## The University of Leeds Algebra Seminar 2019/20   Tuesdays 16:00 Mall 1 (occasionally different time and place)

The seminar will normally take place in Mall 1 at 16:00 on Tuesdays. We will have biscuits and tea (and coffee) in the common room before the seminar.

## Schedule:

 DATE and TIME and PLACE SPEAKER TITLE October 8, 16:00  Mall 1 Bernd Schober   (Universität Oldenburg) Polyhedral invariants for desingularization October 15, 16:00  Mall 1 Paul Johnson   (University of Sheffield) Cancelled. October 22, 16:00  Mall 1 Cesare Ardito   (University of Manchester) Classifying 2-blocks with an elementary abelian defect group October 29, 16:00  Mall 1 Yadira Valdivieso Díaz   (University of Leicester) From the potential to the first Hochschild cohomology group of a cluster tilted algebra November 5, 16:00   Mall 1 Nelly Villamizar   (Swansea University) Spline spaces on polyhedral cells November 12, 15:00   Mall 1 Sinead Lyle (University of Norwich) Decomposition matrices associated to the full transformation semigroup November 12, 16:00   Mall 1 Clemens Koppensteiner (University of Oxford) D-modules in logarithmic geometry November 19, 16:00   Mall 1 Nathan Broomhead (University of Plymouth) Discrete derived categories and their thick subcategories November 26, 16:00   Mall 1 Neil Saunders (Greenwich University) The Exotic Nilpotent Cone and Type C Combinatorics December 3, 16:00   Mall 1 Leonid Monin (University of Bristol) Newton polyhedra theory for overdetermined systems of Laurent polynomials December 10, 15:00   Mall 1 Diego Lobos Maturana (Universidad de Talca) Cellular basis for generalized blob algebras December 10, 16:15   Mall 1 María Cumplido Cabello (Herriot-Watt University) The k-th root problem extraction for braid groups January 28, 16:00   Mall 1 Raphael Bennett-Tennenhaus (University of Leeds) Purity and positive-primitive formulas in triangulated categories: with examples from (and applications to) gentle algebras February 4, 16:00   Mall 1 Dimitra Kosta (University of Glasgow) Unboundedness of Markov complexity of monomial curves in $\mathbb{A}^n$ for $n \geq 4$ February 11, 16:00   Mall 1 Lutz Hille (Universität Münster) Cancelled. February 18, 16:00   Mall 1 Alison Parker (University of Leeds) Tilting modules for the blob algebra February 25, 16:00   Mall 1 Raquel Simoes (Lancaster University) A geometric model for the module category of a gentle algebra March 3, 16:00   Mall 1 Liana Heuberger (Loughborough University) New constructions of Fano 3-folds from mirror symmetry March 10, 16:00   Mall 1 Celeste Damiani (University of Leeds) Alexander invariants for ribbon tangles March 17, 16:00   Mall 1 Rowena Paget (University of Kent) Cancelled. March 24, 15:00   Mall 1 Giulio Belletti (Scuola Normale Superiore Pisa) Cancelled. April 21, 15:00   Virtual Joao Faria Martins (University of Leeds) A rendition and an extension of Quinn's total homotopy TQFT. April 28, 15:00   Virtual Fabian Haiden (University of Oxford) From Hall algebras to legendrian skein algebras May 5, 15:00   Virtual Hipolito Treffinger (University of Leicester) tau-tilting theory and the Brauer-Thrall conjectures June 2, 15:00   Virtual Nils Carqueville (Universität Wien) Introduction to topological quantum field theory

## Abstracts

Bernd Schober (Universität Oldenburg): Polyhedral invariants for desingularization. The goal of my talk is to present a convex geometric viewpoint on singularities and their resolution. More precisely, we discuss how the Newton polyhedron and the Hironaka polyhedron of a Weierstrass polynomial provide invariants of the singularity that reflect how bad" the singularities are. Here, the Hironaka polyhedron is a certain projection of the Newton polyhedron. After a brief introduction to the notions, we study the behaviour of the Hironaka polyhedron under blowing ups for curves and surfaces. Then we explain how this leads to an invariant for desingularization of surfaces in any characteristic that decreases strictly after blowing up a sufficiently nice center. We focus on the ideas and try to hide the technical details as good as possible.
This is joint work with Vincent Cossart.

Cesare Ardito (University of Manchseter): Classifying 2-blocks with an elementary abelian defect group. Donovan's conjecture predicts that given a $p$-group $D$ there are only finitely many Morita equivalence classes of blocks of group algebras with defect group $D$. While the conjecture is still open for a generic $p$-group $D$, it has been proven in 2014 by Eaton, Kessar, Külshammer and Sambale when $D$ is an elementary abelian 2-group, and in 2018 by Eaton and Livesey when $D$ is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classification up to Morita equivalence over a complete discrete valuation ring $\mathcal{O}$ has been achieved for $D$ with rank $3$ or less, and for $D = (C_2)^4$.
I have done $(C_2)^5$, and I have partial results on $(C_2)^6$. I will introduce the topic, give the relevant definitions and then describe the process of classifying this blocks, with a particular focus on the individual tools needed to achieve a complete classification.

Yadira Valdivieso Díaz (University of Leicester): From the potential to the first Hochschild cohomology group of a cluster tilted algebra. The aim of this talk is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.

Nelly Villamizar (Swansea University): Spline spaces on polyhedral cells. In this talk we will give a brief overview of some of the algebraic methods which are used in spline theory. We will give particular attention to the pioneering work of Billera, in which homological methods were introduced for the calculation of dimension formulas. These methods have proved very fruitful for splines on all types of subdivisions, we will attempt to give a flavor for the various results that have been obtained this way, the questions that remain open, and the connections to algebraic geometry that result from these methods.
In particular we shall consider the space of spline functions defined on polyhedral cells. These cells are the union of 3-dimensional polytopes sharing a common vertex, so that the intersection of any two of the polytopes is a face of both. In the talk, we will present new bounds on the dimension of this spline space. We provide a bound on the contribution of the homology term to the dimension count, and prove upper and lower bounds on the ideal of the interior vertex which depend only on combinatorial (or matroidal) information of the cell. We use inverse systems to convert the problem of finding the dimension of ideals generated by powers of linear forms to a computation of dimensions of so-called fat point ideals. The fat point schemes that comes from dualizing polyhedral cells is particularly well-suited and leads to the exact dimension in many cases of interest that will also be presented in the talk.

Sinead Lyle (University of Norwich): Decomposition matrices associated to the full transformation semigroup. The transformation monoid $T_n$ consists of all maps from the set $\{1, 2, \ldots, n\}$ to itself. Consider the algebra $\mathbb{C} T_n$. This algebra has dimension $n^n$ and it is not semisimplefor $n \geq 2$. However it is standardly based (in the sense of Du and Rui) and its representations are controlled by those of its maximal subgroups, the symmetric groups $S_d$ where $1 \leq d \leq n$. In this talk, we shall discuss some of the facts which are known about the representations of the transformation monoid and how they are related to those of the symmetric groups.

Clemens Koppensteiner (University of Oxford): D-modules in logarithmic geometry. Given a smooth variety X with a normal crossings divisor D (or more generally a smooth log variety) we consider the ring of logarithmic differential operators: the subring of differential operators on X generated by vector fields tangent to D. Modules over this ring are called logarithmic D-modules and generalize the classical theory of regular meromorphic connections. They arise naturally when considering compactifications.
We will discuss which parts of the theory of D-modules generalize to the logarithmic setting and how to overcome new challenges arising from the logarithmic structure. In particular, we will define holonomicity and study its interaction with duality. This talk will be very example-focused and will not require any previous knowledge of D-modules or logarithmic geometry. This is joint work with Mattia Talpo.

Nathan Broomhead (University of Plymouth): Discrete derived categories and their thick subcategories. Discrete derived categories, as defined by Vossieck, form a class of triangulated categories in which explicit computations are possible but which are non-trivial enough to manifest interesting behaviour. In this talk, motivated by some small examples, I will explain a geometric model for discrete derived categories, which can be used to explicitly calculate their lattices of thick subcategories in terms of certain generating collections of exceptional and sphere-like objects.

Neil Saunders (Greenwich University): The Exotic Nilpotent Cone and Type C Combinatorics. The exotic nilpotent cone as defined by Kato gives a 'Type A-like' Springer correspondence for Type C. In particular, there is a bijection between the symplectic group orbits on the exotic nilpotent cone and the irreducible representations of the Weyl group of Type C. In this talk, I will outline the various geometric and combinatorial results that follow from this. These results are joint work with Vinoth Nanadakumar and Daniele Rosso, and Arik Wilbert.

Leonid Monin (University of Bristol): Newton polyhedra theory for overdetermined systems of Laurent polynomials. Classical theory of Newton polyhedra calculates topological invariants of a zero set in algebraic torus of a general system of Laurent polynomials in terms of combinatorics of their Newton polyhedra. More precisely, for a fixed polytopes $P_1,...,P_k$ there exists an open dense subset $U$ of the space of Laurent polynomials with Newton polyhedra $P_1,...,P_k$ such that the topological invariant of interest is the same for any system from $U$ and can be computed combinatorially.
It could be that polyhedra $P_1,...,P_k$ are such that the generic system of Laurent polynomials with Newton polytopes $P_1,...,P_k$ does not have any solutions. In this case one can be interested in invariants of generic non-empty zero set. Since in this case consistent systems are not generic, all results of classical Newton polyhedra theory are not applicable to them. In my talk I will explain how to extend theory of Newton polyhedra to the case of overdetermined systems.
If time permits, I will also talk about generalization to systems on spherical homogeneous spaces and other algebraic varieties.

Diego Lobos Maturana (Universidad de Talca): Cellular basis for generalized blob algebras. Abstract

María Cumplido Cabello (Herriot-Watt University): The k-th root problem extraction for braid groups. There are several computational problems in braid groups that have been proposed for their potential applications to cryptography, even if the interest of the subject has decreased, mainly due to the appearance of algorithms which solve the conjugacy problem extremely fast in the generic case. However, there are some other problems in braid groups whose generic-case complexity is still to be studied. This is the case of the k-th root (extraction) problem. In this talk we will see that, generically, finding the k-th root of a braid is very fast, and we will describe an algorithm to do so.

Raphael Bennett-Tennenhaus (University of Leeds): Purity and positive-primitive formulas in triangulated categories: with examples from (and applications to) gentle algebras. In mathematical logic any theory comes equipped with sentences in a fixed language, and a model of that theory is a structure satisfying these sentences. Model theoretic algebra entails studying structures which are algebraic, such as modules over a fixed ring. By Baur's famous elimination of quantifiers, any sentence in the language of modules is equivalent to a Boolean combination of `positive-primitive' (pp) formulas. This motivated the study of pure-monomorphisms: those reflecting the existence of solutions to pp-formulas with constants from the domain. Indeed, any module is elementary equivalent to a so-called pure-injective module: one which is injective with respect to pure-monomorphisms.
These ideas have been generalised beyond the scope of modules. Crawley-Boevey defined notions of purity for any locally finitely presented category, such as various functor categories. Krause then used the Yoneda embedding to inherit these definitions to compactly generated triangulated categories. Since then Garkusha and Prest introduced the appropriate multi-sorted language for these triangulated categories. Examples of such categories include the homotopy category of complexes of projective modules over a finite-dimensional algebra. In my talk I will discuss recent work [arxiv: 1911.07691] in which the gentle algebras of Assem and Skowronski were considered, and indecomposable sigma-pure-injective complexes in these homotopy categories were classified. The proof uses the aforementioned canonical language to exploit ideas and results developed in my PhD thesis.

Dimitra Kosta (University of Glasgow): Unboundedness of Markov complexity of monomial curves in $\mathbb{A}^n$ for $n \geq 4$. Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve $C$ in $\mathbb{A}^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no $d\in \mathbb{N}$ such that $m(C) \leq d$ for all monomial curves $C$ in $\mathbb{A}^4$. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in $\mathbb{A}^n, n \geq 4$.

Alison Parker (University of Leeds): Tilting modules for the blob algebra.

Raquel Simoes (Lancaster University): A geometric model for the module category of a gentle algebra. Gentle algebras can be realised as certain algebras associated to partial triangulations of unpunctured surfaces. Using this fact, we give a geometric model for the module category of any gentle algebras. This is a report on joint work with Karin Baur.

Liana Heuberger (Loughborough University): New constructions of Fano 3-folds from mirror symmetry. Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct previously unknown Fano 3-folds with terminal Gorenstein quotient singularities.
A Laurent polynomial f determines, through its Newton polytope $P$, a toric variety $X_P$, which is in general highly singular. Laurent inversion constructs, from f and some auxiliary data, an embedding of $X_P$ into an ambient toric variety $F$. In many cases this embeds $X_P$ as a complete intersection of line bundles on $F$, and the general section of these line bundles is the Q-Fano 3-fold that we seek.
This is joint work with Tom Coates, Al Kasprzyk and Giuseppe Pitton.

Celeste Damiani (University of Leeds): Alexander invariants for ribbon tangles. Ribbon tangles are proper embeddings of tori and annuli in the 4-dimensional ball, bounding 3-manifolds with only ribbon singularities. We construct an Alexan- der invariant for these objects that induces a functorial generalisation of the Alexander polynomial. This functor is an extension of the Alexander functor for usual tangles defined by Bigelow-Cattabriga-Florens and studied by Florens-Massuyeau. If considered on braid-like ribbon tangles, this functor coincides with the exterior powers of the Burau-Gassner representation. On one hand, we observe that the action of cobordisms on ribbon tangles endows them with a circuit algebra structure over the operad of cobordisms, and we show that the Alexander invariant com- mutes with the circuit algebra's composition. On the other hand, ribbon tangles can be represented by welded tangle diagrams: this allows to give a combinatorial description of the Alexander invariant.

Joao Faria Martins (University of Leeds): A rendition and an extension of Quinn's total homotopy TQFT. I will present recent work towards an extended version of Quinn's total homotopy TQFT. I will start with a explanation of Quinn's TQFT, which will be formulated in the language of homotopy theory. I will then sketch the construction of an extended version of Quinn's TQFT, taking values in the bicategory of algebras, bimodules, and bimodule maps.
Parts of the work are joint with Tim Porter.

Fabian Haiden (University of Oxford): From Hall algebras to legendrian skein algebras. A mysterious relation between Hall algebras of Fukaya categories of surfaces and skein algebras was suggested by recent work of Morton-Samuelson and Samuelson-Cooper. I will discuss how this relation can be made precise using knot theory of legendrian curves and general gluing properties of skein and Hall algebras. Along the way I aim to motivate and review notions such as Hall algebras, Fukaya categories of surfaces, and skein theory. Based on arXiv:1908.10358, arXiv:1910.04182, and an ongoing joint project with Ben Cooper.

Slides of Fabian Haiden's talk.

Hippolito Treffinger (University of Leicester): tau-tilting theory and the Brauer-Thrall conjectures. Back in the 1940's, Brauer and Thrall announced two important results concerning the indecomposable representations of algebras. Since no proof was published in the years that followed the announcement, these statements started to be known as the first and second Brauer-Thrall conjectures. Many tools were introduced in representation theory in order to prove these conjectures and many years passed until a proof of these results was published.
In this talk we will start by motivating the conjectures from basic representation theory. Afterwards, we will show how recent results in tau-tilting theory give rise to modern versions of these conjectures, namely the first and second tau-Brauer-Thrall conjectures, and we will give a proof of the first tau-Brauer-Thrall conjecture.
Time permitting, we will discuss the second tau-Brauer-Thrall conjecture for special biserial algebras.
This is joint work with Sibylle Schroll and Yadira Valdivieso.

Nils Carqueville (Universität Wien): An introduction to topological quantum field theory. I'll review the functorial approach to topological quantum field theory, which can be thought of as representations of (a caricature of) spacetime in algebra. This naturally leads to Frobenius algebras over a field, but also to the representation theory of algebras internal to 2- categories other than Vect. The main part of the talk will be a general introduction; towards the end I'll sketch how this approach leads to new relations in algebra and geometry, and generalises to higher dimensions.

For past algebra seminars see: 2018/19 , 2017/18 .