Seminar Geometric Methods in Mathematics

Andrew Baker (University of Glasgow): Hopf Hecke algebras and Orbit Categories.

Tuesday 13.05.2025:

Marcelo Aguiar (Cornell University): The Eckmann–Hilton argument in duoidal categories

Abstract: We will go over the basics of duoidal categories, illustrating with a number of examples. As monoidal categories provide a context for monoids, duoidal categories provide one for duoids and bimonoids. Our main goal is to discuss a number of versions of the classical Eckmann–Hilton argument which may be formulated in this setting. As an application we will obtain the commutativity of the cup product on the cohomology of a bimonoid with coefficients in a duoid, an extension of a familiar result for group and bialgebra cohomology.

Symposion on Higher Structures for Hopf Algebras (website)

• Monday, 2025-05-19
  • 13:30–14:30: Tony Zorman (TU Dresden): Reconstruction for Lax Module Monads

  • 15:00–16:00: Gabriella Böhm (EMS Press): Bialgebroids: an approach via duoidal categories

  • 16:30–17:30: Ivan Bartulović (TU Dresden): Duplicial functors, entwined coalgebras and Hopf modules

• Tuesday, 2025-05-20:
  • 10:00–11:00: Sebastian Halbig (Universität Marburg): A non-semisimple version of the Kitaev model

  • 13:30–14:30: Niels Kowalzig (Tor Vergata): Brackets and products from centres in extension categories

  • 16:30–17:30: Marvin Dippel (University of Salerno): Global Homotopies for Differential Hochschild Cohomologies

Zbigniew Wojciechowski (TU Dresden): Graduate Lectures on Koszulity

Graduate Lecture 1: From Linear Algebra over the Integers to Koszul duality

Koszul duality is a kind of homological algebra analogue of inverting power series or matrices. In this talk, I will motivate the properties of Koszul algebras and coalgebras through inversion formulas. Most examples will involve path algebras of quivers, so we will invert matrices that count paths between vertices. We will also explore the numerical Koszul criterion and a derived equivalence.

Graduate Lecture 2: Bar and Cobar Constructions, and Koszul Duality for Operads

Instead of inverting power series under multiplication, one can also consider inversion with respect to composition. The composition of power series can be visualized by inserting trees into trees. I will use this perspective to introduce operads as quotients of the tree operad, and explain how to generalize the Koszul complex from Talk 1. This leads to an analogue of the numerical Koszul criterion. Examples will include the associative, commutative, and Lie operads.

Ulrich Krähmer (TU Dresden): Coend of term discussion

I started to think with Tony about Tannaka-Krein and would maybe explain the difference between recovering a Hopf algebra H as a coend with values in Vect vs the braided Hopf algebra that we talked about in Bonn (Tony would say "the Lyubashenkp coend") that you recover from the coend in H-Comod itself. 

Janis Winter (TU Dresden): Geometric Realizations of Simplicial Structures: A Comparison using Crossed Simplicial Groups

Any simplicial complex gives rise to two distinct associated simplicial sets, one being equipped with degreewise actions of the symmetric groups, and one without. While it is easily shown that the geometric realization of the latter simplicial set is homeomorphic to the geometric realization of the simplicial complex itself, it is not trivial to show an analogous homotopy equivalence for the geometric realization of the other. In this talk I explain how the theory of crossed simplicial groups may aid in comparing the geometric realizations of the two simplicial sets. Starting by introducing simplicial structures and their realizations, I will define crossed simplicial groups and look at the key example of the symmetric crossed simplicial group ΔS. We will see that the simplicial sets corresponding to a simplicial complex are in fact ΔS-sets and give an overview of how this could motivate further studies.

Monday
21.10.2024

Guy Boyde, Utrecht University
The homology of Temperley-Lieb algebras
I will report on joint work in progress with Rachael Boyd and Robin Sroka on computing the homology of Temperley-Lieb algebras (in the guise of Tor groups with various coefficients). People have been studying these questions from a topological point of view since about 2019, and the talk will also be a survey, both of what we understand, and of where the remaining mysteries are. Time permitting, I might say a little on what we know about the homology of some related families, such as the Brauer, Jones annular, and Partition algebras. I will expect familiarity with the idea of using projective resolutions to compute Tor, but not much else.

Sarah Brauner, Brown University
Title: Configuration spaces and combinatorial algebras
Abstract: In this talk, I will discuss connections between configuration spaces, an important class of topological space, and combinatorial algebras arising from the theory of reflection groups. In particular, I will present work relating the cohomology rings of some classical configuration spaces—such as the space of n ordered points in Euclidean space—with Solomon’s descent algebra and the peak algebra.
The talk will be centered around two questions. First, how are these objects related? Second, how can studying one inform the other? This is joint, on-going work with Marcelo Aguiar and Vic Reiner.

On the first relative Hochschild cohomology and contracted fundamental group

Monday 18.11.2024

Duplicial functors, entwined coalgebras and Hopf modules
Abstract: Cyclic homology of algebras was independently introduced by Connes and Tsygan in the 1980s. Since then, their constructions have been considered in many versions and generalizations. In this talk we focus on the monadic framework by Böhm and Ştefan, which is later studied in the paper by Kowalzig, Krähmer and Slevin. The ingredients of the setup are two comonads, a distributive law between them and right and left coalgebras over the distributive law. While right coalgebras over it have an easy and known description, no similar was known for left coalgebras over it. After recalling some background and motivation, I will explain how the latter can be obtained under some additional assumptions. Finally, I will discuss how generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples. Based on joint work in progress with John Boiquaye and Ulrich Krähmer.

Severin Barmeier, University of Cologne

From Fukaya categories of orbifold surfaces to representation theory of associative algebras
Abstract: Gentle algebras are a class of quadratic monomial algebras which were studied since 1987 for their tractable representation theory. In 2017 Haiden, Katzarkov and Kontsevich showed that they also arise as formal generators of Fukaya categories of smooth surfaces with boundary, so that the Fukaya category of the surface is equivalent to the derived category of a gentle algebra. When one considers surfaces with orbifold points, there are several questions that arise. (1) How to define the Fukaya category of a singular surface? (2) Does the Fukaya category still admit formal generators? (3) How does it compare to the Fukaya categories of smooth surfaces?We give complete answers to these questions. The answers involve (1) proving an analogue of a conjecture of Kontsevich relating Fukaya categories of noncompact manifolds to cosheaves of A∞ categories, (2) a full classification of the formal generators arising from dissections of orbifold surfaces, and (3) the relationship between deformations of Fukaya categories and partial compactifications as advocated in Seidel's ICM 2002 address.The endomorphism algebras of formal generators form a new class of associative algebras which are generally no longer quadratic nor monomial. We conjecture that this class is closed under derived equivalence which is a rather rare phenomenon.This talk is based on joint work with Sibylle Schroll and Zhengfang Wang (arXiv:2407.16358), and on joint work in progress with the previous two coauthors as well as Cheol-Hyun Cho, Kyoungmo Kim and Kyungmin Rho.

Monday 16.12.2024

Monoidal adjunctions and Drinfeld centers
Abstract: The Drinfeld center (or monoidal center) is a universal braided category that can be associated to any monoidal category. Drinfeld centers often feature interesting structures, like that of modular tensor categories, and applications, such as the construction of link invariants. However, they can be hard to determine. I will discuss an approach based on certain pairs of adjoint functors that generalize restriction and induction between categories of representations of finite groups, as studied in joint projects with Robert Laugwitz and Sebastian Posur.

Ulrich Krähmer, TU Dresden
Clifford Algebras for Quantised Hermitian Symmetric Spaces

As there is still
interest in this, I'll recall the
results from a ten year old paper
with Matt Tucker-Simmons. Therein,
we explain how the Koszul dual of
Berenstein and Zwicknagl's twisted
quantum Schubert cell can be
identified wtth the spinor module of
a quantum flag manifold, and how
this relates the Koszul boundary
operator to the Dolbeault-Dirac
operator. 

Kelly Maggs, MPI-CBG Dresden
A brief history of persistence.

In this talk I will give a broad overview of the historical motivation and main results of persistent homology. This ill consist of:

• The motivating construction: the Rips filtration
• The interval decomposition of persistence modules;
• The interleaving distances and stability theorems.

I will also talk about open questions and challenges in the fields, such as generalisations to multi-parameter persistence. To end, I will briefly discuss applications in both pure mathematics and other fields of science.

Monday
08.04.2024

Christian Lomp, University of Porto 
Actions of Semisimple Hopf algebras
Actions of semisimple Hopf algebras bear striking resemblances to the actions of finite groups. While there are numerous parallels between them, it's important to note that not all actions of semisimple Hopf algebras behave as expected, encountering various obstacles along the way. In this talk I will discuss several questions and their partial solutions, namely as to whether the action of a semisimple Hopf algebra extends to rings of quotients and whether the smash product of a semiprime module algebra and a semisimple Hopf algebra is semiprime. Moreover, I will review the generalized Kac-Paljutkin algebras and their actions on non-commutative domains that do not factor through group algebras.

Ayelet Lindenstrauss, Indiana University
Loday Constructions on Spheres and Tori
(All joint work with Birgit Richter, and parts also with others as detailed during the talk) Given a commutative ring or ring spectrum, there is a functor from finite sets to commutative rings sending a set to the tensor product of copies of the ring (or ring spectrum) indexed by that set.  The Loday construction extends this to simplicial sets.  The simplest nontrivial example is the Loday construction over a circle, which gives Hochschild homology (or, respectively, topological Hochschild homology). I will discuss Loday constructions on higher spheres and on higher tori.  The calculations for spheres are inductive, using the splitting of an n-sphere into two hemispheres, joined along the equator (n-1)-sphere.  The calculations for tori which we know are for cases where the Loday construction on the torus is equivalent to the Loday construction on the wedge of spheres whose suspension is homotopy equivalent to the suspension of the torus.  This kind of stability is unexpected, definitely not generally true, and yet there are surprisingly many cases where it holds.

Monday
29.04.2024
online

Adam Rennie, University of Wollongong
The Levi-Civita connection on noncommutative differential forms
By combining Hilbert module and algebraic techniques, we give necessary and sufficient conditions for the existence of Hermitian and torsion-free connections on noncommutative one-forms, such as those arising from spectral triples. With additional structure we give a sufficient condition for uniqueness. Our methods are constructive, use standard definitions, and allow computation of curvature with comparable difficulty to the differential geometry case. Joint work with Bram Mesland. arXiv:2403.13735  and 2404.07957

27.05.2024
online

Bicommutant categories
Bicommutant categories, initially invented for the purposes of 3d TQFT and 2d CFT, seem to also appear in other domains of math with examples related to group theory, and dynamical systems. In this talk, I will explain why the category Rep(G) of unitary representations of a discrete infinite group G is a bicommutant category, and what this teaches us about Rep(G).

Monday
03.06.2024

Niels Kowalzig, University of Rome Tor Vergata
Higher structures on homology groups
We dualise the classical fact that an operad with multiplication leads
to cohomology groups that form a Gerstenhaber algebra to the context of cooperads. As a result, a cooperad with comultiplication induces a
homology theory that is endowed with the structure of a Gerstenhaber
coalgebra, that is, it comes with a graded cocommutative coproduct which is compatible with a coantisymmetric cobracket in a dual Leibniz sense. As an application, one obtains Gerstenhaber coalgebra structures on Tor groups of bialgebras or Hopf algebras as well as on Hochschild homology for Frobenius algebras. Joint work with Francesca Pratali.

Jean-Simon Pacaud Lemay, Macquarie University
Lifting Trace with Hopf Algebras and Hopf Monads
Hopf algebras and their modules can be defined in an arbitrary symmetric monoidal category. A Hopf algebra H in a symmetric monoidal category X has the special ability of lifting many desirable structures and properties of X to MOD(H), the category of H-modules. Indeed, MOD(H) will be a symmetric monoidal category, and if X is closed, or star-autonomous, or even compact closed, then MOD(H) will be as well. The antipode of H plays a crucial role in lifting these structures. In this talk,
I will explain how Hopf algebras also have the ability of lifting traces [2]. Traced monoidal categories, introduced by Joyal, Street and Verity in [3], are symmetric monoidal categories equipped with a trace operator, which generalizes the classical notion of the trace of matrices in linear algebra. Traced monoidal categories have many applications in mathematics, quantum foundations, and computer science. If X is a traced monoidal category, then for Hopf algebra H, MOD(H) will be a traced symmetric monoidal category [2]. In particular, this means that the trace of an H-module morphism is again an H-module morphism.

We will also consider the special cases of compact closed categories (where the trace is given by duals), or when the monoidal product is a product (where the trace is given by fixpoints) or a coproduct (where
the trace is given by iteration). We will also discuss how this fact also generalizes to the notion of Hopf monads, in the sense of Bruguières, Lack, and Virelizier as introduced in [1].

This is joint work with Masahito Hasegawa, and is based on our paper [2], available at: https://compositionality-journal.org/papers/compositionality-5-10/.

References
[1] A. Bruguieres, S. Lack, and A. Virelizier. Hopf monads on monoidal categories. Advances in Mathematics, 227(2):745–800, 2011.

[2] M. Hasegawa and J.-S. P. Lemay. Traced Monads and Hopf Monads. Compositionality, 5, October
2023.

[3] A. Joyal, R. Street, and D. Verity. Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, 1996.

Justin Barthite, University of Colorado Boulder
Traces and Cotraces in Bicategories
Traces arise in many different places in math: traces of matrices, characters of group representations, and even the Euler characteristic of a CW complex! There are very general notions of trace, expressed in the language of category theory, that capture these examples of traces and whose properties imply familiar results like the Lefschetz fixed point theorem and the induction formula for characters. The formalism of traces doesn't tell the whole story though; there are some constructions that feel trace-like in certain ways but also have a distinct flavor, and what's really needed to explain them from this category-theoretic perspective is a dual notion of "cotrace." I will talk about some of these things that I have been working to understand by developing a theory of bicategorical cotraces.

Monday
16.10.2023
online

Benjamin Haïoun, Paul Sabatier University (Toulouse III)
4-manifold invariants from non-semisimple finite ribbon Hopf algebras
I will discuss a recent construction of (3+1)-TQFTs from finite non-semisimple ribbon categories joint with F. Costantino, N. Geer and B. Batureau-Mirand. For concreteness, I will focus on the underlying 4-manifold invariant coming from categories of representations of a finite dimensional Hopf algebra. I will give explicit formulas from a handle decomposition. At the end of the talk, I will ask the audience for interesting examples.

Oliver Sander, TU Dresden
What do numerical analysts need the Koszul complex for?

Pedro Tamaroff, Humboldt University of Berlin
Differential operators of higher order and their homotopy trivializations
In the classical Batalin–Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology. In this talk, we will explore generalisations of Batalin–Vilkovisky algebras for differential operators of arbitrary order of three different flavours, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. This is joint work with V. Dotsenko and S. Shadrin.

Marco De Renzi, University of Montpellier
Algebraic presentations of cobordisms and TQFTs in dimensions 3 and 4
It has long been known that the category of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra, the circle. This result allows for a complete classification of TQFTs (Topological Quantum Field Theories) in dimension 2. In this talk I will discuss similar algebraic presentations in dimension 3 and 4 due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. I will also present a large family of examples of such algebras and the TQFTs they induce. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.

Monday
11.12.2023

Cole Comfort, University of Lorraine
Graphical algebra and quantum circuits
To a category theorist, quantum circuits are string diagrams for the monoidal category FHilb of finite dimensional Hilbert spaces.  Many people have worked on finding generators and relations presentations for monoidal subcategories of FHilb, called "fragments" of quantum circuits.  Families of graphical languages for different fragments of quantum circuits include the ZX-, ZW- and ZH-calculi.  In this talk we will take a different approach, changing the semantics to particular categories of spans and relations: recovering different fragments of quantum circuits with nondeterministic semantics.  To this end, we review graphical linear and affine algebra: presentations for subcategories of spans and relations where the morphisms are linear or affine subspaces.  When one works over finite fields of prime characteristic, we show how simple fragments of quantum circuits are recovered.  From this point, there are two separate ways in which this semantics can be broadened to capture larger fragments of quantum circuits.  One the one hand, by adding nonlinear behaviour, we obtain a complete axiomatization for "nondeterminstic Boolean circuits".  On the other hand, by doubling affine relations, one obtains a graphical language where the morphisms are affine coisotropic subspaces.  We show how this gives a unified semantics for "stabilizer quantum circuits" as well as linear/affine classical mechanical systems. We discuss how by changing the field, this can be used to give graphical languages for infinite dimensional quantum circuits.

Monday
18.12.2023

Ehud Meir, University of Aberdeen
Geometric methods in braided vector spaces and their Nichols algebras 
The Nichols algebra of a braided vector space is a generalization of both the symmetric algebra and the exterior algebra. It can be realized as a Hopf algebra in some braided monoidal category. By the process of bosonization, Nichols algebras provide an abundance of examples of finite-dimensional non-semisimple Hopf algebras. This raises the question of the finite-dimensionality of Nichols algebras. In this talk, I will explain how to prove that certain Nichols algebras are infinite dimensional by using geometric methods, in particular, reductive-group actions. This is based on a joint work with István Heckenberger and Leandro Vendramin.

Monday
08.01.2024
online

Daniel Gromada, Czech Technical University in Prague
Quantum symmetries and diagrammatic categories
In the talk, I would like to look on the notion of quantum symmetries from the viewpoint of diagrammatic categories. I will explain the correspondence between special Frobenius algebras and partition categories. The most interesting consequence of this correspondence is that, on one hand, all finite quantum spaces with a fixed categorical dimension are mutually quantum isomorphic and, conversely, this provides a classification of fibre functors on the category of all partitions. The diagrammatic approach also allows an easy proof of the well-known fact that all finite spaces (of size at least four) have quantum symmetries. At the end of the talk, I would like to mention my recent result on quantum symmetries of Hadamard matrices, where I use the same technique to show that all Hadamard matrices (of size at least four) have quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic.

Monday
22.01.2024

Anna Rodriguez Rasmussen, Uppsala University
Uniqueness of regular exact Borel subalgebras

Monday
29.01.2024

Manuel Bodirsky, TU Dresden 
(Abstract and concrete) Minions and applications

31.08.2023

Mateusz Stroinski

Daniel Graves

Anna-Katharina Hirmer

19.06.2023

Eamon Quinlan-Gallego

Ján Pulmann

05.06.2023

08.05.2023

Andrew Baker

17.10.2022

Daniel Tubbenhauer

24.10.2022 

Richard Garner

07.11.2022

Chris Heunen

21.11.2022

Andrea Gagna

George Balla

Tony Zorman

Zbigniew Wojciechowski

Speaker

04.04.2022

11.04.2022

25.04.2022

02.05.2022

09.05.2022

16.05.2022

23.05.2022

30.05.2022

02.06.2022

13.06.2022

20.06.2022

27.06.2022

04.07.2022

11.07.2022

08.08.2022

22.08.2022

Robert Laugwitz

Gregory Ginot

Janez Mrcun

Catherine Meussburger

Ben Ward

Michael Batanin

Muriel Livernet

Pierre-Louis Curien

Ivan Bartulović

Bojana Femic

Natalia Iyudu

Ralph Kaufmann

Fosco Loregian

Ieke Moerdijk

Noemi Combe

Hans-Christian Herbig

Speaker

04.10.2021

11.10.2021

18.10.2021

25.10.2021

01.11.2021

08.11.2021

15.11.2021

22.11.2021

29.11.2021

06.12.2021

13.12.2021

20.12.2021

10.01.2022

24.01.2022

31.01.2022

Martin Markl

Lukas Woike

Sebastian Halbig

Myriam Mahaman

Aryan Ghobadi

Laiachi El-Kaoutit

Jeff Giansiracusa

Jens Kaad

Bruce Bartlett

Chelsea Walton

Michael Cuntz

Shahn Majid

Tony Zorman

Stephanie Feilitzsch (Viva)

Myriam Mahaman

Speaker

19.04.2021

26.04.2021

03.05.2021

10.05.2021

17.05.2021

31.05.2021

07.06.2021

14.06.2021

21.06.2021

28.06.2021

05.07.2021

12.07.2021

26.07.2021

Christian Korff

Cigdem Yirtici

internal presentations

Manuel Martins

Nicolas Garrel

Ingo Runkel

Azat Gainutdinov

Yorck Sommerhäuser

Claudia Scheimbauer

Paolo Saracco

Tom Leinster

Martin Hyland

Tony Zorman

Speaker

26.10.2020

02.11.2020

09.11.2020

16.11.2020

23.11.2020

30.11.2020

07.12.2020

14.12.2020

11.01.2021

18.01.2021

25.01.2021

01.02.2021

08.02.2021

15.02.2021

Atabey Kaygun

Lucia Rotheray

Sebastian Halbig

Matthias Valvekens

Friedrich Wagemann

Stephanie Feilitzsch

David Jordan

Tomasz Brzezinski

Anna Kula

Ilya Shaprio

Yuri Berest

Blessing Oni

John Boiquaye

Ivan Angiono