## The University of Leeds Algebra Seminar 2020/21   Tuesdays 15:00 (currently online in zoom)

Until further notice the seminar will take place online at 15:00 on Tuesdays. We will have virtual biscuits and tea (and coffee) before the seminar.

## Schedule:

 DATE and TIME and PLACE SPEAKER TITLE October 6, 15:00  online Dixy Msapato   (University of Leeds) Counting tau-exceptional sequences over Nakayama algebras October 13, 15:00  online Severin Bunk   (Universität Hamburg) Universal Symmetries of Gerbes and Smooth Higher Group Extensions October 20, 15:00  online Ana Ros Camacho   (Cardiff University) Algebraic structures in group-theoretical fusion categories October 27, 15:00  online David Jordan   (University of Edinburgh) Postponed. November 3, 15:00  online Tyler Kelly   (University of Birmingham) Exceptional Collections in Mirror Symmetry November 10, 15:00  online Lutz Hille   (Universität Münster) Moduli of quiver representations, quiver Grassmannians and morphisms November 17, 15:00  online Vincent Koppen   (University of Leeds) On isotypic decompositions for non-semisimple Hopf algebras November 24, 15:00  online Xiuping Su   (University of Bath) Auslander algebras of Nakayama algebras and (quantised) partial flag varieties December 1, 15:00  online David Jordan   (University of Edinburgh) Cluster quantization and topological field theory December 8, 15:00  online Tara Brendle   (Unviersity of Glasgow) The mapping class group of connect sums of $\mathbb{S}^2 \times \mathbb{S}^1$ February 2, 15:00  online Anna Felikson   (Durham University) Mutations of non-integer quivers: finite mutation type February 9, 15:00  online Matthew Pressland   (University of Leeds) The cluster category of a Postnikov diagram February 16, 15:00  online Margarida Melo   (Universita Roma Tre) On the top weight cohomology of the moduli space of abelian varieties February 23, 15:00  online Janina Letz   (Universität Bielefeld) A homotopical characterization of locally complete intersection maps March 2, 15:00  online Man-Wai Cheung   (Harvard University) Compactifications of cluster varieties and convexity March 9, 15:00  online Emine Yildirim   (Queen's University) TBA March 16, 15:00  online Karin Jacobsen   (TBA) TBA March 23, 15:00  online Robert Laugwitz   (University of Nottingham) TBA April 27, 15:00  online Alastair Craw   (University of Bath) TBA May 4, 15:00  online Greg Muller   (University of Oklahoma) TBA

## Abstracts

Dixy Msapato (University of Leeds): Counting tau-exceptional sequences over Nakayama algebras. The notion of a tau-exceptional sequence was introduced by Buan and Marsh in 2018 as a generalisation of exceptional sequences over finite dimensional algebras. In this talk, I will introduce both these notions, and present counting results of tau-exceptional sequences over some classes of Nakayama algebras. In some of these cases we will see obtain closed formulas counting other well known combinatorial objects, and exceptional sequences over some path algebras of Dynkin quivers.

Severin Bunk (Universität Hamburg): Universal Symmetries of Gerbes and Smooth Higher Group Extensions. Gerbes are geometric objects describing the third integer cohomology of a manifold and the B-field in string theory; they can essentially be understood as bundles of categories whose fibre is equivalent to the category of vector spaces. Starting from a hands-on example, I will explain gerbes and their categorical features. The main topic of this talk will then be the study of symmetries of gerbes in a universal manner. We will see that this is completely encoded in an extension of smooth 2-groups. If time permits, in the last part I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of group extensions in $\infty$-topoi.

Ana Ros Camacho (Cardiff University): Algebraic structures in group-theoretical fusion categories. In a categorical sense, Morita equivalence is a useful notion (stemming from ring theory) that allows us to classify collections of objects. It has also nice applications in several topics in (mathematical) physics like e.g. rational conformal field theory. In this talk, we generalize results from Ostrik and Natale that describe Morita equivalence classes of certain algebra objects in pointed fusion categories to the case of group-theoretical fusion categories. These algebra objects also have very good properties that we will describe in detail. We will assume little knowledge of fusion categories. Joint work with the WINART2 team Y. Morales, M. Mueller, J. Plavnik, A. Tabiri and C. Walton..

Tyler Kelly (University of Birmingham): Exceptional Collections in Mirror Symmetry. I will give a few examples of where exceptional collections in categories arise in mirror symmetry and why they are useful. These are a (weaker) categorical version of tilting modules. I aim to focus on the homological algebra and will give a biased view, ending with a few results of D. Favero, D. Kaplan, and myself in this vein.

Lutz Hille (Universität Münster): Moduli of quiver representations, quiver grassmannians and morphisms. Moduli space of quiver representations have been introduced by King following Mumford's celebrated results on geometric invariant theory. So far, only a few moduli spaces are explicitly known. On the other side, quiver grassmannians became later of interest motivated by cluster algebras, however, also here only a few cases are understood. The key problem is the existence of sufficiently many morphisms between those spaces to start induction or to understand topological or K-theoretic properties.
A new idea developed in joint work with Blume produces sufficiently many morphisms, however not between the moduli spaces themself, but between their inverse limit. Similar techniques also work for quiver grassmannians.
The aim of this talk is to define the inverse limit, to give some easy examples and to show the advantage of this modified moduli space. At the end we discuss the geometry of this new spaces.

Vincent Koppen (University of Leeds): On isotypic decompositions for non-semisimple Hopf algebras. In this talk we consider the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this talk we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

Xiuping Su (University of Bath): Auslander algebras of Nakayama algebras and (quantised) partial flag varieties. Auslander algebras are quasi-hereditary algebras, which were introduced by L Scott to study highest weight categories in Lie theory. An important concept in this context is standard modules $\Delta_i$.
In this talk I will discuss certain subcategories of $\Delta-$filtered modules of some Auslander algebras and show that these are stably 2-CY categories. I will also explain how these categories lead to possible categorification of cluster algebras on the (quantised) coordinate rings of partial flag varieties. This talk is based on ongoing joint work with B T Jensen.

David Jordan (University of Edinburgh): Cluster quantization and topological field theory. Quantum cluster algebras are non-commutative algebras characterized by the existence of certain simple "cluster charts" - quantum tori - and mutations - non-commutative birational equivalences of a special form. In the early 2000's Fock and Goncharov introduced an interesting class of quantum cluster algebras associated to a surface, and related these to certain moduli spaces of local systems on the surface.
In this talk I will explain a recasting of Fock-Goncharov's construction in the language of fully extended 4-dimensional topological field theory. Namely, I will explain how to recover their construction as a computation in stratified factorization homology, as introduced by Ayala--Francis--Tanaka, using monadic techniques and representation theory of quantum groups. This endows the quantum cluster algebra with a number of new structures, clarifies its relation to mathematical physics, and points the way to invariants of 3-manifolds built on cluster algebra tools. This is joint work with Ian Le, Gus Schrader, and Sasha Shapiro.

Tara Brendle (University of Glasgow): The mapping class group of connect sums of $\mathbb{S}^2 \times \mathbb{S}^1$. Let $M_n$ denote the connect sum of $n$ copies of $\mathbb{S}^2 \times \mathbb{S}^1$. Laudenbach showed that the mapping class group $\mathrm{Mod}(M_n)$ is an extension of the group Out$(F_n)$ by $(\mathbb{Z}/2)^n$, where the latter group is the "sphere twist" subgroup of $\mathrm{Mod}(M_n)$. In joint work with N. Broaddus and A. Putman, we have shown that in fact this extension splits. In this talk, we will describe the splitting and discuss some simplifications of Laudenbach's original proof that arise from our techniques.

Anna Felikson (Durham University): Mutations of non-integer quivers: finite mutation type . Given a skew-symmetric non-integer (real) matrix, one can construct a quiver with non-integer weights of arrows. Such a quiver can be mutated according to usual rules of quiver mutation introduced within the theory of cluster algebras by Fomin and Zelevinsky. We classify non-integer quivers of finite mutation type and prove that all of them admit some geometric interpretation (either related to orbifolds or to reflection groups). In particular, the reflection group construction gives rise to the notion of non-integer quivers of finite and affine types. We also study exchange graphs of quivers of finite and affine types in rank 3. The talk is based on joint works with Pavel Tumarkin and Philipp Lampe.

Matthew Pressland (University of Leeds): The cluster category of a Postnikov diagram. A Postnikov diagram consists of a collection of strands in the disc, with combinatorial restrictions on their crossings. Such diagrams were used by Postnikov and others to study weakly separated collections in certain matroids called positroids. In this talk I will explain how the diagram determines a cluster algebra structure on a suitable subvariety of the Grassmannian, and simultaneously provides a (Frobenius) categorification of this cluster algebra.

Margarida Melo (Universita Roma Tre): On the top weight cohomology of the moduli space of abelian varieties. In the last few years, tropical methods have been applied quite successfully in understanding several aspects of the geometry of classical algebro-geometric moduli spaces. In particular, in several situations the combinatorics behind compactifications of moduli spaces have been given a tropical modular interpretation. Consequently, one can study different properties of these (compactified) spaces by studying their tropical counterparts.
In this talk, which is based in joint work with Madeleine Brandt, Juliette Bruce, Melody Chan, Gwyneth Moreland and Corey Wolfe, I will illustrate this phenomena for the moduli space Ag of abelian varities of dimension g. In particular, I will show how to apply the tropical understanding of the classical toroidal compactifications of Ag to compute, for small values of g, the top weight cohomology of Ag.
The techniques we use follow the breakthrough results and techniques recently developed by Chan-Galatius-Payne in understanding the topology of the moduli space of curves via tropical geometry.

Janina Letz (Universität Bielefeld): A homotopical characterization of locally complete intersection maps. This talk is about locally complete intersection maps of commutative noetherian rings. Results of Dwyer, Greenlees and Iyengar, and Pollitz characterize the complete intersection property for a noetherian ring in terms of the structure, as a triangulated category, of the bounded derived category of the ring. I present a similar characterization for locally complete intersection maps. This is joint work with Briggs, Iyengar, and Pollitz.

Man-Wai Cheung (Harvard University): Compactifications of cluster varieties and convexity. Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational "mutation" maps. They can be seen as a generalization of the toric varieties. In toric geometry, projective toric varieties can be described by polytopes. We will see how to generalize the polytope construction to cluster convexity which satisfies piecewise linear structure. As an application, we will see the non-integral vertex in the Newton Okounkov body of Grassmannian comes from broken line convexity. We will also see links to the symplectic geometry and application to mirror symmetry. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

Emine Yildirim (Queen's University): TBA. TBA.

Karin Jacobsen (Aarhus University): TBA. TBA.

Robert Laugwitz (University of Nottingham): TBA. TBA.

Alastair Craw (University of Bath): TBA. TBA.

Greg Muller (University of Oklahoma): TBA. TBA.

For past algebra seminars see: 2019/20 , 2018/19 , 2017/18 .