Seminar Geometric Methods in Mathematics

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
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.

Réamonn Ó Buachalla, Charles University in Prague
Nested Pairs of Quantum Homogeneous Spaces
Abstract: In recent work Brzezinski and Szymanski formulated a q-deformation of the classical $2$-sphere fibration of the full flag manifold of $SU_3$ over the complex projective plane. Explicitly, they described the full quantum flag manifold $\mathcal{O}_q(\mathrm{F}_3)$ as the total space of a quantum fibration over the quantum projective plane, with a Podle\'s sphere fibre $\OO_q(S^2)$. The authors put this non-principal quantum fibration forward as a motivating example for a proposed theory of noncommutative fibrations with quantum homogeneous fibers. In this talk I will describe a simple but effective framework for producing more examples of noncommutative fibrations with quantum homogeneous fibers. Following Brzezinski and Szymanski, variations on Takeuchi's equivalence and Schneider's descent theorem will be established. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples, and a Dynkin diagram recipe for constructing quantum fibrations is given.

Hans-Christian Herbig, Universidade Federal do Rio de Janeiro
Symplectic forms, connections and curvature in the presence of
singularities
Abstract: We construct symplectic forms, Levi-Civita connection and Riemannian curvature on the double cone (seen as an affine variety) and on the single cone (seen as a differential space in the sense of Sikorski).
Levi-Civita connection and Riemannian curvature are constructed as well for the real orbit space of the Coxeter group A2. The principles employed permit to do this for finite quotient singularities. Collaboration with William Osnayder Clavijo Esquivel, Christopher Seaton.

On the first relative Hochschild cohomology and contracted fundamental group

Monday 18.11.2024

Jonas Nehme, Universität Bonn
Understanding representations of Lie superalgebras via Schur—Weyl duality
Abstract: I will explain how one can use Schur—Weyl duality for Lie superalgebras to obtain an explicit descriptions of the endomorphism ring of a projective generator.
I will demonstrate this approach for the periplectic Lie superalgebra, although the methods can be applied to any simple Lie superalgebra.

Friday, 13.12.2024

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.

Zbigniew Wojciechowski, TU Dresden
Diagrammatics for representations of the smallest quantum symmetric pair
Representations are good, but diagrams are better. I explain how to do diagrammatics for type I and II representations of Uq(sl2) and how to extend that diagrammatics to module category diagrammatics for the coideal subalgebra Uq'(gl1), which forms with Uq(sl2) the smallest quantum symmetric pair. If time allows I explain some combinatorial applications of our diagrammatics, involving counting dimensions of Hom spaces.

Boris Shoikhet, Euler International Mathematical Institute
A generalisation of the Davydov-Yetter complex and deformations of monoidal categories
We introduce a new complex associated with the following data. Let k be a field, and let C,D be k-linear (dg) bicategories (k a field), F,G: C-->D be strong functors, \eta,\theta: F-->G be strong natural transformations. Then our complex A(C,D)(F,G)(\eta,\theta) can be thought of as the complex of derived modifications (that is, it is a derived version of classical 3-morphisms in the 3-category Bicat, known as modifications). (It is the n=2 version of the well-known complex associated with 1-dg-categories C,D, and dg functors F,G: C-->D, equal to the Hochschild cochain complex Hoch(C, M_{F,G}), where the C-bimodule M_{F,G}=D(F(-),G(=)), which plays the role od derived natural transformations). This complex A(C,D)(F,G)(\eta,\theta) is a totalisation of a functor \Theta_2-->(complexes). For the case of monoidal category C (a bicategory with 1 object), it is shown that H^3 of our complex A(C,C) for F=G=id, \eta=\theta=id is equal to the classes of infinitesimal deformations of C. Some higher structures on these complexes are constructed. It is expected that there is a e_3-algebra structure on A(C,C) for any k-linear bicategory C. (Joint work with P. Panero). 

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

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.

Liao Wang, University of Bonn
Annular webs and K-matrices
We define a diagrammatic category which captures the structure of a braided module category generated by a braided vector space and a solution of the reflection equation or boundary Yang-Baxter equation. Our diagrammatic category often behaves like a monoidal category, which is explained via a realization as annular webs. As an example, this category admits a functor to the category of representations of various quantum symmetric pair coideal subalgebras. Quantum symmetric pairs are introduced in the 1990s by Gail Letzter as generalizations of Drinfeld-Jimbo quantum groups. Balagovic-Kolb constructed a universal K-matrices which act on representations of the coideal subalgebra and solves the reflection equation. We explicitly compute the K-matrices associated to type AIII quantum symmetric pairs and extend the theory to type II representations of quantum gl Γ. We expect the annular web category controls this representation category, which we will explore in future works. 

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.

Monday
24.06.2024, 
5pm
online

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
01.07.2024
online

Sean Sanford, Ohio State University
Invertible Fusion Categories
A tensor category $\mathcal C$ over a field $\mathbb K$ is said to be invertible if there is a tensor category $\mathcal D$ such that $\mathcal C\boxtimes\mathcal D$ is Morita equivalent to $\mathrm{Vec}_{\mathbb K}$. When $\mathbb K$ is algebraically closed, it is well-known that the only invertible fusion category is $\mathrm{Vec}_{\mathbb K}$, and any invertible multi-fusion category is Morita equivalent to $\mathrm{Vec}_{\mathbb K}$.  In contrast, we show that for general $\mathbb K$, the invertible multi-fusion categories over $\mathbb K$ are classified (up to Morita equivalence) by $H^3(\mathbb K;\mathbb G_m)$, the third Galois cohomology of the absolute Galois group of $\mathbb K$.  This group of invertible fusion categories is a higher-dimensional analogue of the Brauer group of (which is isomorphic to $H^2(\mathbb K;\mathbb G_m)$).  As an application, we show that the Morita classification of fusion categories by their Drinfel’d centers breaks when $H^3(\mathbb K;\mathbb G_m)$ is nontrivial.

Monday 
08.07.2024

Mateusz Stroinski, Uppsala University
Lax module functors, bialgebras and tensor categories
In this talk, I will explain how one can extend the reconstruction theory for module categories over tensor categories to the non-rigid case, with particular focus on categories associated to (non-Hopf!) bialgebras. The crucial tool towards that end is lax module functors, which I will describe explicitly in the bialgebraic case. I will then show the additional applications of this formalism in the rigid setting of tensor categories, extending the Etingof–Ostrik reconstruction theory and answering a conjecture of Etingof–Ostrik, regarding a generalization of Skryabin's theorem on projectivity for comodule algebras. Time permitting, I will show how the Hausser–Nill theorem and the Bruguières–Lack–Virelizier generalization of Sweedler's theorem on Hopf modules can be reinterpreted, reproven and, in the first case, also generalized, using our methods. These results are joint work in progress with Tony Zorman.

Julius Benner, TU Dresden (master's viva)
Lie–Rinehart algebras and the Poincaré–Birkhoff–Witt theorem

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.

Wednesday
18.10.2023

John Boiquaye, University of Ghana
PhD Defense

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
08.01.2024
online

Adriana Balan, University Politehnica of Bucharest
On 2-categorical aspects of (quasi)bialgebras
Over the past decades, a plethora of results have been obtained concerning the structure of various categories of (co)algebras, bialgebras and Hopf algebras. These include among others, completeness and cocompleteness, local presentability, whether the respective category is abelian, monoidal, Tannaka reconstruction theory, and so on.
In this talk we shall bring forward some 2-categorical aspects of the above-mentioned structures, starting from Day and Street’s observation that (dual) quasi-bialgebras are the pseudomonoids in the monoidal 2-category of coalgebras, and (dual) quasi-Hopf algebras are instances of autonomous pseudomonoids in the monoidal bicategory of coalgebras and bicomodules.

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