London Algebra Colloquium

Abstract: For integers $k,l,m > 1$, the ordinary {\em triangle group\/} $\Delta^+(k,l,m)$
is the group with presentation $\langle\, x,y \ | \ x^k = y^l = (xy)^m = 1 \,\rangle$,
or equivalently $\langle\, x,y,z \ | \ x^k = y^l = z^m = xyz = 1 \,\rangle$.
Such groups play an important role in the study of large automorphism groups
of algebraic curves and compact Riemann surfaces, and of regular maps on
orientable and non-orientable surfaces.

Much of the early part of this study (dating back over 100 years to work by Dyck,
Klein and Burnside, for example) considered only small quotients of triangle groups,
and subsequent work (by Macbeath, Higman and others) concentrated on finite
simple quotients (such as $\PSL(2,q)$, the alternating groups,
and even some of the exceptional finite simple groups).

But surprisingly, the recent determination of all orientably-regular maps of
`small’ genus (now up to genus 1501) has shown that finite simple groups and
finite insoluble groups account respectively for less than 0.1\% and less than
7\% of the associated quotients of ordinary triangle groups (with quotient order
up to about $100000$), while finite soluble quotients account for over 93\%.

Very little attention has been paid to soluble quotients, and in this talk I will
describe some recent research (with my PhD student Darius Young) that helps to
correct this relative lack of attention. In particular, I will explain how every
non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient
of derived length $c$ for some $c \le 3$, and has infinitely many such quotients
of derived length $d$ for every $d > c$.

Also I will report on some work by Darius in the last few weeks in which he proved
that the natural density (in the positive integers) of the set of orders of finite
quotients of a triangle group $\Delta^+(k,l,m)$ is {\bf zero} for every triple $(k,l,m)$.

This work completed an investigation by Larsen (2001), but avoids resorting to the
classification of finite simple groups.
It also led to the following (proved by MC \& Gabriel Verret): the natural density
of the orders of finite quotients of the free product
$C_p * C_q = \langle\, x,y \ |\ x^p = y^q = 1\,\rangle$
is zero whenever $p$ or $q$ is odd, but is positive whenever $p$ and $q$ are even.

Furthermore, this had unexpected consequences for the densities of orders of finite
arc-transitive graphs of given valency, and also led to a much more general theorem
about possibilities for the density of orders of finite quotients of a given
finitely-generated infinite group.

Abstract: This is a talk about a relative version of Koszul duality, and how it can tell us about the homological algebra of (seemingly) non-Koszul local rings. It’s all joint work with James Cameron, Janina Letz, and Josh Pollitz.

I’ll start by reminding everyone how classical Koszul duality works (at least some parts of it). Then I’ll talk about this relative definition: A homomorphism of commutative local rings is Koszul its derived fibre F is formal and the homology H(F) is a Koszul algebra in the classical sense. The goal is first to make sense of this, then to convince you that this is a good definition with interesting consequences, and that it actually happens all the time.

After all that, I’ll hopefully give some applications to the asymptotic homological algebra of local rings, by taking the classical Koszul resolutions written down by Priddy and generalising them with some A-infinity tricks.

Abstract: One of the most fundamental ways to understand the action of an algebraic group on a variety, is to determine its generic stabilizer, if it exists. Recent work by Guralnick and Lawther has settled the problem for actions on subspaces of irreducible modules. I will talk about an extension of these results to actions on subspaces with a form, focusing on some of the computational methods involved.

Abstract:In this talk I will give an introduction to analogues to the classical finiteness conditions FP_n for totally disconnected locally compact groups (tdlc). I will present a construction of non-discrete tdlc groups of arbitrary finiteness length. As a bi-product we also obtain a new collection of (discrete) Thompson-like groups which contains, for all positive integers n, groups of type FP_n but not of type FP_{n+1}. This is joint work with I. Castellano, B. Marchionna, and Y. Santos-Rego.

Abstract: Let p be an odd prime. In this talk we will discuss a part of the generation-2 proof of the CFSG that deals with groups of both even- and p-type. This is joint work with R. Lyons and R. Solomon.

Abstract: A group G is said to be equationally Noetherian if every system of equations over G is equivalent to a finite subsystem. It is easy to see that all linear groups are equationally Noetherian. Deep work of Sela and Groves–Hull uses geometric methods to show that equational Noetherianity holds in groups which admit sufficiently well-behaved actions on non-positively curved spaces. In this talk, I will show how such geometric methods can be extended to encompass wider classes of groups. As a corollary, we will see that all free-by-cyclic groups are equationally Noetherian. This is based on joint work with Motiejus Valiunas.

Abstract: A finite group is never the union of conjugates of a proper subgroup. Equivalently, a nontrivial finite transitive permutation group always contains a derangement (a fixed-point-free permutation). This is a theorem of Jordan (1872) and it follows from a simple counting argument. I will discuss several variations on this theorem that have attracted attention over the decades, highlighting connections with number theory, graph theory and extremal combinatorics. I will then focus on the conjecture that a finite group is never the union of conjugates of two proper subgroups of the same size and discuss joint work with David Ellis towards this. Along the way, I will discuss recent work with Martin Liebeck on representations of extensions of simple groups, generalising work of Feit and Tits (1978).

Abstract: Originating with the familiar representation-theoretic concept, we have nowadays come to understand $G$-complete reducibility as a geometric property, characterised using cocharacters of $G$ and closure of $G$-orbits on certain varieties, and as a combinatorial property, characterised via the action of subgroups on the spherical building of $G$. This allows a number of generalisations. When a subgroup $K$ of $G$ acts on $G^n$, we are led to ‘relative complete reducibility.’ In the presence of a Frobenius endomorphism $\sigma$ acting on $G$ and its building, we arrive at so-called ‘$\sigma$-complete reducibility’. And finally, we can replace $G$ by groups such as Kac-Moody groups: Infinite-dimensional analogues with a twin building, in which the notion of opposition (and hence complete reducibility) can still be defined and studied.

This talk consists of joint work with Michael Bate, Ben Martin, Gerhard Roehrle and their former students Chris Attenborough, Falk Bannuscher, Maike Gruchot and Tomohiro Uchiyama, and incipient joint work with my PhD student Harvey Sykes.

Abstract: Let G be a finite transitive permutation group on a set X and recall that an element in G is a derangement if it has no fixed points on X. Let D(G) be the set of derangements in G and define d(G) = |D(G)|/|G|. By combining recent work of Larsen, Shalev and Tiep with a theorem of Fulman and Guralnick, it follows that G = D(G)^2 and d(G) > 0.016 for all sufficiently large simple transitive groups G.

In this talk, I will report on some recent joint work with Marco Fusari, which seeks to extend both results in several directions. For example, we prove that G = D(G)^2 and d(G) is at least 89/325 for all finite simple primitive groups with soluble point stabilisers, and we show that the lower bound on d(G) is best possible. In addition, we prove that every finite simple transitive group can be generated by two conjugate derangements.

Abstract: We discuss the algebraic structure of KLR algebras by way of the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. Combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Instead of looking only at the sets of Dyck tableaux (which enumerate the q-decomposition numbers) we look at the relationships for passing between these Dyck tableaux. In fact, this “meta-Kazhdan-Lusztig combinatorics” is sufficiently rich as to completely determine the Ext-quiver and relations presentation of these algebras.

Abstract:

Abstract: We say that a field equipped with a valuation and derivation is differentially henselian if the valuation is henselian, and the derivation satisfies a certain genericity property. They are examples of differentially large fields, a notion introduced by León Sánchez and Tressl as a differential analogue for large (also known as ample) fields. We will give a brief introduction to the notions of differential largeness and henselian valued fields, followed by results on characterising such fields in terms of differential algebras followed by an application of the Weil descent to prove properties about algebraic extensions.

Abstract: A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measure preserving transformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We demonstrate paradoxical colouring rules defined via u.s.c. convex valued correspondences (if the colours b and c are acceptable by the rule than so are all convex combinations of b and c). This connects measure theoretic paradoxes to problems of optimisation and shows that there is a continuous mapping from bounded group-invariant measurable functions to itself that doesn’t have a fixed point (but does has a fixed point in non measurable functions).

Abstract: The chromatic filtration gives a powerful organising principle for the stable homotopy category, and thereby the stable homotopy groups of spheres. It is a longstanding heuristic that the structure of each graded piece should be controlled by the action of the automorphism group of a Lubin-Tate formal group. In this talk I will give an introduction to these objects, and discuss recent work to make this heuristic precise via the condensed mathematics of Clausen-Scholze. I’ll also use these results to deduce some new computational results. I will not assume prior knowledge of higher algebra or stable homotopy theory.

Abstract: A classical result of model theory asserts that a particular choice of topological group characterises a first order, ω-categorical theory up to the level of bi-interpretability – a notion of when two theories are ‘equivalent’. Recent work by Ben Yaacov has shown that, by generalising to a topological groupoid, the ω-categoricity assumption can be dropped.

However, in many ways, bi-interpretability is too rigid a notion of ‘equivalence’. In the topos-theoretic approach to first-order logic, it is more natural to consider Morita equivalence of theories, which can then be compared to similar notions of Morita equivalence for topological and algebraic structures.

We present work in progress, establishing a bi-equivalence between Grothendieck toposes (with enough points) and a bi-category of fractions on topological groupoids (as opposed to localic, or pointfree, groupoids, as in Moerdijk’s approach). As a consequence, we deduce that two first-order theories are Morita equivalent if and only if any pair of their representing topological groupoids admits a co-span of weak equivalences.

Abstract: We shall report on a class of
noncommutative products of finite-dimensional Euclidean spaces
. They are described by natural families of coordinate algebras which are quadratic and are associated with an -matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence the algebras enjoy a list of nice properties, being regular of finite global dimension. There is a natural description of generalized Clifford algebras .

Notably, we have eight-dimensional noncommutative euclidean spaces with particularly well behaved ones having deformation parameter . Quotients include seven-spheres as well as noncommutative quaternionic tori . There is invariance for an action of on the torus in parallel with the action of on a `complex’ noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. There are also natural noncommutative principal fibration.

Abstract: When studying algebraic objects with units, it is natural to assume morphisms between them preserve these units. When we study non-unital objects, however, we find that in many contexts, simply dropping the unitality constraint from our definition of a morphism gives us too many ‘degenerate’ morphisms. Instead we need a sort of approximate unitality variously labelled ‘regularity’ or ‘non-degeneracy’, and often we find that this approximate unitality gives an extension of our morphism to a ‘large unitalization’ of the non-unital object. Combining existing results with new work, we look at a variety of settings where this happens, including regular modules over non-unital rings, enriched semicategories as studied by Borceux et al, and the theory of multiplier algebras of operator algebras.

Abstract: The representation type of a finite-dimensional k-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules. Intuitively, a finite-dimensional k-algebra is of tame representation type if we can classify its finite-dimensional modules and wild representation type if its module category contains a copy of the category of finite-dimensional modules of all other finite-dimensional k-algebras. An archetypical (although not finite-dimensional) tame algebra is k[x]. The structure theorem for finitely generated modules over a PID describes its finite-dimensional modules. Drozd’s famous dichotomy theorem states that all finite-dimensional algebras are either wild or tame.

The (theory of) a class of modules is said to be decidable if there is an algorithm which given a sentence in the language of modules (a sentence is a particular kind of statement about modules) answers whether it is true in all modules in that class. A long-standing conjecture of Mike Prest claims that the (theory of) the class of all modules over a finite-dimensional algebra is decidable theory if and only if it is of tame representation type. The reverse direction of this conjecture is often hard to prove even in particular examples. One difficulty is that the conjecture talks about all modules rather than just finite-dimensional ones. In this talk I will present work in progress around and in support of a new conjecture, inspired by Prest’s conjecture, which claims that the (theory of) the class of finite-dimensional modules over a finite-dimensional algebra is decidable if and only if it is of tame representation type.

No background knowledge in logic or model theory will be assumed.

Abstract: Given a finite group G and a prime p, the set of complex valued irreducible characters of G is naturally partitioned into families called p-blocks. The p-block distribution of characters plays a central role in the modular representation theory of finite groups and giving an explicit description of this distribution for finite (quasi)-simple groups has been a long ongoing endeavour. I will talk about recent joint work with Gunter Malle on the p-block distribution of characters of finite quasi-simple exceptional groups of Lie type and on an application to a conjecture of G.R. Robinson on defects of characters.

Abstract: After giving the formulation of the Hodge conjecture for tropical varieties, we will present a proof in the case the tropical variety admits a triangulation with rational coordinates. This provides a partial answer to a question of Kontsevich. The proof uses Hodge theoretic properties of tropical varieties established in companion work, that we will review in the talk.

Based on joint work with Matthieu Piquerez.

Abstract: A group Γ is sofic if elements of Γ can be distinguished by almost-actions on finite sets. It is a major unsolved problem to determine whether all groups are sofic. One approach to this problem which has gained much recent attention is that of “permutation stability”, that is, showing that almost-actions of a group are controlled by its actions. We introduce a “local” generalization of permutation stability, under which actions are replaced by partial actions. We exhibit an uncountable family of groups which are locally permutation stable but not permutation stable, coming from topological dynamics. The proof is based on a criterion for local stability of amenable groups, in terms of invariant random subgroups.

Abstract: A group automorphism $\varphi: \Gamma \to \Gamma$ induces the action $g\cdot x= gx\varphi(g)^{-1}$ on $\Gamma$. The orbits of such action are called twisted conjugacy classes (also known as Reidemeister classes). In the past few years, the sizes of such classes have been intensively investigated. One of the main goals in the area is to classify groups where all classes are infinite. For groups not having such property, one is interested in the possible sizes of the classes. In this talk, we will discuss how zeta functions of groups can be used to determine these sizes for certain nilpotent groups.

Abstract: Percolation theory is an important branch of probability and statistical physics. Originally intended to study the passage of water through a porous stone, and subsequently used to explain the behaviour of magnets at high temperatures, it is also a source of beautiful pure mathematics. A classical percolation model is as follows: start with an infinite graph (e.g. the integer lattice in R^2), and then either delete or retain each edge independently at random, with retention probability p, say. What is the probability that the resulting random subgraph have an infinite connected component? Duminil-Copin, Goswami, Raoufi, Severo and Yadin (Duke Math. J., 2020) recently classified those infinite Cayley graphs for which there exists p < 1 such that there is almost surely an infinite connected component, verifying a conjecture of Benjamini and Schramm from 1996. In this talk, I will describe joint work with Tom Hutchcroft in which we prove an analogous result for finite Cayley graphs, essentially verifying a conjecture of Benjamini from 2001. There are many different ingredients; I will concentrate on the more algebraic ones.

Abstract: In the 1970s, C. T. C. Wall compiled a list of nine problems concerning the topology of group presentations including the Generalised Andrews-Curtis conjecture and Wall’s D2 problem. These problems are often stated as conjectures, but it is widely believed that there exist counterexamples to most (if not all) of them. Whilst counterexamples have been proposed for many of these problems, the major obstacle lies in the development of the algebraic techniques necessary to show they work. Over the last few years, I have been exploring links between these problems and various aspects of integral representation theory such as the classification of stably free $\mathbb{Z}[G]$-modules. The aim of the talk will be to give a light introduction to this topic and explain some recent progress on these problems which includes a counterexample to one and an affirmative answer to another in a special case.

Abstract:  One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.

Abstract: For a reductive group scheme G over a DVR O with finite residue field F_q, Lusztig defined (virtual) representations R_T^{\theta} of the finite group G(O/p^r), where p^r is a power of the maximal ideal p of O and \theta is a character of T(O/p^r), where T is a maximal torus. These representations are direct higher-level analogues of the classical Deligne–Lusztig representations of reductive groups over finite fields, obtained for r = 1. Just like their classical analogues, these representations are irreducible (up to sign) whenever \theta is suitably “regular”.

On the other hand, prior to Lusztig’s cohomological construction, Gérardin had defined representations of the same groups (with certain restrictions) attached to the same set of parameters \theta, using non-cohomological explicit methods, such as Clifford theory. Thus, in 2004 Lusztig raised the problem of whether the higher level Deligne–Lusztig representations R_T^{\theta} are the same as those of Gérardin. This was proved in 2016 in joint work with Chen for even powers r, yielding as a consequence a formula for the dimension of a higher-level Deligne–Lusztig representation.
The case when r is odd is much harder and requires several additional ideas. I will talk about recent progress on this which comes very close to answering Lusztig’s problem and is expected to lead to a full answer. In particular, it is now known that for the groups GL_n(O_r), the representations R_T^{\theta} (when irreducible) are precisely the regular semisimple representations. This is joint work with Zhe Chen.

Abstract: I will introduce one-relator groups, briefly survey the state-of-the-art, and discuss some recent developments obtained in a collaboration with Marco Linton.

Abstract: For a permutation group G we can define two numerical invariants, the maximal irredundant base size and the relational complexity, denoted I(G) and RC(G) respectively. Roughly speaking the maximal irredundant base size is the size of the “worst” base for G, and relational complexity is a measure of when a local property extends to a global one.

We begin by defining these numerical invariants and then cover some examples which illustrate the “nice” behaviour of I(G) and the “nasty” behaviour of RC(G). We then discuss how we can bound RC(G) by studying “nicer” numerical invariants. Finally we sketch a calculation of the exact value of relational complexity for a particular group.

Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on ΩxΩ component wise. Define the orbitals to be the orbits of G on ΩxΩ. The diagonal orbital is the orbital of the form ∆ = {(α, α)|α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α,β)∈ Γ with α,β∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

Abstract: Let G be the general linear or symplectic group over an algebraically closed field of characteristic p>0.
I will discuss how cap and cap-curl diagrams can be used to compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for G, under the assumption that p is bigger than the greatest hook length in the partitions involved.
This also gives decomposition numbers for the Brauer algebra under the same assumptions.
If time allows, I will define certain Kazhdan-Lusztig polynomials and obtain that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott.
Our work combines ideas from work of Cox- De Visscher, Brundan-Stroppel, and Shalile with techniques from the representation theory of reductive groups.

Abstract: A transfer system is a graph on a lattice satisfying certain restriction and composition properties. They were first studied on the lattice of subgroups of a finite group in order to examine equivariant homotopy commutativity, which then unlocked a wealth of links to combinatorial methods. On a finite total order [n], transfer systems can be used to classify different homotopy theories on [n]. The talk will involve plenty of examples and not assume any background knowledge.

Abstract: Cocycles appearing in group (co)homology are manifestations of geometric objects such as torsors and gerbes. Likewise, cocycle calculations can often be interpreted as geometric constructions with these objects. Category theory provides a convenient tool to translate back and forth between the geometry side and the cocycle side. This also serves to illustrate how certain complicated formulas pop up in mathematical physics (eg, topological quantum field theory), twisted K-theory, etc.
In this talk, I will discuss the category theoretic viewpoint on (twisted)
representation theory of finite groups and using ideas from calculus derive old and new formulas that were initially motivated by TQFT.

Abstract: Kazhdan-Lusztig polynomials are remarkable polynomials associated to pairs of elements in a Coxeter group W. They describe the base change between the standard and Kazhdan-Lusztig bases for the corresponding Hecke algebra. They were discovered by Kazhdan and Lusztig in 1979 and have found applications throughout representation theory and geometry. In 1987, Deodhar introduced the parabolic Kazhdan-Lusztig polynomials associated to a Coxeter group W and a standard parabolic subgroup P. These describe the base change between the standard and Kazhdan-Lusztig bases for the anti-spherical module for the Hecke algebra. (We recover the original definition of Kazhdan and Lusztig by taking the trivial parabolic subgroup).
(Anti-spherical) Hecke categories first rose to mathematical celebrity as the centrepiece of the proof of the (parabolic) Kazhdan-Lusztig positivity conjecture. The Hecke category categorifies the Hecke algebra and the anti-spherical Hecke category categorifies the anti-spherical module. More precisely, it was shown by Elias-Williamson (and Libedinsky-Williamson) that the (parabolic) Kazhdan-Lusztig polynomials are precisely the graded decomposition numbers for the (anti-spherical) Hecke categories over fields of characteristic zero, hence proving positivity of their coefficients.
The (anti-spherical) Hecke categories can be defined over any field. Their graded decomposition numbers over fields of positive characteristic p, the so-called (parabolic) p-Kazhdan-Lusztig polynomials, have been shown to have deep connections with the modular representation theory of reductive groups and symmetric groups. However, these polynomials are notoriously difficult to compute.
Unlike in the case of the ordinary (parabolic) Kazhdan-Lusztig polynomials, there is not even a recursive algorithm to compute them in general.
In this talk, I will discuss the representation of the anti-spherical Hecke categories for (W,P) a Hermitian symmetric pair, over an arbitrary field. In particular, I will explain why the decomposition numbers are characteristic free in this case.
This is joint work with C. Bowman, A. Hazi and E. Norton.

Abstract: Khovanov-Sazdanovic recently introduced a class of monoidal categories obtained as quotients of categories of 2-d cobordisms based on evaluating closed genus surfaces according to a rational power series . Examples captured by this construction are Deligne’s interpolation categories of . After a general introduction on interpolation categories in the style of Deligne, I will talk about joint work with J. Flake and S. Posur where we describe the associated graded category of . The main results is a classification of indecomposable object in over arbitrary fields. Specifying to the case of over a field of characteristic , the associated graded Grothendieck ring is an analogue of the ring of symmetric functions obtained from modular representation theory.

Abstract: I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a first-order definable group G, living in the field, which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but here assuming also that the group is a definable subgroup of GL(n, K).

Abstract: In this talk I will describe a framework for developing analytic algebra and geometry using bornological modules. I will explain why bornological modules are very suitable for doing algebra, and how using the tools of relative algebraic geometry one can use them to define derived global analytic stacks. I will give some examples and also discuss how this approach relates to condensed mathematics. This is joint work with Federico Bambozzi, Oren Ben-Bassat, and Jack Kelly.

Abstract:

Abstract: We provide axioms that guarantee a category is equivalent to that of Hilbert spaces, either with continuous linear maps or with linear contractions. The axioms are purely categorical and do not presuppose any analytical structure such as probabilities, convexity, complex numbers, continuity, or dimension. We’ll discuss the axioms, sketch the proofs of the theorems, and survey open questions, further directions, and context. (Based on arxiv:2109.07418 and arXiv:2211.02688.)

Abstract: In recent work with He, Lee, and Pozdnyakov, we observed an unexpected phenomenon involving the average value of as varies over the set of elliptic curves with conductor in certain intervals. In this talk, I will outline how this was first observed as a consequence of principal component analysis (a technique from of data science), discuss some generalisations to other arithmetic objects (such as Dirichlet characters, modular forms, and genus 2 curves), and explore some frequently asked questions (for example, what happens if we order by height instead of conductor).

Abstract: Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this talk, I will show how surface combinatorics yield approaches to cluster structures on the coordinate ring of the Grassmannians. I will also point out an expected link to root systems associated to the Grassmannian cluster categories.

Abstract: Roughly speaking a Beauville surface is a complex surface defined by letting a finite group G act on a product of two complex curves. These have numerous nice properties and are easier to study than most complex surfaces since their definition can be entirely internalized into G. This naturally raises the question of what the genus of the corresponding surface is. One natural question posed by Zvonkin concerns minimizing the genus and this was largely answered by Jones and Pierro’s work on Hurwitz groups. But not every group is Hurwitz – what about the rest?

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 -categories. Some of this is joint work with M. Szymik (Sheffield) other with A. Phimister (Leicester).

Abstract: In this talk I will give an overview of RoCK blocks and their significance in attacking Broue’s abelian defect group conjecture. I will then go on to talk about RoCK blocks for double covers of symmetric groups and the unexpected difficulties one encounters when compared to the symmetric group situation studied by Chuang-Kessar. The main tool used in overcoming these difficulties is that of Quiver Hecke superalgebras, mirroring the work of A. Evseev who used KLR algebras in the symmetric group setting. (This is ongoing joint work with A. Kleshchev.)

Abstract: The core of a finite-dimensional modular representation $M$ of a finite group is its largest non-projective summand. Dave Benson and Peter Symonds introduced a new invariant for $M$. This invariant is a result of studying the asymptotics of the core of tensor powers of $M$. In this talk, we will see some interesting properties of this invariant. We obtain a closed formula for computing this invariant for signed permutation modules of the symmetric groups. We prove that the sequence of dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic. When $M$ is $\Omega ^+$-algebraic then these dimension sequences are eventually linearly recursive. This partially answers a conjecture by Benson and Symonds.

Abstract: In this work, we develop a new method to adjoin roots to ring spectra and show that this process results in interesting splittings in algebraic K-theory.
In the first part of my talk, I will provide motivation for algebraic K-theory and highly structured ring spectra. After this, I will discuss trace methods, a program that provides computational tools for algebraic K-theory, and introduce our results.
This is joint work with Tasos Moulinos and Christian Ausoni.

Abstract: Dade’s Conjecture plays an important role in modular representation theory of finite groups as it implies most of the so called Local-Global conjectures. In 2017 Spaeth introduced a strengthening of Dade’s Conjecture, known as the Character Triple Conjecture, and showed that if this new statement holds for quasisimple groups, then Dade’s Conjecture holds for every finite groups. In this talk we consider this problem for quasisimple groups of Lie type in nondefining characteristic. First, by extending results of Broué-Malle-Michel and Cabanes-Enguehard we show how to understand the distribution of characters into blocks for groups of Lie type by providing a description of the so-called Brauer-Lusztig blocks in terms of generalized Harish-Chandra series. Moreover, we show how to obtain a parametrization of generalized Harish-Chandra theory which is compatible with Clifford thery and with the action of automorphisms. Finally, we apply these results and show how this parametrization can be used to prove the Character Triple Conjecture for quasisimple groups of Lie type. This reduces the verification of the Character Triple Conjecture to certain questions on the extendibility of characters of Levi subgroups that are being studied by other authors.

Abstract: Say we are given only the ring structure of a group ring RG of a finite group G over a commutative ring R. Can we then find the isomorphism type of G as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of kG as a ring determine the isomorphism type of G as a group, if G is a p-group and k a field of characteristic p?

After giving an overview of some history of general isomorphism problems and the state of knowledge on the modular formulation, I will present recent results on the problem which include a counterexample to the Modular Isomorphism Problem.

Abstract:Let B be a block algebra of a finite group with respect to an algebraically closed field of positive characteristic.
We show that the number of irreducible ordinary characters of B can be bounded by the Cartan matrix of (a Brauer correspondent of) B. An inductive method of Usami-Puig, often allows the construction of Cartan matrices on a local level. An computer implementation of their method leads to new evidence for Brauer’s k(B)-Conjecture.

Abstract: After recalling the basic definitions of category theory and block theory of finite groups, I will introduce the notion of functorial equivalence of blocks, developed in a recent joint work with Deniz Yilmaz.

For a commutative ring $R$, and a field $k$ of characteristic $p>0$, we introduce the category of diagonal $p$-permutation functors over $R$. To a pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $F_{G,b}$, and we say that two such pairs $(G,b)$ and $(H,c)$ are functorially equivalent over $R$ if the functors $F_{G,b}$ and $F_{H,c}$ are isomorphic.

We show that the category of diagonal $p$-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a full description of the simple functors, and explicit formulas for their multiplicities as summands of $F_{G,b}$. It follows that functorial equivalence preserves the defect groups of blocks and their number of simple modules. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan’s finiteness conjecture.

Abstract: Let G be a finite group and k a field of positive characteristic p dividing |G|. Endotrivial kG-modules are finitely generated kG-modules M such that the set of k-linear transformations End_k(M) forms a permutation kG-module. The stable isomorphism classes of endotrivial kG-modules form a finitely generated abelian group. In this talk, we will review the background and we will present the main results. We will end with a selection of examples on endotrivial modules for Very Important Groups.

Abstract: Let G be a linear algebraic group over an algebraically closed field k. A major strand of algebraic group theory is to study the subgroup structure of G: can we describe the subgroups H of G (up to conjugacy) and understand how they fit together? The problem becomes more tractable if we put extra hypotheses on H. For instance, we have a good understanding of the set of connected reductive subgroups H when G is simple. Suppose G is connected and reductive. A subgroup H of G is said to be G-irreducible if it is not contained in any proper parabolic subgroup of G. Recently we proved the following result: if H is a connected reductive subgroup of G that contains a regular unipotent element of G then H is G-irreducible. This result was proved by Testerman and Zalesski and later extended by Malle and Testerman. Our proof is short and carries over nicely to the case when H or G is nonconnected. We have also proved analogous results for Lie algebras and finite groups of Lie type. I will discuss these results and sketch the ideas behind the proofs. This is joint work with Michael Bate and Gerhard Röhrle.

Abstract: Let G be a finite primitive permutation group acting on a set X with nontrivial point stabiliser G_x. We say that G is extremely primitive if G_x acts primitively on every orbit in X \ {x}. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. After surveying previous results, we will discuss joint work with Tim Burness towards completing this classification dealing with the almost simple groups with socle an exceptional group of Lie type. We will describe the various techniques used in the proof and, discuss the results we proved on bases for primitive actions of exceptional groups.

Abstract: If p and q are different primes and G is a finite group, it is not generally reasonable to expect meaningful interactions between the p-representation theory of G and its q-representation theory. However there are some exceptions, specially when we deal with principal blocks. For instance, if B_p (G) denotes the principal p-block of G, Bessenrodt, Navarro and Olsson proved that Irr(B_p(G))=Irr(B_q(G)) if and only if p and q do not divide |G|. Another nice interaction between B_p(G) and B_q(G) is studied in works by Malle and Navarro and by Liu, Willems, Wang, and Zhang, who prove a version of Brauer’s Height Zero conjecture for principal blocks and two primes. In this talk we give an overview of this kind of results and we propose some problems involving principal blocks for different primes.

Abstract: A triple of finite groups (H,M,K), usually written H> M<K, is called a primitive amalgam if M is a subgroup of both H and K, and each of the following holds:
(i) Whenever A is a normal subgroup of H contained in M, we have N_K(A)=M; and (ii) whenever B is a normal subgroup of K contained in M, we have N_H(B)=M.

Primitive amalgams arise naturally in many different contexts across pure mathematics, from Tutte’s study of vertex-transitive groups of automorphisms of finite, connected, trivalent graphs; to Thompson’s classification of simple N-groups; to Sims’ study of point stabilizers in primitive permutation groups, and beyond. In this talk, we will discuss some recent progress on the central conjecture from the theory of primitive amalgams: the Goldschmidt–Sims conjecture. Joint work with L. Pyber.

Abstract: Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will discuss these conjectures and their relationship to other conjectures and properties of groups. I will then explain how modern solvers for Boolean satisfiability can be applied to them, producing the first counterexample to the unit conjecture.

Abstract: Until recently all known sharply 2-transitive groups were of the form K x K^* for some (near-)field K. I will explain recent constructions of sharply 2-transitive groups without abelian normal subgroups, how to build simple sharply 2-transitive groups and discuss other conditions one might ask of such permutation groups.

Abstract: This talk will be a discussion of infinite groups which share properties with finite groups, either in their actions (strongly bounded groups) or in their relationship to proper subgroups (Jonsson groups). There will be a historical review and exposition of some recent constructions of such groups. Includes joint work with Saharon Shelah.

Abstract: A basic problem in modular representation theory of finite groups is to understand decomposition numbers, that is, how an irreducible representation of a group in characteristic 0 decomposes into irreducible representations over a field of positive characteristic. This problem is open even for symmetric groups. I will discuss the case of a finite group of Lie type B or C in non-defining characteristic. The combinatorics of higher-level Fock spaces plays an important role in this setting, as in the representation theory of type B Hecke algebras at roots of unity. This allowed Olivier Dudas and I to determine some new decomposition numbers of these groups. Based on recent and ongoing joint work with Olivier Dudas.

Abstract: The last two talks I gave at various places were on the maximal subgroups of ²E₆(q) and the maximal subgroups of E₇(q), so this is the next obvious step. In this talk I will discuss the programme to classify the maximal subgroups of the finite simple groups E₈(q), and the progress so far made. If time permits, some indications of the new difficulties that present themselves with E₈, rather than smaller groups, will be discussed.

Abstract: When studying decomposition matrices for spin representations of symmetric groups a problem, which does not arise for the non-spin case, is given by pairs of irreducible representations labeled by the same partition. This problem can be avoided by considering generalised decomposition matrices instead. Even for generalised decomposition matrices however not much is known. For example not even the form of the generalised decomposition matrix is known in general. In this talk I will present some results on such matrices.

Abstract: The Hecke category recently plays very important role in modular representation theory. Here the Hecke category means a categorification of the Hecke algebra. There are several realizations of the Hecke category. In this talk, I will explain a new realization. The realization is motivated by the theory of Soergel bimodules. I will also explain some applications of this realization.

Abstract: I will first give a brief survey of some previous results with Louise Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras.

I will then move on to discussing a recent project with Rob Muth and Louise Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to give all composition factors (including their grading shifts), and show that in most characteristics, these Specht modules are in fact semisimple. In some other small characteristics, we can explicitly determine their structures, including some in which the modules are `almost semisimple’. I will present this story, with some running examples that will help the audience keep track of what’s going on.

Abstract: A difference ring is a ring with a distinguished endomorphism. Such objects can be associated with recurrence relations/difference equations, recursively defined sequences, dynamical systems, functional equations and many other contexts.

We develop a Galois theory of difference ring extensions modelled on Janelidze’s categorical theory, where the relevant extensions are classified in terms of difference Galois groupoids. Given that the space of connected components of a difference ring can be a profinite space with a continuous self-map, the considerations take on a topological dynamics flavour, and we discuss some connections with symbolic dynamics.

Disclaimer: this theory is unrelated to Picard-Vessiot style Galois theory of linear difference equations.

Abstract: A main topic in the representation theory of finite groups is to understand how much information about the structure of Sylow subgroups can be obtained from the character table of a group.
I will explain how to detect, after an easy inspection of the character table of a group G, whether or not a Sylow 2-subgroup of G is generated by 2 elements. This talk is based on joint works in collaboration with Gabriel Navarro, Noelia Rizo and Mandi Schaeffer Fry.

Abstract: The family of tilting modules plays a crucial role in the representation theory of the special linear group SL_n and the quantum group of the corresponding Lie algebra, and one ideally aims to understand their structure completely. In this talk, I will discuss recent progress on tilting modules for SL₂, where we study them as objects inside a monoidal category governed by well-known Temperley-Lieb diagrammatics. We work in the generalised setting based on two characteristic parameters, namely the characteristic of the underlying field and a root of unity. In this setting, we determine all decompositions of tensor products of simple tilting modules into indecomposable tilting modules. We are then able to explicitly describe the morphisms that project onto these indecomposable summands in some of these cases. These morphisms are known as Jones-Wenzl projectors, which we generalise to this arbitrary setting (and have recently been defined independently by Martin and Spencer). We give a description of the decomposition of the tensor product of these arbitrary Jones-Wenzl projectors into orthogonal, primitive idempotents (in our known cases).

This is joint work with Daniel Tubbenhauer, Paul Wedrich and Jieru Zhu.

Abstract: (work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic l. There is an algorithm to compute them for GL(n,q) when l is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group GU(n,q). I will first explain how one can produce a “natural” self-equivalence in the case of GL(n,q) coming from the topology of the Hilbert scheme of the complex plane . The combinatorial part of this equivalence is related to Macdonald’s theory of symmetric functions and gives (q,t)-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between GU(n,q) and GL(n,–q).

Abstract: The partition algebra is closely connected to the symmetric group. In fact, it can be defined as the centraliser algebra of the diagonal action of the symmetric group on the tensor product of the natural permutation module.
In this talk I will explain how one can generalise much of the combinatorics used to study the symmetric group to the partition algebra.
I will also discuss how this can help us shed new light on the mysterious Kronecker coefficients which appear in the representation theory of the symmetric group. This is based on joint work with C. Bowman, J. Enyang and R. Orellana and recent results by S. Creedon.

Abstract: In this talk I will discuss a relative to Thompson’s group $F$, the group $F_\tau,$ which is the group of piecewise linear homeomorphisms of $[0,1]$ with breakpoints in $\mathbb{Z}[\tau]$ and slopes powers of $\tau,$ where $\tau = \frac{\sqrt5 -1}{2}$ is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type $F_\infty.$ Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to show that the commutator subgroup is simple and give a unique normal form for its elements. The surprising feature is that the $T$- and $V$-versions of these groups are not simple, however. This is joint work with J. Burillo and L. Reeves.

Abstract: The fusion system of a finite group G at a prime p is a category whose objects are the subgroups of a fixed Sylow p-subgroup S, and where the morphisms are the conjugation homomorphisms induced by the elements of G. The notion of a saturated fusion system is
abstracted from this standard example, and provides a coarse representation of what is meant by the p-local structure of a finite group. Once the group G is abstracted away, there appear many exotic fusion systems not arising in the above fashion. Exotic fusion systems are prevalent at odd primes, but only a single one-parameter family of “simple” fusion systems at the prime 2 are currently known. These are closely related to the groups Spin_7(q), q odd, and were first considered by Solomon and Benson, although not as fusion systems per se. I’ll explain some “coincidences” that allow the Benson-Solomon systems Sol(q) to exist, and then discuss various results about these systems as time allows. The results are related to the questions: How “close” to a group is Sol(q)? Are there any more exotic systems constructed in some direct fashion from the existence of Sol(q)? How many 2-modular “simple modules” would the principal 2-block of Sol(q) have if it were a group? In various combinations, this is joint work with E. Henke, A. Libman, and J. Semeraro.

Abstract: Many maximal subgroups of finite symmetric groups arise as stabilisers of some structure on the point set: for example the maximal intransitive permutation groups are subset stabilisers. The primitive groups of diagonal type for a long time have seemed exceptional in this respect. Csaba Schneider and I have introduced diagonal structures which, for the first time, give a combinatorial interpretation to these primitive groups of simple diagonal type. In further work together also with Peter Cameron and Rosemary Bailey, we’ve exhibited these groups as automorphism groups of `diagonal graphs’.

This talk is about joint work with Pavel Etingof and Victor Ostrik. A theorem of Deligne says that in characteristic zero, any symmetric tensor category “of moderate growth” admits a tensor functor to vector spaces or to super (i.e., Z/2-graded) vector spaces. In prime characteristic, this is not true, but one may ask whether there is a good list of “incompressible” symmetric tensor categories to which they they do all map. We construct an infinite ascending chain of finite symmetric tensor categories in characteristic p, all of which are incompressible. The constructions are based on the theory of tilting modules over the algebraic group SL(2). It is possible that this is the complete list, but we have not proved that.

Abstract: One of the distinguished features of the representation theory of finite groups is the ability to take a representation in characteristic zero and reduce it to obtain a representation over a fixed field of positive characteristic (a modular representation). If one starts with a representation that is irreducible in characteristic zero then its modular reduction can fail to be irreducible. The decomposition matrix encodes the multiplicities of the modular irreducible representations in this reduction.

In this talk I will present recent joint work with Olivier Brunat and Olivier Dudas establishing a fundamental property of the decomposition matrix for finite reductive groups, namely that it has a unitriangular shape. The solution to this problem involves the interplay between Lusztig’s geometric theory of character sheaves and a family of representations whose construction was originally proposed by Kawanaka.

Abstract: Clifford’s theorem explores the relationship between the irreducible modules of a group G and those of a normal subgroup N, over an arbitrary field F. In particular it applies to irreducible Brauer characters. Our focus here is on irreducible 2-Brauer characters.

We begin by showing that if \theta is an irreducible 2-Brauer character of N, then G has a real-valued irreducible 2-Brauer character over \theta if and only if \theta is G-conjugate to its complex conjugate.

Now suppose that \theta is real-valued. Then it is a remarkable fact that \theta has a unique real extension to its stabilizer in G. So G has a unique real-valued irreducible 2-Brauer character \mu such that \theta occurs with odd multiplicity in the restriction to N of \mu.

Next let \phi be a real-valued irreducible 2-Brauer character of G. Fong’s Lemma asserts that \phi is the Brauer character of a symplectic representation of G. However it is a delicate question to determine
whether \phi has orthogonal type. Suppose not and also that N is not contained in the kernel of \phi. Then we show that the restriction to N of \phi is a sum of distinct real-valued non-orthogonal irreducible
2-Brauer character of N.

Finally we discuss a consequence for blocks. Recall that a block of G is weakly regular with respect to N if its central character vanishes off N. Now let b be a real 2-block of N. We show that set of 2-blocks of G which lie over b and which are weakly regular with respect to N contains a unique real 2-block.

Abstract: Let F be a finitely generated group and G be a reductive algebraic group. The study of homomorphic images of F in G has a long and distinguished history, having applications to representation theory, generating sets of finite simple groups, Hurwitz surfaces, regular maps and hypermaps, the inverse Galois problem, differential geometry, and more besides.

The space Hom(F,G) is an algebraic variety with a natural G-action. In joint work with Ben Martin (Aberdeen), using algebraic geometry and geometric invariant theory we are able to prove a ‘rigidity’ result: under natural hypotheses, the G-orbits of certain interesting homomorphisms are both closed and open in an appropriate subvariety of Hom(F,G).

As an application, if F is generated by torsion elements which multiply to 1, if G is defined over the finite field F_q, and if a certain dimension bound holds for conjugacy classes of G, then only finitely many groups of Lie type G(q^e) are quotients of F. This proves and generalises a 2010 conjecture of C. Marion on triangle groups.

Abstract: In this talk I will discuss the representation theory of the general linear Lie algebra over a field of positive characteristic. The irreducible representations factor through certain quotients of the enveloping algebra, known as reduced enveloping algebras. It turns out that these reduced enveloping algebras may be described completely by examining a finite collection of such algebras, labelled by the conjugacy classes of nilpotent matrices of rank n. Premet has shown that each of these reduced enveloping algebras is actually Morita equivalent to an algebra known as a restricted finite W-algebra. The main result of this talk is a joint work with Simon Goodwin, in which we show that these restricted finite W-algebras can be described explicitly as certain subquotients of a Yangian.​

Abstract: Let G be a finite permutation group on a set X. A base for G is a subset Y of X such that G_(Y), the pointwise-stabilizer of Y in G, is trivial. There has been a long history of studying how small a base can be for different classes of group G. We will discuss some variants of this study, particularly focusing on upper bounds for primitive groups: in particular, we want to know how big a minimal base can be, how big an irredundant base can be, and how big an independent set can be. (The precise definition of these three notions will be given in the seminar.)

Our interest in these statistics stems from their connection to another statistic — the relational complexity of a finite permutation group. This last statistic was introduced in the 1990’s by Greg Cherlin in work applying certain model theoretic ideas of Lachlan. In particular the relational complexity of a permutation group gives an idea of the “efficiency” with which the group can be represented as the automorphism group of a homogeneous relational structure.

Abstract: The GGS-groups were some of the early positive answers to the famous Burnside problem. These groups act on infinite rooted trees and are easy to describe, plus possess interesting properties. A natural aspect of these groups to study is their maximal subgroups, and in particular, whether these groups have maximal subgroups of infinite index. It was proved by Pervova in 2005 that the torsion GGS-groups do not have maximal subgroups of infinite index. In this talk, I will consider the remaining non-torsion GGS-groups. This is joint work with Dominik Francoeur.

Abstract: A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices SL(2,R). In general, given a semisimple Lie group G over some local field F, one may ask which lattices in G attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel’s result, Lubotzky determined that a lattice of minimal covolume in SL(2,F) with F=F_q((t)) is given by the so-called characteristic p modular group SL(2,F_q[1/t]). He noted that, in contrast with Siegel’s lattice, the quotient by SL(2,F_q[1/t]) was not compact, and asked what the typical situation should be: “for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice? “.

In the talk, we will review some of the known results, and then discuss the case of SL(n,R}) for n > 2. It turns out that, up to automorphism, the unique lattice of minimal covolume in SL(n,R) (n > 2) is SL(n,Z). In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.

A complex reflection is a complex linear map given by a matrix A for which (A-I) has rank 1. In this talk I will describe an algorithm for finding polyhedra associated to certain groups acting on C^3 generated by three complex reflections. In many cases these polyhedra may be geometrised in such a way that they are fundamental polyhedra and the group is discrete.
An application of this algorithm is that it gives fundamental domains for all known (commensurability classes) of non-arithmetic lattices in PU(2,1).

The classification of finite subgroups of SO(3) is well known: these are either cyclic or dihedral groups or one of the symmetry groups of the Platonic solids. In the 19th century, Felix Klein investigated the orbit spaces of those groups and their double covers, the so-called binary polyhedral groups. This investigation is at the origin of singularity theory.
Quite surprisingly, in 1979, John McKay found a direct relationship between the resolution of the singularities of the orbit spaces and the representation theory of the finite group one starts from. This “classical McKay correspondence” is manifested, in particular, by the ubiquitious Coxeter-Dynkin diagrams.
In this talk I will first review the history of this fascinating result, and then give an outlook on recent joint work with Ragnar-Olaf Buchweitz and Colin Ingalls about a McKay correspondence for finite reflection groups in GL(n,C).

Associahedra are remarkable CW-complexes introduced by Stasheff in 1960s to encode a homotopically coherent notion of associativity. They have been realised as polytopes with integer coordinates in several different ways over the past few decades. I shall explain that the realisations of associahedra due to Loday lead to toric varieties of particular merit. These varieties have been already identified with “brick manifolds” arising when studying subword complexes for Coxeter groups (Escobar, 2014). It turns out that they also arise as “wonderful models” in the sense of de Concini and Procesi for certain subspace arrangements. Guided by that geometric picture, I shall argue that in some sense these varieties give a “noncommutative version” of Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points, in that they give rise to remarkable algebraic structures resembling cohomological field theories of Kontsevich and Manin. This is a joint work with Sergey Shadrin and Bruno Vallette.

Let I be an ideal of the ring of Laurent polynomials with coefficients in a real-valued field. The fundamental theorem of tropical algebraic geometry states the equality between the tropicalisation of the variety V (I) and the tropical variety associated to the tropicalisation of the ideal I.
In this talk I show this result for a differential ideal J of the ring of differential polynomials
K[[t]]{x_{1} ,…, x_{n} }, where K is an uncountable algebraically closed field of characteristic zero.
I show the equality between the tropicalisation of the set of solutions of J, and the set of solutions of tropicalisation of J.

Abstract: I will introduce Chern-Schwartz-MacPherson cycles of an arbitrary matroid M, which are a special collection of balanced polyhedral fans associated to M. These CSM cycles are of special significance in tropical geometry, and they satisfy very interesting combinatorics. In the case the matroid M arises from a complex hyperplane arrangement A, these cycles naturally represent the CSM class of the complement of A. This is joint work with Lucía López de Medrano and Kristin Shaw.

Abstract: In recent years, gentle algebras have been connected to many different areas of mathematics such as cluster theory, nodal stacky curves and homological mirror symmetry. In this talk we will give a geometric model of the bounded derived category of gentles algebras developed in joint work with Pierre-Guy Plamondon and Sebastian Opper. Our model is based on the representation theory of gentle algebras. By work of Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk this gives a model of the partially wrapped Fukaya category of surfaces with stops.

Abstract: Given a representation of an abstract group G, one can always define a representation of a subgroup H of G, by simply restricting the action of the group to the subgroup. This procedure yields a functor called restriction. In the other direction, given a representation of a subgroup H of G, there is a recipe for defining a representation of G from the representation of H. This also gives a functor called induction. A classic result in the representation theory of abstract groups is the adjunction relation between induction and restriction known as Frobenius reciprocity. The aim of this talk is to explain under what conditions we have an analogue of Frobenius reciprocity in the setting of continuous represention for a topological group G and a closed subgroup H in three different categories: discrete representations, linear complete representation and linearly compact representations.

Abstract: Generalizing work of Maulik and Okounkov, we explain how to use certain intertwiners of highest-weight modules for Virasoro algebras to produce solutions to the Yang-Baxter equation. The proof uses the geometry of the Hilbert scheme of points on a surface.

Abstract: Let G be a finite abelian group of order n. An (n,k,λ) m-External Difference Family (EDF)is a collection of m disjoint subsets of G each of size k, with the property that each nonzero group element occurs precisely λ times as a difference between group elements in two different subsets from the collection.  Motivated by an application to the construction of weak algebraic manipulation detection codes, a reciprocally-weight EDF (RWEDF) is defined to be a generalisation of an EDF in which the subsets may have different sizes, and the differences are counted with a weighting given by the reciprocal of the set sizes.

In this talk I will discuss some interesting structural properties of RWEDFs with certain parameters, and describe a construction of an infinite families of nontrivial RWEDFs.

Abstract: A balanced presentation of a group is one with an equal number of generators and relators. Since presentations with more generators than relators define infinite groups, balanced presentations present a borderline situation where both finite and infinite groups can be found. It is of interest to find which balanced presentations can define finite groups, and what groups can arise. We consider groups defined by balanced presentations with the property that each relator is of the form R(x,y) where R is some fixed word in two generators. Examples of such groups include Right Angled Artin Groups, Higman groups, and cyclically presented groups in which the relators involve exactly two generators. To each such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Extending work of Pride, we show that if the graph is triangle-free then the corresponding group cannot be trivial or finite of rank greater than 2. This is joint work with Johannes Cuno.

Abstract: The table of marks of a finite group G characterises the actions of G on the transitive G-sets, which are in bijection to the conjugacy classes of subgroups of G. Thus the table of marks provides a complete classification of the permutation representations of a finite group G up to equivalence.
In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix.
This is joint work with Goetz Pfeiffer.

Abstract: I will present joint work with Markus Linckelmann, Justin Lynd, and Jason Semeraro connecting local-global relationships (known and conjectural) in the modular representation theory of finite groups to the theory of fusion systems.

Abstract: The new results described in this talk were proved jointly work with Charles Leedham-Green and Eamonn O’Brien.
Over the past 30 years, an algorithm CompositionTree has been developed for enabling practical computation in large matrix groups over finite fields. The principal aim is to find a composition series and membership test for an input group G ≤ GL(d,q). This has been implemented in Magma and performs well in practice.
In 2009, Babai, Beals and Seress published a polynomial time algorithm (assuming oracles for integer factorization and discrete logs) with the same aims for odd q. But this is not suitable for implementation.
We have been asked whether it is feasible to show that CompositionTree can be easily adapted to run in polynomial time, and we can now prove that this is possible with a few provisos.
The main new idea is that we can modify our black box algorithms for constructive recogniton of the finite nonabelian simple groups so that, if the input group is not simple, then a nontrivial element in a proper normal subgroup is output.

Abstract: The Specht modules are the key players in the representation theory of the symmetric groups. If the characteristic of the field is different than 2, it is well-known that these modules are indecomposable. In characteristic 2 there exist decomposable Specht modules and the first example of such a module was found by Gordon James in the 70s. Surprisingly enough, only a few other examples of such modules have been discovered since then. In this talk I will present many new families of decomposable Specht modules and describe explicitly their indecomposable summands. This is a joint work with Stephen Donkin.

Abstract: Minimal representations are an important class of “small” infinite dimensional unitary representations of Lie groups. They are characterised by the fact that their annihilator ideal is equal to the Joseph ideal. Two prominent examples are the metaplectic representations of Mp(2n) (a double cover of Sp(2n)) and the minimal representation of the indefinite orthogonal group O(p,q).
There exists a unified framework to construct the minimal representation of a Lie group associated to a simple Jordan algebra.
In this talk I will construct a generalisation of the minimal representation of so(p,q) to the orthosymplectic Lie superalgebra osp(p,q|2n) using Jordan superalgebras. This representation also has an annihilator ideal equal to a Joseph-like ideal. I will also mention the obstacles which prevent a straightforward generalisation to other Lie superalgebras.

Abstract: Famous Deligne’s conjecture, which is now a theorem, claims that Hochschild complex of an associative algebra admits an action an operad weakly equivalent to the little 2-cubes operad.
Davydov-Yetter deformation complex of a tensor category is, in a sense, a categorification of Hochschild complex. It is natural to ask if there is an analogue of Deligne’s statement on this context.
We show that a similar action exists but instead of little 2-cubes we get little 3-cubes action. The proof is combinatorial and relies on liftings of certain paths on a commutative lattice to paths of restricted complexity on a noncommutative lattice.
This is a joint work with Alexei Davydov.

Abstract: The representation dimension of an algebra is a finite integer that is supposed to indicate how complicated an algebra’s module category is. This dimension was first introduce by Auslander in 1971 and is, in general, notoriously hard to compute. This measure is related to the representation type of an algebra: an algebra has finite representation type if and only if it’s representation dimension is less than 3.
Separable equivalence is an equivalence relation on finite dimensional algebras. Over a field of a characteristic p, a group algebra is separably equivalent to the group algebra of its Sylow p-subgroup. We use this relationship between a group and its Sylows to put an upper bound on the representation dimension of a group algebra for any finite group with a elementary-abelian Sylow subgroup.

Abstract: Difference algebra studies algebraic structures equipped with an endomorphism/difference operator, and difference algebraic varieties are defined by systems of difference polynomial equations over difference rings and fields. In this talk, we will:

• argue that twisted groups of Lie type are best viewed as difference algebraic groups;
• develop the cohomology theory of difference algebraic groups;
• compute the cohomology in a number of interesting cases, and discuss its applications.

Abstract: Singularity categories are triangulated categories occurring as invariants associated to singular algebras. For hypersurface singularities these categories can be realised via matrix factorisations, and in this case Knorrer periodicity constructs equivalences between the singularity categories of many different hypersurfaces.
I will discuss these ideas, and talk about how equivalences of singularity categories in the non-hypersurface (and non-Gorenstein) setting can be constructed by considering quasi-hereditary noncommutative resolutions produced from certain geometric situations. In addition, Ringel duality has a very explicit description and interpretation for these quasi-hereditary algebras.

Abstract: The relevance of the McKay conjecture in the representation theory of finite groups led to the study of the decomposition into irreducible constituents of the restriction of characters to Sylow p-subgroups.
I will present some recent results on the topic.

Abstract: Dress’s theory of Mackey functors is a successful axiomatization of the representation theory of finite groups, capturing the formal aspects of such classical invariants as the character ring or group (co)homology. But, typically, each such invariant is only a partial shadow (consisting of abelian groups and homomorphisms) of a richer structure (consisting of additive, abelian or triangulated categories and suitable functors between them).
In joint work with Paul Balmer, we develop a theory of “Mackey 2-functors” in order to study this higher structure, thus explaining certain phenomena which, though invisible to classical Mackey functors, occur throughout equivariant mathematics.
In this talk I will provide examples of Mackey 2-functors, such as derived and stable module categories in representation theory or equivariant stable homotopy categories in topology, I will motivate our axioms and explain the first results of the theory.

Abstract: In this talk I will introduce the notion of silting objects and mutation of silting objects. I will then show how the combinatorics of silting mutation can give one information regarding the structure of the space of stability conditions. In particular, I will show how a certain discreteness of this mutation theory enables one to employ techniques of Qiu and Woolf to obtain the contractibility of the space of stability conditions for a class of mainstream algebraic examples, the so-called silting-discrete algebras. This talk will be a discussion of joint work with Nathan Broomhead, David Ploog, Manuel Saorin and Alexandra Zvonareva.

Abstract: For finite balanced presentations of quaternion groups Q_{4n}, n>5, it is unknown if the kernel of the associated matrix is always generated by a single element. A positive answer for any value of n>5 would resolve one of the most fundamental and longstanding questions in topology: Is cohomological dimension the same as geometric dimension for finite cell complexes.

I will explain the background to this, contrast with the situation for dihedral groups which is completely understood, and explain my recent incremental result for the case of two generators and two relators.

Abstract: In this talk, we will discuss a pro-p group G whose finite quotient groups give your “favourite” Sylow p-subgroups of GL_n(q) for all positive integers n, where q is a power of p.

Elaborating on work by Weir in the 50s and recent results by Bier and Holubowski, we will dip into the subgroup structure of G.

Time permitting, we will also discuss field extensions, a p-adic variant of G and Hausdorff dimensions of some closed subgroups.

Abstract: The existence of a grading on a ring often makes computations a lot easier. In particular this is true for the computation of homotopy invariants. For example one can readily compute such invariants for the stable categories of graded modules over connected graded self-injective algebras. Using work of Tabuada, we’ll show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. As another illustration of these ideas we’ll show that cluster categories of Dynkin type A_n, for even n, are “A^1-homotopy phantoms”. All this is based on joint work with Greg Stevenson.

Abstract: We explain how to construct an explicit topological model for every two-block Springer fiber of type D. These so-called topological Springer fibers are homeomorphic to their corresponding algebro-geometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic Kazhdan-Lusztig polynomials of type D with respect to a maximal parabolic of type A. As an application it is discussed how the topological Springer fibers can be used to reconstruct the famous Springer representation in an elementary and combinatorial way.

Abstract: The Jantzen sum formula is a classical result in the representation theory of the symmetric and general linear groups that describes a natural filtration of the modular reductions of the simple modules of these groups. Analogues of this result exist for many algebras including the cyclotomic Hecke algebras of type A. Quite remarkably, the cyclotomic Hecke algebras of type A are now know to admit a Z-grading because they are isomorphic to cyclotomic KLR algebras. I will explain how to give an easy proof of the Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A using the KLR grading. I will discuss some consequences and applications of this approach.

Abstract: Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry we introduce a technique called “conservative descent” in order to establish such decompositions locally. The decompositions we have in mind are those for projective bundles, blow-ups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semi-orthogonal decompositions for Brauer-Severi varieties.
This is joint work with Daniel Bergh (Copenhagen).