Emine Yıldırım

pmtey@leeds.ac.uk

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The University of Leeds Algebra Seminar 2023/24

Tuesdays 15:00 in MALL

Co-organised with my amazing colleague Francesca Fedele.

Date Speaker Title
October 3 Farrokh Atai (University of Leeds) Vertex operator algebra and super-Jack polynomials
October 17 Alison Parker (University of Leeds) Some representation theory of Kadar-Martin-Yu algebras
October 24 Bradley Ryan (University of Leeds) Double Affine Hecke Algebras and Character Varieties
October 31 Sam Hannah (Cardiff University) Classifying Frobenius Algebras in Dijkgraaf-Witten Categories
November 14 Jeffrey Hicks (University of Edinburgh) Resolutions from the perspective of symplectic geometry
November 21 Jack Romo (University of Leeds) Homotopy Bicategories of Complete 2-fold Segal Spaces
November 22 Michael Tsironis (Vrije Universiteit Amsterdam) Skein relations for punctured surfaces
November 28 Katherine Ormeño Bastias (Universidad de Talca) Seminormal forms for the Temperley-Lieb algebra
December 5 Michael Wibmer (University of Leeds) Difference algebraic groups
December 12 Sofia Franchini (Lancaster University) Torsion pairs in the completions of Igusa-Todorov discrete cluster categories
February 6 Nivedita Viswanathan (Brunel University London) Higher Fano Manifolds
February 13 Federico Campanini (Université catholique de Louvain) An overview on pretorsion theories
February 20 Aslak Bakke Buan (Norwegian University of Science and Technology) Mutation of tau-exceptional sequences
February 27 Gordana Todorov (Northeastern University) Dynkin diagrams and higher: Auslander algebras, Cluster Tilting objects and Preprojective Algebras
March 5 Anders Sten Kortegård (Aarhus University) Derived equivalences of self-injective 2-Calabi—Yau titled algebras
March 12 Amit Hazi (University of Leeds) Existence and rotatability of the two-colored Jones–Wenzl projector
March 19 (at 2PM) David Terrence Nkansah (Aarhus University) Nakayama functors are wannabe Serre functors
April 23 Patrick Kinnear (University of Edinburgh) Non-semisimple Crane-Yetter varying over the character stack
April 30 Nicholas Williams (Lancaster University) Donaldson–Thomas invariants for the Bridgeland–Smith correspondence
May 7 Martin Palmer (Institutul de Matematică Simion Stoilow al Academiei Române) Do the dual Miller-Morita-Mumford classes vanish in the homology of the big mapping class group?
May 21 Tathagata Ghosh (University of Leeds) Instantons and Quivers: An overview
June 4 Cristina Palmer-Anghel (University of Leeds) Universal coloured Alexander invariant via configurations on ovals in the disc
June 11 Christine Lee (Texas State University) A topological model for the HOMFLY-PT polynomial

Abstracts:


Christine Lee (Texas State University)

A topological model for the HOMFLY-PT polynomial

A topological model for a polynomial knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and SL_n polynomials, and our construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.


Cristina Palmer-Anghel (University of Leeds)

Universal coloured Alexander invariant via configurations on ovals in the disc

The coloured Jones and Alexander polynomials are quantum invariants that come from representation theory. There are important open problems in quantum topology regarding their geometric information. Our goal is to describe these invariants from a topological viewpoint, as intersections between submanifolds in configuration spaces. We show that the Nth coloured Jones and Alexander polynomials of a knot can be read off from Lagrangian intersections in a fixed configuration space.

At the asymptotic level, we geometrically construct a universal ADO invariant for links, as a limit of invariants given by intersections in configuration spaces. The parallel question of providing an invariant unifying the colored Jones invariants is the subject of the universal Habiro invariant for knots. The universal ADO invariant that we construct recovers all of the coloured Alexander invariants (in particular, the Alexander polynomial in the first term).


Tathagata Ghosh (University of Leeds)

Instantons and Quivers: An overview

Highly motivated by the interplay among physics, geometry and algebra, in this completely expository but very ambitious talk I aim to review the contributions of instantons in the algebraic study of quivers. Instantons solve the Yang-Mills equations, and are very important objects of study in Quantum field theory and particle physics.

In this talk I will very briefly discuss the ADHM constructions for instantons, instantons on ALE spaces, and as an immediate consequence, quiver varieties! If time permits, I may mention some applications of Quiver varieties in cluster algebras and quantum cluster algebras.


Martin Palmer (Institutul de Matematică Simion Stoilow al Academiei Române)

Do the dual Miller-Morita-Mumford classes vanish in the homology of the big mapping class group?

The Mumford conjecture – a consequence of the Madsen-Weiss theorem – describes the rational homology of the mapping class groups Mod(Σ(g,1)) in the limit as g goes to infinity, in terms of the dual Miller-Morita-Mumford (MMM) classes. Instead of taking the colimit of the mapping class groups, one may instead take the colimit of the surfaces Σ(g,1) themselves, to obtain an infinite-type surface Σ(∞), and consider its mapping class group Mod(Σ(∞)), called the “big mapping class group”. The structure of its homology is very mysterious, and very large: it is uncountably generated in every positive degree. There is a natural homomorphism from the colimit of Mod(Σ(g,1)) to Mod(Σ(∞)), and one may wonder what its effect is on homology; in particular whether the dual MMM classes vanish on Mod(Σ(∞)). This is a special case of a more general question for any infinite-type surface S: does its mapping class group Mod(S) admit non-zero homology classes supported on a compact subsurface of S? We will give a complete answer to this question when S has non-zero genus (including the case S=Σ(∞)) and a partial answer when S has genus zero. This represents joint work with Xiaolei Wu.


Nicholas Williams (Lancaster University)

Donaldson–Thomas invariants for the Bridgeland–Smith correspondence

Celebrated work of Bridgeland and Smith shows a correspondence between quadratic differentials on Riemann surfaces and stability conditions on certain 3-Calabi–Yau triangulated categories. Part of this correspondence is that finite-length trajectories of the quadratic differential correspond to stable objects of phase 1. Speaking roughly, these stable objects are then counted by an associated Donaldson–Thomas invariant. Work of Iwaki and Kidwai predicts particular values for these Donaldson–Thomas invariants according to the different types of finite-length trajectories, based on the output of topological recursion. We show that the category recently studied by Christ, Haiden, and Qiu produces Donaldson–Thomas invariants matching these predictions. In this talk I will give a gentle introduction to quadratic differentials, explain the construction relating them to stability conditions, and, time permitting, indicate the proof of our main result. This is joint work with Omar Kidwai.


Patrick Kinnear (University of Edinburgh)

Non-semisimple Crane-Yetter varying over the character stack

Under the cobordism hypothesis, it is known that the category Rep_q(G) of representations of Lusztig’s quantum group at a root of unity defines a fully extended 4-dimensional TQFT. We explain in this talk how to construct from Rep_q(G) a relative TQFT Z. Our theory is defined relative to 4d G-gauge theory at good odd roots of unity (and relative to G^L-gauge theory at good even roots of unity). As we explain, Z assigns (higher) quasicoherent sheaves on the G-character stack. It is known that de-equivariantizing the G-action on Z recovers the invertible theory defined by the non-semisimple representation category Rep u_q for the small quantum group (which we call the non-semisimple Crane-Yetter theory). Our main result is that Z itself is invertible relative to 4d G-gauge theory. Our results imply that to a 3-manifold Z assigns a line bundle on the character stack. To a surface we show that Z assigns an invertible sheaf of categories with global sections the skein category: this gives a global, stacky version of the skein algebra unicity theorem for higher rank groups. In the recently advanced perspective on topological symmetry in QFT, one can regard Z as a boundary wall symmetry defect of G-gauge theory, which is known to be invertible after de-equivariantizing, or gauging. Our main result says that the defect itself is invertible before de-equivariantizing.


David Terrence Nkansah (Aarhus University)

Nakayama functors are wannabe Serre functors

Classic Auslander-Reiten theory is a neat tool used to paint a portrait of the category of modules over an Artinian ring. Nakayama functors play an important role in this painting. In this talk, we will construct Nakayama functors and Auslander–Reiten translates, on proper abelian subcategories, homologically through approximation theory. These abelian subcategories, defined by Jørgensen in 2022, are generalisations of hearts of t-structures. As a consequence, we will prove that our proper abelian subcategories are dualising k-varieties, that they have enough projectives if and only if they have enough injectives and give a new proof of the existence of Auslander-Reiten translates in the module category of an algebra.


Amit Hazi (University of Leeds)

Existence and rotatability of the two-colored Jones–Wenzl projector

The two-colored Temperley-Lieb algebra is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. In this talk, I will give conditions for the existence and rotatability of the two-colored Jones-Wenzl projector in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe’s category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.


Anders Sten Kortegård (Aarhus University)

Derived equivalences of self-injective 2-Calabi—Yau titled algebras

A Frobenius category is a kind of exact category to which we can associate a triangulated category, called the stable category. We will consider a k-linear Frobenius category E, with a “nice” associated stable category C. Given two “good maximal rigid” objects in E their endomorphism algebras are always derived equivalent. In this talk, we look at the corresponding endomorphism algebras in the stable category C, and give a criterion for when these will become derived equivalent. We will then give an example of how we can produce such derived equivalent endomorphism algebras using Postnikov diagrams.


Gordana Todorov (Northeastern University)

Dynkin diagrams and higher: Auslander algebras, Cluster Tilting objects and Preprojective Algebras

A well-known theorem of Maurice Auslander about artin algebras describes the correspondence between {algebras A of finite representation type} and {algebras B, with gl.dim.B ≤ 2 ≤ dom.dim.B}, (now called Auslander algebras).

Higher Auslander algebras were introduced by Osamu Iyama. This talk will be about families of higher Auslander algebras and related notions of higher Cluster Tilting Objects and also higher Preprojective algebras, all constructed from the fundamental domains of cluster categories of Dynkin quivers.


Aslak Bakke Buan (Norwegian University of Science and Technology)

Mutation of tau-exceptional sequences.

We first recall classical notions of exceptional sequences and their mutations for module categories of hereditary algebras (e.g. quiver algebras). Next, we discuss a generalization to all finite dimensional algebras, motivated by tau-tilting theory, by Adachi-Iyama-Reiten; by Jasso’s reduction techniques for such modules and corresponding torsion pairs; and by the introduction of signed exceptional sequences by Igusa-Todorov.

The interplay between theories for tau-tilting modules, torsion pairs and wide subcategories is central for our discussions.

This is based on joint work with Eric J. Hanson and Bethany R. Marsh


Federico Campanini (Université catholique de Louvain)

An overview on pretorsion theories

Pretorsion theories are defined as “non-pointed torsion theories”, where the zero object and the zero morphisms are replaced by a class of “trivial” objects and a suitable ideal of morphisms respectively. Thus, the notion of pretorsion theory can be defined in any arbitrary category C, starting from a pair (T ,F) of full replete subcategories of C where T and F consist of the classes of “torsion” and “torsion-free” objects, and whose intersection defines the class of “trivial objects”.

In this talk, we shall first present some key background on torsion and pretorsion theories, describing some examples in the category of topological spaces, (internal) preorders, and in Cat, the category of small categories. We then show two ways of obtaining pretorsion theories starting from torsion theories, so that many new examples of pretorsion theories can be given in pointed categories. Lattices and chains of torsion theories are widely studied topics and they are the perfect framework for applying our result. Then, we shall show how to obtain pretorsion theories “extending” a torsion theory with a Serre subcategory. We shall discuss some applications in representation theory and in the framework of recollements of abelian categories.

Based on joint works with F. Borceux, M. Gran, W. Tholen [1, 2] and F. Fedele [3].

[1] F. Borceux, F. Campanini and M. Gran, Pretorsion theories in lextensive categories, to appear on Israel J. Math (2023). https://arxiv.org/abs/2205.11054

[2] F. Borceux, F. Campanini, M. Gran and W. Tholen, Groupoids and skeletal categories form a pretorsion theory in Cat, accepted on Adv. Math (2023). https://arxiv.org/abs/2207.08487

[3] F. Campanini, F. Fedele, Building pretorsion theories from torsion theories, preprint. https://arxiv.org/abs/2310.00316

[4] A. Facchini, C.A. Finocchiaro and M. Gran, Pretorsion theories in general categories, J. Pure Appl. Algebra 225 (2) (2021) 106503.


Nivedita Viswanathan (Brunel University London)

Higher Fano Manifolds

Fano manifolds are smooth varieties that have a positive first Chern class, c_1(T_X). This positivity condition on Fano manifolds has geometric implications such as them being covered by rational curves and in fact being rationally connected. In a quest to explore higher dimensional analogues of such geometric properties of objects in hand, we study Fano manifolds further by imposing stronger positivity restrictions on them, that is by assuming positivity of higher Chern characters. These are called Higher Fano Manifolds. While examples of such higher Fano manifolds have been described in recent literature, very few generalized results exist. In this talk, I will explain a combinatorial strategy that we use in our work to prove that projective spaces are the only higher Fano manifolds among toric projective manifolds. This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Enrica Mazzon and Svetlana Makarova.


Sofia Franchini (Lancaster University)

Torsion pairs in the completions of Igusa-Todorov discrete cluster categories

Given an infinity-gon having a finite number of two-sided accumulation points, Igusa and Todorov defined a discrete cluster category of type A-infinity. The completion of this category was defined by Paquette and Yıldırım. Both categories are triangulated, have cluster-tilting subcategories, and have nice geometric models: their indecomposable objects can be regarded as arcs, or limits of arcs, of the infinity-gon. These geometric models allow to classify some important classes of subcategories using arc combinatorics. In this talk we will introduce the Igusa-Todorov discrete cluster category and the Paquette Yıldırım completion, and we will discuss the classification of the torsion pairs, t-structures, and co-t-structures in the completion.


Michael Wibmer (University of Leeds)

Difference algebraic groups

Difference algebraic groups are a generalization of algebraic groups. Instead of just algebraic equations, one allows difference algebraic equations as the defining equations. These groups naturally occur as Galois groups in certain Galois theories and they have a rich structure theory, loosely similar to the structure theory of algebraic groups. This talk is an introduction to difference algebraic groups.


Katherine Ormeño Bastias (Universidad de Talca)

Seminormal forms for the Temperley-Lieb algebra

Over the last few years, Mathas introduced seminormal forms for a cellular algebra endowed with a family of Jucys-Murphy elements. The seminormal forms are orthogonal primitive idempotents and eigenvectors for the Jucys-Murphy elements under certain conditions. In this framework, there is a dichotomy: the separated case (semisimple) and the non-separated case (non-semisimple). Following this dichotomy, we study the cellular basis of the Temperley-Lieb algebra and construct the seminormal forms for Temperley-Lieb using the Jones-Wenzl idempotents and some combinatorial tools for the separated case. Additionally, we briefly explore what occurs in the non-separated case. This is joint work with my PhD supervisor, Steen Ryom-Hansen.


Note: Alongside with Tuesday talk, there is another talk on Wednesday at 2PM in LT15. Details below.

Michael Tsironis (Vrije Universiteit Amsterdam)

Skein relations for punctured surfaces

Given a marked bordered surface S and a triangulation T, one can associate a Cluster algebra and a Jacobian algebra. In the classical setting where all the marked points lie on the boundary, snake graph calculus is a very helpful combinatorial tool, which was used by İlke Çanakçı and Ralf Schifler (2015) to give an alternative proof of the so-called skein relations, which are some identities in terms of cluster variables. Recently, Jon Wilson (2020) constructed a new type of snake graph, the so-called loop graph, which allows one to easily work in the setting, where we are also allowing marked points (puntures) in the interior of the surface S. In this talk, we will describe how one can associate a module in the Jacobian algebra, to every given loop graph, and how we can use this construction to prove skein relations in this broader setting.


Jack Romo (University of Leeds)

Homotopy Bicategories of Complete 2-fold Segal Spaces

Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models where composition operations are given, like Bénabou’s bicategories, tricategories following Gurski and the models of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, where a contractible space of possible composites is provided instead, including those of Tamsamani and Paoli, along with quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell, Nikolaus and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of (\infty, n)-category for general n.

In this talk, I will present my contribution to the problem of taking homotopy bicategories of non-algebraic (\infty, 2)-categories. The procedure I present of translating homotopies to coherence isomorphisms and ‘2-homotopies’ to coherence conditions seems to have more general application, which I will discuss if time permits.


Jeffrey Hicks (University of Edinburgh)

Resolutions from the perspective of symplectic geometry

One of the great insights of homological algebra is that one can resolve a complicated object into a complex of simpler ones. This talk will look at resolutions from several different perspectives - commutative algebra, algebraic geometry, topology, and differential geometry. We will then see how resolutions appear naturally in symplectic geometry, focusing on examples in real dimension 2 (so we may draw everything on the board - no background in symplectic geometry is needed for the talk!). Time permitting, I’ll discuss some applications to toric varieties:

All work is joint with Andrew Hanlon and Oleg Lazarev.


Sam Hannah (Cardiff University)

Classifying Frobenius Algebras in Dijkgraaf-Witten Categories

Algebra objects are interesting objects in category theory, with particular uses in classifying modules over tensor categories. Detecting and classifying algebra objects provides an important goal in understanding the representation theory of such categories. Dijkgraaf-Witten categories are an example of tensor categories associated to a finite group G and a 3-cocycle on G. During this talk, I will present a classification of Frobenius algebras in these categories, generalising existing results by Davydov-Simmons to a field of arbitrary characteristic. This is done by means of a Frobenius Monoidal Functor. Joint work with Ana Ros Camacho (Cardiff) and Robert Laugwitz (Nottingham).


Bradley Ryan (University of Leeds)

Double Affine Hecke Algebras and Character Varieties

The double affine Hecke algebra (DAHA) was introduced by Ivan Cherednik in order to prove Macdonald’s conjectures. Developments from Noumi, Sahi and Stokman extended Cherednik’s definition to produce a DAHA with connections to Koornwinder polynomials. It has been conjectured that the centre of this DAHA at the “classical level” is isomorphic to the algebra of functions on a particular character variety. In this talk, we introduce this DAHA and motivate a proof of this conjecture. This is based on joint work-in-progress with my PhD supervisor Oleg Chalykh.


Alison Parker (University of Leeds)

Some representation theory of Kadar-Martin-Yu algebras

Kadar-Martin-Yu introduced a new chain of subalgebras of the Brauer algebra. These algebras start with Temperley-Lieb and end with the Brauer algebra and build in representation theoretic intensity. This gives a new tool to tackle the long standing problem of understanding the representation theory of the Brauer algebra. We present an introduction to these new algebras and some results about their representation theory. This is joint work with my PhD student N. M. Alraddadi.


Farrokh Atai (University of Leeds)

Vertex operator algebra and super-Jack polynomials

The fruitful interplay between representation theory and the theory of special functions has led to many interesting discoveries in mathematical physics. Moreover, it is known that representations of certain infinite dimensional algebras can be used to give a mathematical formulation of some quantum field theories.

In this talk, I will recall how highest weight representations of certain affine Kac-Moody algebras are given by field operators and symmetric functions. Using the field operators, we can construct vertex operator with interesting applications in mathematical physics. By considering a one-parameter extension of these vertex operators, we also find that they are closely related to the integrable Calogero-Moser-Sutherland models. In particular, we find that there is a one-to-one correspondence between the space spanned by these vertex operators and the so-called super-Jack polynomials. If time permits, I will also discuss how this approach can be used to construct a Laurent polynomial generalization of the super-Jack polynomials.


2023 Leeds Algebra Seminars