## The University of Leeds Algebra Seminar 2018/19   Tuesdays 15:00 Mall 1 (occasionally different time and place)

The seminar will normally take place in Mall 1 at 15:00 on Tuesdays. After the seminar we will have biscuits and tea (and coffee) in the common room.

## Schedule:

 DATE and TIME and PLACE SPEAKER TITLE October 2, 15:00   Mall 1 Amit Shah   (University of Leeds) Partial cluster-tilted algebras, quasi-abelian hearts and Auslander-Reiten theory October 9, 15:00   EC Stoner SR 7.70 Isobel Webster   (University of Leeds) A lattice isomorphism theorem for cluster groups of type A October 16, 15:00  Mall 1 Jenny August   (University of Edinburgh) The Tilting Theory of Contraction Algebras October 23, 15:00  Mall 1 Celeste Damiani   (University of Leeds) Remarkable quotients of virtual braid groups November 6, 16:00  Mall 1 Alan Mcleay   (Université du Luxembourg) Mapping class groups, covers, and braids November 13, 16:00  Mall 1 Janet Page   (University of Bristol) Symbolic vs. ordinary powers and the containment problem for Hibi rings November 20, 16:00   Mall 1 Milena Hering   (University of Edinburgh) The F-splitting ratio of a toric variety November 27, 16:00   Mall 1 Marina Logares (University of Plymouth) Integrable systems and Higgs bundles December 11, 16:00  Mall 1 Dylan Allegretti (University of Sheffield) Quiver representations, cluster varieties, and categorification of canonical bases December 12, 14:00   Mall 2 Markus Szymik (NTNU) Symmetry groups of algebraic structures and their homology January 29, 16:00   Mall 1 Theo Raedschelders (University of Glasgow) Cancelled January 29, 16:00   Mall 1 Eleonore Faber (University of Leeds) The magic square of reflections and rotations and the McKay correspondence February 5, 16:00   Mall 1 Billy Woods (University of Glasgow) Completed group algebras and skew power series rings February 12, 16:00   Mall 1 Nelly Villamizar (Swansea University) Cancelled February 12, 16:00   Mall 1 João Faria Martins (University of Leeds) Wirtinger 2-relations for the fundamental 2-group of the complement of a knotted surface in $S^4$ February 19, 16:00   Mall 1 School colloquium February 26, 16:00   Mall 1 Sven-Ake Wegner (Teesside University) Is functional analysis a special case of tilting theory? March 5, 16:00   Mall 1 Fatemeh Mohammadi (University of Bristol) Toric degenerations of Grassmannians March 12, 16:00   Mall 1 Mattia Talpo (Imperial College) Topological realization of varieties over C((t)) via log geometry March 19, 16:00   Mall 1 Ilke Canakci (Newcastle University) Generalised friezes and the weak Ptolemy map March 26, 16:00   Mall 1 Paolo Bellingeri (University of Caen) Virtual braids and permutations April 30, 16:00   Mall 1 Alex Fink (Queen Mary University London) Matrix orbit closures and their classes May 7, 16:00   Mall 1 Vincent Gélinas (Trinity College Dublin) The A-infinity structure on Ext and homological properties of finite dimensional algebras May 14, 16:00   Hillary Place SR (G.18) Hamid Ahmadinezhad (Loughborough University) Birational rigidity of 3-folds and where the concept sits in birational classification

## Abstracts

Amit Shah (University of Leeds): Partial cluster-tilted algebras, quasi-abelian hearts and Auslander-Reiten theory. A partial cluster-tilted algebra is defined as an endomorphism ring of a rigid object R of a cluster category C (associated to a finite-dimensional hereditary algebra over a field). The representation theory of this kind of algebra is harder to get to than the representation theory of the related cluster-tilted algebra. Buan and Marsh showed that if A is the endomorphism ring of a rigid object R in C , then one needs to perform two steps to get to the module category mod A. First, you quotient C by the subcategory X=Ker(How(R,-)), and then you localise (in the sense of Gabriel and ZIsman) the integral category C/X at the class of regular morphisms. The resultant category is equivalent to mod A. In order to understand mod A from C, my recent strategy has been to better understand C/X. In this talk, I will show that one can use Nakaoka’s theory of twin cotorsion pairs to show that C/X is a quasi-abelian category (i.e. has kernels and cokernels, and pullbacks of cokernels are cokernels and pushouts of kernels are kernels), and then I will give a characterisation of Auslander-Reiten sequences in such a category. At the beginning I will quickly define the cluster category, and throughout the talk there will be a simple running example to demonstrate what’s happening.

Isobel Webster (University of Leeds): A lattice isomorphism theorem for cluster groups of type A Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers that are mutationally equivalent to oriented simply-laced Dynkin diagrams, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and we are interested in whether the cluster group presentations possess comparable properties. I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the mutation- Dynkin type A case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups.

Jenny August (University of Edinburgh): The Tilting Theory of Contraction Algebras. Contraction algebras are a class of finite dimensional, symmetric algebras introduced by Donovan and Wemyss as a tool to study the minimal model program in geometry. In this talk, I will give an introduction to these algebras, before then going on to describe how an associated hyperplane arrangement (a simple picture) coming from the two-term tilting theory controls something that, for a general algebra, is considered extremely complicated; namely, the entire derived equivalence class of such an algebra.

Celeste Damiani (University of Leeds): Remarkable quotients of virtual braid groups. Virtual braid groups are one of the most famous generalisations of braid groups. Among their remarkable quotients we have loop/welded braid groups $LB_n$ and unrestricted virtual braid groups $UVB_n$. We describe the group structure of unrestricted virtual braid groups as right-angled Artin groups, and explore some applications to fused links, such as the definition of a fused link group through a representation for $UVB_n$ in the group of automorphisms of the free $2$-step nilpotent group of rank $n$.

Alan Mcleay (Université du Luxembourg): Mapping class groups, covers, and braids. The mapping class group of a surface is the group of isotopy classes of boundary preserving homeomorphisms of the surface. Given a finite sheeted covering space between surfaces, we may ask what relationship, if any, exists between the two mapping class groups? In joint work with Tyrone Ghaswala we investigate this question for surfaces with non-empty boundary. I will discuss a classical theorem of Birman-Hilden and give new insight into a family of covering spaces related to the Burau representation of braid groups.

Janet Page (University of Bristol): Symbolic vs. ordinary powers and the containment problem for Hibi rings. The relationship between symbolic powers and ordinary powers of prime ideals is an active area of research in commutative algebra. It is easily verified that n-th ordinary power of a prime ideal is contained in its $n$-th symbolic power, and typically we strive for containments in the other direction. In characteristic 0, Ein, Lazarsfeld, and Smith showed there is a uniform containment which holds for every prime ideal in a regular ring $R$. Namely, they showed that there is a $d$ (in this case the dimension of $R$) such that for every prime ideal $\mathfrak{p}$ in $R$, we have that the $dn$-th symbolic power of $\mathfrak{p}$ is contained in its $n$-th ordinary power. Since then, this result has been extended to characteristic $p$ by Hochster and Huneke and much more recently to mixed characteristic by Ma and Schwede, but there are still few results known in the non regular case. In this talk, I will review the background of this problem and discuss new results (joint with Daniel Smolkin and Kevin Tucker) in this direction, by introducing a class of toric rings called Hibi rings, which can be combinatorially defined in terms of finite posets.

Milena Hering (University of Edinburgh): The F-splitting ratio of a toric variety. The Frobenius morphism is a useful tool in the study of commutative rings and algebraic varieties. One of its uses is to give a measurement of how bad the singularities of a ring are. This measurement is called the the F-splitting ratio, which agrees with the F-signature for normal rings. The F-signature of a normal toric ring was computed by Von Korff. I will give give an introduction to these notions and present the computation of the F-splitting ratio of a seminormal toric ring. This is joint work with Kevin Tucker.

Marina Logares (University of Plymouth): Integrable systems and Higgs bundles. The moduli space of Higgs bundles is a rich geometric object which lies in the interface of algebraic geometry, differential geometry and mathematical physics. One of its properties is that it carries the structure of an algebraic completely integrable system known as the Hitchin system. We will talk about the Hitchin system and discuss a variation of it providing a complex partially integrable system. This talk is based on joint work with I. Biswas and A. Peón-Nieto.

Dylan Allegretti (University of Sheffield): Quiver representations, cluster varieties, and categorification of canonical bases. Associated to a compact oriented surface with marked points on its boundary is an interesting class of finite-dimensional algebras. These algebras are examples of gentle algebras, and their representation theory has been studied by many authors in connection with the theory of cluster algebras. An important fact about these algebras is that their indecomposable modules come in two types: string modules, which correspond to arcs connecting marked points on the surface, and band modules, which correspond to closed loops on the surface. Thanks to the work of many mathematicians, the string modules are known to categorify generators of a cluster algebra. In this talk, I will explain how, by including band modules in this story, one can define a family of graded vector spaces which categorify Fock and Goncharov's canonical basis for the algebra of functions on an associated cluster variety. These vector spaces are of interest in mathematical physics, where they are expected to provide a mathematical definition of the space of framed BPS states from the work of Gaiotto, Moore, and Neitzke.

Markus Szymik (NTNU): Symmetry groups of algebraic structures and their homology. The symmetric groups, the general linear groups, and the automorphism groups of free groups are examples of families of groups that arise as symmetry groups of algebraic structures but that are also dear to topologists. There are many other less obvious examples of interest. For instance, in joint work with Nathalie Wahl, this point of view has led to the computation of the homology of the Higman-Thompson groups. I will survey a general context and some other geometric examples in this talk.

Eleonore Faber (University of Leeds): The magic square of reflections and rotations and the McKay correspondence. 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).

Billy Woods (University of Glasgow): Completed group algebras and skew power series rings . The main focus of this talk will be on the ring-theoretic properties of Iwasawa algebras - that is, appropriately completed group algebras of certain nice profinite groups. I'll give some of their basic structural properties, explain how you should think about them, and detail some strong parallels between the worlds of Iwasawa algebras and other, more classical objects. Unfortunately, many seemingly basic questions on Iwasawa algebras remain almost unanswered, such as "what are their prime ideals?". I'll talk about recent research in this direction, and (time permitting) outline some potentially fruitful approaches in the theory of skew power series rings.

João Faria Martins (University of Leeds): Wirtinger 2-relations for the fundamental 2-group of the complement of a knotted surface in $S^4$. Crossed modules made their way to the mathematical literature via the work of JHC Whitehead, as they appear naturally in homotopy theory. In particular crossed modules serve as algebraic models for spaces $X$ for which $\pi_i(X)=0$ for $i >2$. Crossed modules had a recent reappearance in mathematics, stemming out of the fact that they are equivalent to 2-groups: higher order (categorified) analogues of groups.
In this talk I will show a method to calculate the fundamental 2-group of the complement of a knotted surface $\Sigma$ in S^4, given a diagram describing a hyperbolic splitting of $\Sigma$. In particular I will show how calculate the fundamental crossed module of the complement of $\Sigma$ via a set of higher order (categorified) Wirtinger relations. I will hence give a solution to R.H. Fox's old problem of whether the second homotopy group of the complement of a knotted surface can be determined from a movie-presentation of it.

References:
João Faria Martins: The Fundamental Crossed Module of the Complement of a Knotted Surface. Trans.Am.Math.Soc. 361 (2009) 4593-4630.
Ralph H. Fox: Some problems in knot theory, in "Topology of 3-Manifolds and Related Topics," edited by M. K. Fort, Jr., Prentice-Hall, Inc., Englewood Cliffs, N.J., (1962), 168-176.

Sven-Ake Wegner (Teesside University): Is functional analysis a special case of tilting theory? The objects of functional analysis together with the corresponding morphisms don't form abelian categories. Classical examples, e.g., the category of Banach spaces, satisfy almost all axioms of an abelian category but the canonical morphism between the cokernel of the kernel and the kernel of the cokernel of a given map fails in general to be an isomorphism. In 2003, Bondal and van den Bergh showed that there is a correspondence between (co-)tilting torsion pairs and so-called quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical t-structure on its derived category. In this talk we discuss examples of categories appearing in functional analysis that are not quasiabelian but which carry a natural exact structure. The derived category is thus defined. We are interested in an extension of the aforementioned equivalence as this could be a step towards a successful categorification of certain analytic problems.

Fatemeh Mohammadi (University of Bristol): Toric degenerations of Grassmannians. Many toric degenerations and integrable systems of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can also be seen to arise from the cones of the tropicalisation of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians and prove a necessary condition for such an initial degeneration to be toric. I will present several combinatorial conjectures and computational challenges around this problem. This is based on joint works with Kristin Shaw and with Oliver Clarke.

Mattia Talpo (Imperial College): Topological realization of varieties over C((t)) via log geometry. I will describe some work in progress with Piotr Achinger about defining a "Betti realization" functor for varieties over the formal punctured disk Spec C((t)), i.e. defined by polynomials with coefficients in the field of formal Laurent series in one variable. Our construction is via "good models" over the power series ring C[[t]] and the "Kato-Nakayama" construction in logarithmic geometry, that I will review during the talk.

Ilke Canakci (Newcastle University): Generalised friezes and the weak Ptolemy map. Frieze patterns, introduced by Conway, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Frieze patterns with positive integer values are of a special interest since they are in one-to-one correspondence with triangulations of polygons by Conway--Coxeter. Remarkably, this established a connection to cluster algebras–predating them by 30 years– and to cluster categories. Several generalisations of frieze patterns are known. Joint with Jørgensen, we associated frieze patterns to dissections of polygons where the entries are over a (commutative) ring. Furthermore, we introduced an explicit combinatorial formula for the entries of these friezes by generalising the 'T-path formula' of Schiffler which was introduced to give explicit formulas for cluster variables for cluster algebras of type A.

Paolo Bellingeri (University of Caen): Virtual braids and permutations . Let $VB_n$ the virtual braid group on $n$ strands and $S_n$ the symmetric group on $n$ elements. We determine all possible homomorphisms between:
- $VB_n$ and $VB_m$
- $VB_n$ and $S_m$
- $S_n$ and $VB_m$
when $n>4$ and $n\geq m$. As corollaries we get several results on virtual braid groups, in particular we compute their outer group and we show that virtual braid groups are hopfian and co-hopfian. The approach is completely different from Artin and Lin ones for classical braids and permutations, and it is based on Basse-Serre theory of amalgamated products of groups. This is a joint work with Luis Paris.

Alex Fink (Queen Mary University London): Matrix orbit closures and their classes. If an ordered point configuration in projective space is represented by a matrix of coordinates, the resulting matrix is determined up to the action of the general linear group on one side and the torus of diagonal matrices on the other. We study orbits of matrices under the action of the product of these groups, and their Zariski closures, as well as their images in quotients of the space of matrices like the Grassmannian. The main question is what properties of these varieties are determined by the matroid of the point configuration; the main result is that their finely graded Hilbert functions are so determined.
The results of mine are joint with Andy Berget, apart from some which are instead joint with David Speyer.

Vincent Gélinas (Trinity College Dublin): The A-infinity structure on Ext and homological properties of finite dimensional algebras. We'll talk about applications of the A-infinity structure on the Koszul dual (= Ext algebra of the simples) of a finite dimensional path algebra. Time willing, we'll go over some of: the philosophy of minimal free DGA models replacing minimal projective resolutions, computing cohomology of iterated Serre functors, or testing the finite generation (Fg) conditions for Hochschild cohomology of Snashall-Solberg. On that later point, in the case of monomial relations we will use the A-infinity structure of Tamaroff to sketch a recent proof that a finite dimensional monomial algebra satisfies Fg if and only if it is Gorenstein. This last part is ongoing joint work with V. Dotsenko and P. Tamaroff.

Hamid Ahmadinezhad (Loughborough University): Birational rigidity of 3-folds and where the concept sits in birational classification. This talk will be a gentle introduction to birational geometry with a quick survey on some recent aspects of birational classification of complex varieties in dimension 3. Keywords are birational geometry, minimal model programme, Fano varieties, birational rigidity.

For past alegebra seminars see: 2017/18 .