Seminars Semester 1 2025-26

23 September 2025

Speaker: Anna Felikson (University of Durham)

Title: Polytopal realizations of non-crystallographic associahedra

Abstract: 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.

30 September 2025

Speaker: Chris Milionis (Newcastle)

Title: The center of the BMW algebra via Jucys-Murphy elements

Abstract: Classical Schur-Weyl duality of type A over C for Lie algebras connects the category of f.d.representations of the symmetric group with the category of f.d. modules of the universal enveloping algebra of sl_n. For simple Lie algebras of type B,C,D, the Schur-Weyl dual is the Brauer algebra, a so called one parameter diagrammatic algebra Br_n(δ), with δ any complex number. For quantum groups of type B,C,D, the Schur-Weyl duals are the BMW algebras, a two parameter diagrammatic algebra B_n(q,t) with q,t any complex numbers. As in the case of symmetric groups (and many other diagrammatic algebras), these algebras have a remarkable family of commuting elements which are usually called the Jucys-Murphy elements.

 In this talk, we will introduce the BMW algebras, their connection to quantum groups of types B,C,D, give a brief overview of its representation theory and describe its center for q not a root of unity, and for almost all values of t as a subalgebra of the symmetric Laurent polynomials in the Jucys-Murphy elements.

 As a first corollary of these results, the Jucys-Murphy elements in these cases can be used to give an Okounkov-Vershik like approach to the f.d. representations of B_n(q,t).

As a second corollary, for t=q^{2a}, where a is a nonnegative integer making B_n(q,t) non-semisimple, one can give an alternative characterization for when two simple modules lie in the same block.

07 October 2025

Speaker: Thorsten Heidersdorf (Newcastle)

Title: Koszulity for semi-infinite highest weight categories

Abstract: Koszul algebras are positively graded algebras with very amenable homological properties. Typical examples are the polynomial ring over a field or the exterior and symmetric algebra of a vector space. A category is called Koszul if it has a grading with which it is equivalent to the category of graded modules over a Koszul algebra. Koszulity is a very nice property but often very difficult to check. I will give a criterion which allows to check Koszulity in case the category is a semi-infinite highest weight category (which is a structure that appears often in representation theory).

14 October 2025

Speaker: Iordannis Romainis (University of Edinburgh)

Title: Finiteness and holonomicity of skein modules

Abstract: Skein modules are topological invariants assigned to 3-manifolds, dependent on a quantum parameter q and an algebraic reductive group G. For closed 3-manifolds at generic q they were conjectured by Witten to be finite-dimensional—a statement later proved by Gunningham, Jordan, and Safronov. In this talk, I present joint work with David Jordan on a generalization of this conjecture to 3-manifolds with boundary. In this setting, the finiteness property is replaced by the condition that the skein module is holonomic over the boundary skein algebra. This includes finite generation and a Lagrangian support condition. 

21 October 2025

Speaker: Federica Gavazzi (Heriot-Watt)

Title: Algebraic and Topological aspects of virtual Artin groups

Abstract: Virtual Artin groups were introduced a few years ago by Bellingeri, Paris, and Thiel with the aim of generalizing the well-studied structure of virtual braids to all Artin groups. In this talk, we will present two possible perspectives for studying these groups: an algebraic one and a topological one. From an algebraic and group-theoretic point of view, we will investigate the rigidity of these groups, specifically addressing the question of whether they can be decomposed into a direct product of two proper subgroups. The answer to this question provides interesting insights into the automorphism groups of these structures. Additionally, we will briefly discuss some topological aspects of these fascinating groups. In particular, we will explore the construction of K(π,1) spaces for certain subgroups of virtual Artin groups, linking them to a famous conjecture and existing constructions.

28 October 2025

Speaker: Simon Wood (Cardiff University)

Title: Tensor product structures on categories of vertex operator algebra modules

Abstract: Vertex operator algebras encode the symmetries of conformally invariant quantum field theories. They are generalisations of unital commutative associative algebras, where the commutative and associative properties are broken in a subtle way. It is therefore natural that these algebras admit modules and tensor products of modules. The most studied and in a sense maximally nice family of vertex operator algebras are called rational and their categories of modules form modular tensor categories. In this talk I will present recent work on the kinds of structures one should expect on categories of modules over non-rational vertex operator algebras. No prior knowledge of vertex operator algebras will be assumed.

11 November 2025

Speaker: Damien Gaboriau (École Normale Supérieure de Lyon)

Title: Dynamics on the Space of Subgroups

Abstract: To every countable group G one can attach a curious and rather mysterious topological object: the space of all its subgroups, denoted Sub(G). It is compact and totally disconnected. The group G then acts on this space by conjugation, giving rise to a natural dynamical system.  

One can explore this system by peeling away the isolated points of Sub(G) step by step, a process that eventually reveals its perfect kernel—the irreducible core that cannot be broken down any further—and a numerical invariant known as the Cantor–Bendixson rank.  

In this talk, I will describe what these objects mean, why they are interesting, and what kinds of dynamics one can observe: from orderly behavior to more surprising phenomena of chaos.

Along the way, we will look at concrete families of groups — such as abelian groups, hyperbolic groups and the more subtle (and sometimes rather tricky) Baumslag—Solitar groups — where these phenomena can be seen in action.

This relies on joint works with P. Azuelos, S. Bontemps, A. Carderi, F. Le Maître, and Y. Stalder.

18 November 2025

Speaker: Alexis Marchand (Kyoto University)

Title: Sharp spectral gaps for stable commutator length from negative curvature

Abstract: Stable commutator length (scl) is a measure of homological complexity of group elements, with connections to many topics in geometric topology, including quasimorphisms, bounded cohomology, and simplicial volume. The goal of this talk is to shed light on some of its relations with non-positive curvature. We will present a geometric method to prove sharp lower bounds for scl, giving a new proof of a theorem of Heuer on the spectral gap of scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl. Time permitting, we will show how the proof works in the special case of free groups, recovering a theorem of Duncan and Howie.

25 November 2025

Speaker: Ruadhaí Dervan (University of Glasgow)

Title: K-stability and moduli of higher-dimensional varieties

Abstract: A basic goal of algebraic geometry is to construct geometric spaces (called moduli spaces) which parametrise classes of algebraic varieties, such as curves. The problem has been solved in dimension one, through the classical construction of the moduli space of “stable” curves. In higher dimensions, while many new difficulties arise, the theory of K-stability is conjecturally the right tool to construct such moduli spaces. This has all been worked out for a special class of higher-dimensional varieties, called Fano varieties, but we know very little for more general varieties. I will discuss some recent progress in this direction using a variant of K-stability, partially joint work with Rémi Reboulet.

02 December 2025

Speaker: Dmitri Nikshych (University of New Hampshire)

Title: Witt Reduction of Braided Fusion Categories

Abstract: Braided fusion categories can be viewed as “quantum analogues’’ of metric Lie algebras—those equipped with a symmetric invariant form. This goes back to Drinfeld’s work on quasi-Hopf algebras from 1980’s. This analogy allows one to adapt familiar linear-algebraic constructions to the categorical setting.

I will describe a notion of Witt reduction (or localization) for braided fusion categories over C, obtained by factoring out certain characteristic Tannakian subcategories. The resulting reduction gives rise to a complete invariant of the category that admits an interpretation in terms of orthogonal representations of finite groups over finite fields. This leads to new approaches to the  classification of fusion categories and to natural group-theoretic questions.

09 December 2025

Speaker: Dimitra Kosta (University of Edinburgh)

Title: On strongly robust toric ideals: when Markov and Graver bases coincide.

Abstract: In the first part of the talk, I will give an introduction to Markov bases of toric ideals, which are one of the first connections between statistics and commutative algebra.  In the second part of the talk, I will discuss recent work on strongly robust toric ideals. A toric ideal is called strongly robust when the Graver basis is a minimal system of generators. In the talk, I will explain how to build a strongly robust simplicial complex which determines the strongly robust property of toric ideals. I will then discuss our results on the strongly robust property in the case of monomial curves, as well as codimension 2 toric ideals and configurations in general position. This is joint work with A. Thoma and M. Vladoiu.

Past Seminars in 2024-25

1 October 2024

Speaker: Chris Milionis (Newcastle University)

Title: The Okounkov Vershik Approach to finite dimensional representations of S_n

Abstract: The classical approach to the representation theory of the symmetric group does not take into account many algebraic features that S_n enjoys, for example its connection to Lie algebras. We will present an approach pioneered by Okounkov and Vershik, which mimics the highest weight theory of finite dimensional semisimple Lie algebras over \mathbb{C}. This heavily depends on some special elements of the group algebra \mathbb{C}S_n called Young-Jucys-Murphy(YJM) elements, used to define an analogue of the Cartan subalgebra.

8 October 2024

Speaker: Nora Szakacs (University of Manchester)

Title: The large scale geometry of inverse semigroups and their maximal group images

Abstract: The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of C*-algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric distortion of the natural map from the former to the latter. This work is joint with Mark Kambites.

15 October 2024

Speaker: Bruce Westbury (University of Warwick)

Title: A uniform R-matrix for the exceptional series

Abstract: This talk combines R-matrices and the exceptional series of simple Lie algebras. An R-matrix is a solution to the Yang-Baxter equation (with spectral parameter) and they can be constructed using the representation theory of quantum groups. There is evidence that the adjoint representations of the exceptional simple Lie algebras behave uniformly just as do the defining representations of classical groups (SO,Sp,SL). In this talk we will give a uniform formula for the R-matrices associated to the adjoint representations in the exceptional series. On one hand this is further evidence supporting the idea of the exceptional series and on the other gives new explicit formulae for R-matrices.

22 October 2024

Speaker: James Waldron (Newcastle University)

Title: Skew Hecke algebras

Abstract: Let G be a finite group, H a subgroup of G, and R a ring equipped with an action of G. To this data one can associate a ‘skew Hecke ring’. Variants of this construction have appeared in topology in the theory of Elliptic Cohomology, and in mathematical physics in the theory of Quantum Reference Frames.

I will explain a number of common generalisations of basic results concerning skew group rings and Hecke rings of finite groups. If there is time I will discuss modules over skew Hecke rings, and some potential links to other areas.

The talk will mostly be based on arXiv:2311.09038 . Everything is joint work with Leon Loveridge at USN Norway.

29 October 2024

Speaker: Martin Kerin (University of Durham)

Title: Double disk bundles

Abstract: When searching for examples satisfying certain geometric properties,
it is often convenient to examine manifolds constructed by gluing simple pieces
together. One common example of such a construction involves gluing disk
bundles together along their common boundary. On the other hand, many
geometric phenomena impose strong topological conditions on the underlying
manifold, such as the existence of a decomposition into a union of disk bundles
(glued along a common boundary).

Given that they arise frequently from these two different viewpoints, it thus
makes sense to study manifolds which decompose as a union of disk bundles in
their own right. In this talk, I will report on joint work with J. DeVito in this direction.

5 November 2024

No seminar

12 November 2024

No seminar

19 November 2024

Speaker: Andrew Duncan (Newcastle University)

Title: Formal languages and word problems for groups: beyond context free.

Abstract. Given a finite group presentation the problem of deciding if a word in the generators represents the identity element of G, or not, is in general undecidable: there is no algorithm to decide if a given word in the generators belongs to the set of all words representing the identity element of G. This set is called the “word problem” of G.

There is a well developed theory that attempts to classify group presentations by pinning down the nature of their word problems. For example it’s well known that a group has a word problem recognisable by a finite state automaton (that is a “regular” word problem) if and only if the group is finite.

At the next level, the word problem is recognised by an automaton with single memory stack (so “context free”) if and only if the group has a finite index free subgroup.

Between context free and non-recursive, not so much is yet known.

I will talk about a generalisation of context free to “multiple context free”, say something about properties of these sets of words and about which groups are known to have, or not to have, multiple context free word problems.

26 November 2024

Speaker: Ilaria Collazo (University of Leeds)

Title: Classifying Bijective Set-theoretic Solutions to the Pentagon Equation

Abstract: In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions.

3 December 2024

Speaker: Stefan Kolb (University of Newcastle)

Title: Representations of very non-standard quantum so(2N-1)

Abstract: Letzter’s theory of quantum symmetric pairs provides new quantum deformations of the Lie algebra so(n-1) considered as a Lie subalgebra of so(n). These deformations are realized as coideal subalgebras B of the Drinfeld-Jimbo quantum enveloping algebra U=Uq(so(n)). For even n=2N the algebra B has an obvious Cartan subalgebra which makes it possible to mimic quantum group constructions.

In this talk I will discuss this example as an illustration of Letzter’s theory. I will outline a Poincare-Birkhoff-Witt Theorem and the classification of finite-dimensional irreducible representations of B in this case. This part of the talk is based on Jake Stephen’s PhD thesis. Time permitting, I will discuss semi-simplicity and more recent developments.

10 December 2024

Speaker: Matthew Cordes (Heriot-Watt)

Title: Coxeter groups with connected Morse boundary

Abstract: Given a finitely generated (infinite) group, one can cook up a topological space called the Morse boundary that encodes the possible “hyperbolic” directions of some (any) Cayley graph of the group. The topology of the Morse boundary can be challenging to understand, even for simple examples. In this talk, I will focus on a basic topological property: connectivity and on a well-studied class of CAT(0) groups: Coxeter groups. I will discuss a set of criteria that guarantees that the Morse boundary of a Coxeter group is connected. In particular, when we restrict to the right-angled case, we get a full characterization of right-angled Coxeter groups with connected Morse boundary. This is joint work with Ivan Levcovitz.

28 January 2025

Speaker: Bart Vlaar (Beijing Institute of Mathematical Sciences and Applications)

Title: Recent advances for tensor K-matrices

Abstract: Solutions of ordinary (type A) braid relations arise naturally in representations of quasitriangular bialgebras. One can also consider (tensor) K-matrices, solutions of cylindrical (type B) braid relations. They appear in a more refined algebraic context, involving a suitable pair of quasitriangular bialgebra A and coideal subalgebra B. The key family of examples is given by quantum symmetric pairs (A = Drinfeld-Jimbo quantum group, B = q-deformed fixed-point subalgebra). I will review some recent joint work with Andrea Appel and show that tensor K-matrices for suitable bialgebras canonically yield ring homomorphisms from the Grothendieck ring of finite-dimensional A-modules to a (commutative subalgebra of) B. This is of particular interest if A is a quantum group of affine type.

04 February 2025

Speaker: Stuart Hall (University of Newcastle)

Title: Something about Jordan Algebras

Abstract: Jordan Algebras were introduced in the 1930s  as part of an attempt  to find useful algebraic settings for doing quantum mechanics.  Sadly for the physicists, there turns out to be a very short list of these objects with only one ‘nonstandard/exceptional’ example appearing in the theory. This will be a non-technical talk where I’ll say something about the list of Jordan Algebras, and then try to explain a link recently found by my coauthor Paul Schwahn between Jordan Algebras and infinitesimally deformable Einstein symmetric spaces.  

11 February 2025

Speaker: Philipp Bader (University of Glasgow)

Title: Comparing Teichmüller and curve graph translation lengths

Abstract: The mapping class group of a closed surface S acts on the Teichmüller space, as well as the curve graph of S. Pseudo-Anosov elements act in a particularly nice way: for a pseudo-Anosov, there exists a geodesic in both Teichmüller space and the curve graph, such that (a power of) the pseudo Anosov preserves that geodesic and acts on it by translation. Measuring these translation lengths yields two real numbers as invariants for the pseudo-Anosov. 

In this talk, we will introduce the above notions and explore the relationship between the two invariants. In particular, we’ll show the different behaviour of the two for a family of pseudo-Anosovs.

18 February 2025

Speaker: JeongHyeong Park (Sungkyunkwan University, Korea)

Title: Recent developments on weakly-Einstein manifolds

Abstract: A Riemannian manifold (M, g) is said to be weakly Einstein
if the tensor $R_iabcR^abcj$ is a scalar multiple of the metric tensor gij.
The definition and several applications of weakly Einstein manifolds were
introduced by Euh, Park, and Sekigawa in our study of a curvature identity
that holds on any 4-dimensional Riemannian manifold. In dimension $4$,
any Einstein manifold is weakly Einstein (which was the motivation for the definition).
Similarly to Einstein manifolds, weakly Einstein compact manifolds admit a variational
definition: they are critical points of the integral of the squared norm of the curvature
tensor on the space of metrics of fixed volume with parallel Ricci tensor.
In this talk, we discuss our recent progress on weakly Einstein manifolds.
(This is joint work with Y. Euh, J. Kim, S. Kim and Y. Nikolayevsky.)

25 February 2025 - MOVED TO ROOM KGVI.1.12.

Speaker: Ulrich Krähmer (University of Dresden)

Title: The ring of differential operators on a monomial curve is a Hopf algebroid

Abstract: The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid (I’ll introduce these notions of course). If the semigroup is
symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode).
Based on joint work with Myriam Mahaman.

4 March 2025

Speaker: Liao Wang (University of Bonn), Monday 4-5pm in ARMB.1.04

Title: Tensor product decomposition of quantum gl_2 \times gl_2

Abstract: Quantum symmetric pairs were introduced by Gail Letzter in the 90s as a quantization of the embedding g^theta into g, where g is a simple Lie algebra and theta is a Lie algebra involution of g. It turns out that usually the only  possible quantization as a coideal of  g^theta is not a Hopf algebra.
 The upside of this quantization is that one can restrict the quantum group representations. The downside however, is that despite a classification and explicit Serre type presentation, the calculation is very lengthy and technical. Using Schur—Weyl duality and the theory of Ariki—Koike algebras, there is an abstract classification of simple submodules of tensor powers of the natural representation. Furthermore, by calculating common eigenspaces of Jucyc—Murphy elements (using sagemath), we calculated explicit  decomposition of small tensor powers of the natural representation into irreducibles. We found operators that mimic the two copies of sl_2, and we are hoping to
 find a nice description of irreducible representations there. 

11 March 2025

Speaker: cancelled

18 March 2025

Speaker: cancelled

25 March 2025

Speaker: cancelled

29 April 2025

Speaker: Collin Bleak (University of St. Andrews)

Title: Embedding certain automatic groups into the rational group R

Abstract: We introduce left-continuous automatic groups as a subclass of the well-known automatic groups.  The class of left-continuous automatic groups is a bit mysterious as a subclass of the automatic groups, but, we know at least that the class contains the CAT(0) Cubical Complex groups (CCC groups), which themselves represent a broad class of groups of topical interest.  Our main theorem states that all left-continuous automatic groups embed as subgroups of the rational group R, a group introduced in 2000 by Grigorchuk, Nekrashevych, and Suschanskii.  We will provide definitions and examples of these various groups along the way, and have some discussion as well of different forms of boundaries of groups.  We also discuss how similar embedding theorems have been of use in larger programs of discovery.  Joint work with Belk, Chatterji, Matucci and Perego. 

06 May 2025

Speaker: Maxim Nazarov (University of York)

Title: Yangian of the periplectic Lie superalgebra

Abstract: Yangians are deformations of the enveloping algebras of Lie algebras
of polynomial functions of one variable with values in any
finite-dimensional simple Lie algebra. They were defined 40 ago by
Drinfeld, who generalised examples that emerged in Quantum Physics
about 5 years before. Since then, Yangians have been extensively
studied by mathematicians and physicists. They also found remarkable
applications in Lie Theory and Quantum Field Theory.

Yangians can be defined for all classical Lie superalgebras as well.
The theory of these Yangians began unfolding only in the last years.
The aim of this talk is to present recent results on the Yangian
corresponding to the periplectic Lie superalgebra. The latter is a
superanalogue of the orthogonal and the symplectic Lie algebra at the
same time.