how a new symmetry property of cobordism maps in framed instanton homology constrains the geography of these invariants, with applications to homology cobordism and to the Dehn surgery realization problem for many rational homology 3-spheres.

BMC03 Topology, Clyde 1, Tuesday

Simon Smith (Lincoln)

Local to global behaviour of groups acting on trees

Groups acting on trees play a fundamental role in the theory of groups.Bass--Serre Theory, and in particular the notion of a graph of groups, is a powerful tool for decomposing groups acting on trees, but its usefulness for constructing non-discrete groups acting on trees is, in some situations, severely limited. Such groups play an important role in the theory of infinite permutation groups, and the theory of t.d.l.c. groups, as they are a rich source of examples of simple groups. For these groups proving simplicity is often tricky,and relies on the group satisfying some sort of independence property, usually Tits' independence property (P).

An alternative, but complementary, approach to the study of groups acting on trees has recently emerged based on local actions, that is, the action of a vertex stabiliser on the neighbouring vertices. In many situations this alternative is better suited to constructing non-discrete groups acting on trees. The beginnings of this approach can be found in a 2000 paper of M.~Burger and Sh.~Mozes, in which the authors use this local-to-global' approach to construct an interesting class of (virtually) simple t.d.l.c. groups acting on trees, all with Tits independence property (P). The majority of new constructions of compactly generated simple t.d.l.c. groups have used ideas inspired by this work.

More recently, the Burger--Mozes construction has been generalised (by the speaker) to obtain a new product (called the box product) of permutation groups, and this was used to prove that there are $2^{\aleph_0}$ isomorphism types of non-discrete, compactly generated, simple t.d.l.c. groups. It also formed an integral part of the recent classification of subdegree-finite primitive permutation groups.

This generalisation also uses a local-to-global' approach to construct groups acting on trees with Tits independence property (P).

In joint work with Colin Reid, we have developed a general method for describing and classifying all actions of groups on trees with property (P). This is done using an object called a local action diagram, akin to a graph of groups, but for local actions. Our work can be seen as a local action' complement to Bass-Serre theory. Under this framework, the Burger--Mozes construction corresponds to a local action diagram consisting of a single vertex with a set of loops, each with its own reverse, and the box product construction corresponds to a local action diagram consisting of a pair of vertices and no loops. Any connected graph can form the basis of a local action diagram and each local action diagram gives rise to a unique (up to conjugacy) group with property (P).

Furthermore, for a group $G$ with property (P), one can easily determinewhether or not $G$ has certain properties (e.g. compact generation and simplicity) directly from its local action diagram.

Efthymios Sofos (Glasgow)

Prime values of integer polynomials and random Diophantine equations

Schinzel's hypothesis is a central conjecture in number theory which states that integer polynomials satisfying the obvious necessary assumptions represent primes. It is completely open in all cases except when the polynomial has degree 1 . In joint work with Alexei Skorobogatov we settle the conjecture in 100% of the cases (when polynomials are ordered by height of coefficients). Furthermore, we explore consequences for integer solutions to random Diophantine equations.

BMC02 Number Theory, Bute 2, Tuesday

Ioan Stanciu (Oxford)

Primitive ideals in the affinoid enveloping algebra of a semisimple Lie algebra

I will start by defining the affinoid enveloping algebra of a semisimple Lie algebra and explain the connection with the Iwasawa algebra and the classical enveloping algebra. Next, I will review the characterisation of primitive ideals in the classical enveloping algebra of a semisimple Lie algebra and explain how one can use geometric representation theory to obtain an affinoid version of Duflo's theorem. Finally, I will talk about how a large class of two-sided ideals in the affinoid enveloping algebra is controlled by two-sided ideals in the classical enveloping algebra

BMC01 Algebra and Representation Theory, Bute 1, Wednesday

Greg Stevenson (Glasgow)

Points of cochains on BG and their tangent vectors

Given a prime ideal of a commutative noetherian ring we can attach to it a residue field k and a k-vector space of tangent vectors to the corresponding point in the associated affine scheme. I'll explain how to package this in a homotopy invariant fashion and thus extend these concepts to the cochains on the classifying space of a finite group (amongst other settings). This is based on joint work with Paul Balmer and Henning Krause, and with James Cameron.

BMC03 Topology, Clyde 1, Wednesday

Catharina Stroppel (Bonn)

Verlinde rings and DAHA actions

In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A.

I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.

BMC01 Algebra and Representation Theory, Bute 1, Wednesday

Andrew Sutherland (MIT)

Stronger arithmetic equivalence

Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, Galois closure, and isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphisms of some or all of these invariants without forcing an isomorphism of number fields, along with explicit examples and some open questions. These results also have applications to the construction of curves with isomorphic Jacobians (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler).

BMC02 Number Theory, Bute 2, Tuesday

Louise Sutton (Manchester)

Decomposable Specht modules

The representation theory of the symmetric groups and associated Hecke algebras can be studied via the KLR algebras, whose representations are constructed as graded Specht modules. In this talk, we review a catalogue of results on decomposable Specht modules in level 1 - the study of which began in the 80s by Murphy on Specht modules indexed by hooks for the symmetric group. We move on to discuss recent work in level 2 of the KLR algebras on decomposable Specht modules - namely those indexed by a pair of hooks - in which we describe the structure of large families of semisimple and close-to-semisimple Specht modules.

BMC01 Algebra and Representation Theory, Bute 1, Wednesday

Lewis Topley (Birmingham)

Finite W-algebras and the orbit method

In a recent work Losev constructed a version of the orbit method for semisimple Lie algebras: this is a map from the set of coajoint orbits to the primitive spectrum of the enveloping algebra. For classical Lie algebras the map is known to be injective and Losev conjectured that the image should consist of primitive ideals obtained from one dimensional representations of W-algebras via Skryabin's equivalence. In this talk I will explain my proof of this conjecture. One of the key steps is the use of Dirac reduction to obtain new presentations of the semiclassical limits of finite W-algebras.

BMC01 Algebra and Representation Theory, Bute 1, Tuesday

Gareth Tracey (Oxford)

Invariable generation and the Chebotarev invariant of a finite group

Given a finite group $X$, a classical approach to proving that $X$ is the Galois group of a Galois extension $K/\mathbb{Q}$ can be described roughly as follows:\\

(1) prove that $\Gal(K/\mathbb{Q})$ is contained in $X$ by using known properties of the extension (for example, the Galois group of an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ embeds into the symmetric group $\Sym(n)$);\\

(2) try to prove that $X = \Gal(K/\mathbb{Q})$ by computing the Frobenius automorphisms modulo successive primes, which gives conjugacy classes in $\Gal(K/\mathbb{Q})$, and hence in $X$. If these conjugacy classes can only occur in the case $\Gal(K/\mathbb{Q})=X$, then we are done. The \emph{Chebotarev invariant} of $X$ can roughly be described as the efficiency of this `algorithm'.

In this talk we will define the Chebotarev invariant precisely, and describe some new results concerning its asymptotic behaviour.

Daniel Tubbenhauer (Bonn)

2-representation theory of Soergel bimodules

This talk is a friendly introduction to 2-representation theory, with the emphasis on example such as 2-representations of Soergel bimodules (the categorical analog of representations of Hecke algebras).

BMC01 Algebra and Representation Theory, Bute 1, Tuesday

Michel van Garrel (Birmingham)

Stable maps to Looijenga pairs

Start with a rational surface Y with a decomposition of its anticanonical divisor into at least 2 smooth nef components. We associate 5 enumerative theories to such Looijenga pairs: 1) all genus stable log maps with maximal tangency to each boundary component; 2) the all genus open Gromov-Witten theory of a toric Calabi-Yau threefold associated to the Looijenga pair; 3) genus 0 stable maps to the local Calabi-Yau surface obtained by twisting Y by the sum of the line bundles dual to the components of the boundary; 4) the Donaldson-Thomas theory of a symmetric quiver specified by the Looijenga pair and 5) BPS invariants associated to the various curve counting theories. In joint work with Pierrick Bousseau and Andrea Brini, we provide closed-form solutions to essentially all of these invariants and show that the theories are equivalent. I will focus on the relationship between 1) and 2) and present how the well-understood BPS theory of 2) leads to refinements of sheaf counting invariants of local Calabi-Yau fourfolds.

BMC07 Algebraic Geometry, Bute 2, Wednesday

Alina Vdovina (Newcastle)

Buildings, C*-algebras and new higher-dimensional analogues of the Thompson groups

We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their $K$-theory. The underlying building structure allows explicit computation of the $K$-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the $K$-theory ofC*-algebras gives new invariants to recognize non-isomorphic groups.

BMC04 Operator Algebras, Clyde 2, Tuesday

Hanneke Wiersema (Kings College London)

On a BSD-type formula for L-values of Artin twists of elliptic curves

In this talk we will discuss the possible existence of a BSD-type formula for L-functions of elliptic curves twisted by Artin representations. After outlining some expected properties of these L-functions, we will present arithmetic applications and some explicit examples. This is joint work with Vladimir Dokchitser and Robert Evans.

BMC02 Number Theory, Bute 1, Thursday

Angela Wu (UCL)

Weinstein handlebodies for complements of smoother toric divisors

In this talk, I will introduce you to two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which come with a handle decomposition compatible with their symplectic structure. I will then show you an algorithm which produces the Weinstein handlebody diagram for the complement of a smoothed toric divisor in a "centered" toric 4-manifold. This is based on joint work with Acu, Capovilla-Searle, Gadbled, Marinković, Murphy, and Starkston.

BMC03 Topology, Clyde 1, Tuesday

Runlian Xia (Glasgow)

Non-commutative Hilbert transforms and Cotlar-type identities

The Hilbert transform $H$ is a basic example of a Fourier multiplier. Its behaviour on Fourier series is the following:

$$\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},$$ with $m(n)=-i\,{\rm sgn} (n)$. Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.

We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$-spaces. The pioneering work in this direction is due to Mei and Ricard, in which they prove $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity on von Neumann algebras. In this talk, we introduce a new form of Cotlar identities for groups that are not necessarily free products and study their validity for lattices of ${\rm SL}_2(\mathbb{C})$ and some other groups acting on trees.

Joint work with Adrián González and Javier Parcet.

BMC04 Operator Algebras, Clyde 2, Wednesday

Makoto Yamashita (Oslo)

Homology and K-theory of torsion free ample groupoids

The problem of connecting the integral homology to the K-groups of C*-algebra for ample groupoids was recently popularized by Matui. As an answer for this, we construct a spectral sequence starting from the groupoid homology which ends at the K-groups when the groupoid satisfies the Baum-Connes conjecture and has torsion free stabilizers. The construction crucially relies on the Meyer-Nest theory of triangulated structure on equivariant KK-categories. The same technique allows us to incorporate Putnam’s homology theory for Smale spaces in place of groupoid homology.

BMC04 Operator Algebras, Clyde 2, Wednesday