Seminar Geometric Methods in Mathematics

Link: https://bbb.tu-dresden.de/b/ulr-ha4-xul-uoo

This is the website of the research seminar of Uli (Prof. Dr. Ulrich Krähmer) and his group. The talks usually cover topics such as homological algebra and category theory, noncommutative algebra and geometry, or Hopf algebras and their generalisations. The target audience consists of PhD and MSc students who write their thesis in one of these areas. Of course everyone is welcome, please contact us to be added to the email list if you want to join.

There will be a blend of external and internal speakers giving formal and informal talks, or we just read and discuss as a group, sometimes focussing for a few weeks on one topic. The only rule is that we meet each week in term time.

Unless otherwise indicated, the seminar takes place every Monday at 3:00 PM (CEST/CET). It is streamed on BigBlueButton: https://bbb.tu-dresden.de/b/ulr-ha4-xul-uoo. We will try to stream even the offline talks, so that people outside of Dresden will not miss out on anything.
In person talks will most likely happen in the room Z21/250.

Please register for the Mailing-List via: https://mailman.zih.tu-dresden.de/groups/listinfo/math-geometric-method

The speakers for the summer term 2023 are:

Thursday 31.08.2023	Josefin Bernhard, TU Dresden (Master's thesis defence) Hopf algebra actions on coordinate rings
Monday 10.07.2023	Mateusz Stroinski, Uppsala University Module categories via Tambara modules: reconstruction and enrichment Let C be a closed monoidal category and let PC be the category of C-presheaves. The following two results are very useful in the study of a module category M over C: ("Tannaka") Reconstruction: M is equivalent to the category of module objects over an algebra object in PC; Action via enrichment: to M, one can associate a category S(M) enriched over PC, and viceversa. In my recent work, I show that one can remove the assumption that C is closed by replacing PC with TC, the category of Tambara modules over C. Tambara modules can be thought of as profunctors of module categories; they were first introduced by Tambara in 2006, but have been studied in greater detail only in the last couple of years. In this talk, I will give a more detailed account of the above results, including the definitions and constructions involved. To make the above more digestible and visual, and less category-intense, I will use group actions, monoid actions and their action graphs as an illustrating special case; the above can be thought of as a "categorification" thereof. This talk is based on https://arxiv.org/abs/2210.13443
Monday 03.07.2023	Daniel Graves, University of Leeds Categorifying equivariant monoids PROPs are "product and permutation categories". They encode structure borne by objects in a symmetric monoidal category. In this talk I will discuss how the PROP that indexes the structure of a monoid in a symmetric monoidal category is closely related to the theory of crossed simplicial groups. I will then report on recent work which generalizes this in two ways. I'll talk about how we can obtain PROPs encoding the structure of monoids with a group action. I will also talk about how we can obtain analogous indexing categories in the setting of braided monoidal categories.
Monday 26.06.2023	Anna-Katharina Hirmer, Friedrich-Alexander-Universität Erlangen-Nürnberg Generalised Kitaev models from Hopf monoids: topological invariance and examples Quantum double models were introduced by Kitaev to obtain a realistic model for a topological quantum computer. They are based on a directed ribbon graph and a finite-dimensional semisimple Hopf algebra. The ground state of these models is a topological invariant of a surface, i.e. only depends on the homeomorphism class of the oriented surface but not the ribbon graph. Meusburger and Voß generalised part of the construction from Hopf algebras to pivotal Hopf monoids in symmetric monoidal categories. We explain the construction of the ground state for involutive Hopf monoids and show that it is topological invariant. We explicitly describe this construction for Hopf monoids in Set, Top, Cat and SSet.
Monday 19.06.2023 ON ZOOM Passcode: 177540	Eamon Quinlan-Gallego, University of Utah Finiteness properties of local cohomology If R is a commutative noetherian ring and J is an ideal of R, the local cohomology modules of R with support in J tend to have surprising finiteness properties despite being not finitely generated. In this talk I will survey some results in this direction and, if time permits, I will talk about some recent work.
Monday 12.06.2023 online	Ján Pulmann, University of Edinburgh Connected sum and Beilinson-Drinfeld algebras Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. We show how this structure gives rise to a commutative product on the space of functions associated to the modular operad. Together with the usual non-commutative BV structure present on this space of functions, the commutative product forms a Beilinson-Drinfeld algebra, a close cousin of a BV algebra. Based on a join work with Martin Doubek, Branislav Jurco and Lada Peksova.
Monday 05.06.2023	Nima Rasekh, Max Planck Institute for Mathematics Bonn Formalizing (Double-)Categories In recent years mathematical formalization has experienced a meteoric growth with applications ranging from computing stable homotopy groups of spheres to condensed mathematics. In this talk we will focus on the formalization of the concept of a category and in particular surprising mathematical challenges that one encounters in this setting. If time permits I will also discuss ongoing joint work with Benedikt Ahrens, Paige North and Niels van der Weide aimed at generalizing this formalization to double categories. No knowledge of type theory, mathematical formalization or proof assistants will be assumed for this talk and mathematicians of all stripes are welcome to attend!
Monday 15.05.2023 online	Estanislao Herscovich, Université Grenoble Alpes Double quasi-Poisson algebras are pre-Calabi–Yau
Monday 08.05.2023 20:15 online	Ben Elias, University of Oregon Categorial Diagonalization
Monday 24.04.2023 online	Andrew Baker, University of Glasgow Locally Frobenius Hopf algebras Nearly Frobenius algebras were introduced by Moore and Peterson and later renamed P-algebras by Margolis; they are graded Hopf algebras which are unions of finite dimensional subHopf algebras. The main examples were related to the Steenrod algebra for a prime. I will outline a non-graded analogue of the theory which accomplishes most of the same goals. An important class of examples is provided by group algebras of locally finite groups. Locally Frobenius Hopf algebras are coherent but not Noetherian and the best outcomes are for coherent modules. Finite dimensional modules are never coherent so they are at the opposite extreme. I will outline some ideas on cohomological aspects and stable module categories for these Hopf algebras which may lead to future work.
Monday 17.04.2023	Friedrich Wagemann, University of Nantes Crossed modules of algebras over an operad This is joint work with Salim Rivière and Johan Leray. Equivalence classes of crossed modules of Lie algebras or associative algebras are well-known to be in bijection with cohomology classes in degree 3 for Chevalley-Eilenberg resp. Hochschild cohomology. We generalize this correspondence to algebras over an operad (with a fixed resolution, for example, a Koszul operad). We show that the Homotopy Transfer Theorem can be interpreted as associating to a crossed module a cocycle, and the Rectification Theorem as associating to a cocycle a crossed module.

The speakers for the winter term 2022/23 were:

Monday 17.10.2022 10:00 online	Daniel Tubbenhauer, University of Sydney Representations of monoidal categories There is a MathOverflow question with title "Why aren't representations of monoids studied so much?". Well, I do not know, but it can't be because the theory is bad. In fact, there is a well-developed theory that is about 80 years old, and this theory is almost as smooth and beautiful as group representation theory itself. There are even categorifications of it: What categorifies monoids? For example monoidal categories. And indeed the representation theory of monoids categorifies to representation theory of monoidal categories. This talk is a friendly overview of the representation theory of monoidal categories with no prior knowledge about this theory assumed.
Monday 24.10.2022 09:00 online	Richard Garner, Macquarie University Cofree cocommutative coalgebras and abstract differentiation When k is an algebraically closed field of characteristic zero, there is a simple formula for the cofree cocommutative coalgebra on a given k-vector space. When one moves beyond this case, it is well known that describing the cofree cocommutative coalgebra gets considerably harder. What is perhaps less well known is that the formula which works in the well-behaved case, still makes sense in general, and still describes something with a universal property. This something is related to cartesian differential categories, which provide a category-theoretic formalisation of the key properties of the category of smooth maps between Euclidean spaces. The goal of this talk is to explain some of these interrelated concepts. We assume nothing more than a basic knowledge of coalgebras and category theory. This is joint work with JS Lemay
Monday 07.11.2022	John Bourke, Masaryk University A skew approach to enrichment for Gray-categories
Monday 14.11.2022	Julian Holstein, Universität Hamburg Koszul duality and dg categories as coalgebras
Tuesday 15.11.2022 online	Chris Heunen, University of Edinburgh Sheaf representation of monoidal categories Wouldn't it be great if monoidal categories were nice and easy? They are! We will discuss how a monoidal category embeds into a "nice" one, and how a "nice" monoidal category consists of global sections of a sheaf of "easy" monoidal categories. This theorem cleanly separates "spatial" and "temporal" directions of monoidal categories. More precisely, "nice" means that certain morphisms into the tensor unit have joins that are respected by tensor products, namely those morphisms that are central and idempotent with respect to the tensor product. By "easy" we mean that the topological space of which these central idempotents form the opens is a lot like a singleton space, namely local. Categorifying flabby sheaves in the appropriate way makes the representation functorial from monoidal categories to schemes of local monoidal categories. The representation respects many properties of monoidal categories: if a monoidal category is closed/traced/complete/Boolean, then so are its local stalks. As instances we recover the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, the representation by germs of open subsets of a topological space, and the Takahashi representation of Hilbert modules as continuous fields of Hilbert spaces
Monday 21.11.2022 online	Misha Feigin, University of Glasgow On Dunkl angular momenta algebra
Monday 28.11.2022	Joanna Meinel, Universität Bonn How to find highest weight vectors in tensor powers
Monday 05.12.2022	Andrea Gagna, The Czech Academy of Sciences A rough overview on the Geometric Langlands correspondence The geometric Langlands correspondence can be seen at the same time as: The function field analogue of the arithmetic Langlands correspondence. A vast generalization of the Riemann–Hilbert correspondence. A (geometric) extension of the Fourier–Pontryagin duality. In this talk I will give a rather naive overview of the statement of the Geometric Langlands correspondence, as presented by Arinkin and Gaitsgory, touching on all the three points above to help give a solid intuition of the problem at hand.
Monday 12.12.2022 online	Vanessa Miemietz, University of East Anglia Hopf algebras and symmetric bimodules
Monday 19.12.2022	George Balla, RWTH Aachen Symplectic PBW degenerate flag varieties
Monday 16.01.2023	Tony Zorman, TU Dresden Duality in Monoidal Categories There are at least three categorical gadgets that capture various notions of mathematical duality: closed monoidal, -autonomous, and rigid monoidal categories. These concepts are all interlinked—rigid monoidal categories are always -autonomous, and they in turn are necessarily closed monoidal. In fact, the resulting internal-hom is very well-behaved. We will explore connections in the other direction: does the shape of the internal-hom of a closed monoidal category already characterise rigidity or *-autonomy? This talk is based on joint work with Sebastian Halbig.
Monday 23.01.2023	Zbigniew Wojciechowski, TU Dresden Tensor product decompositions over a coideal subalgebra of U_q(gl2) and Type D Jones-Wenzl Projectors A classical problem is the decomposition of tensor powers of the natural representation k^n viewed as module over GL_n or its lie algebra counterpart gl_n. Answering this problem gives rise to beautiful diagrammatics and combinatorics via Schur-Weyl duality of S_n and GL_n, which can be quantized and categorified. In the talk I will discuss a non-trivial analogue of the story, where we replace S_n by the type B/D Weyl group (resp. their Hecke algebras) and GL_n by a certain coideal subalgebra of U_q(gl_2n). Moreover we will discuss idempotents in a Temperley-Lieb quotient of the Hecke algebra and an approximation theorem for those via the type D full twist.
Monday 30.01.2023	Ulrich Krähmer, TU Dresden Incidence Hopf Algebras

The speakers for the summer term 2022 were:

Date

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

The speakers for the winter term 2021/22 were:

Date

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

The speakers for the summer term 2021 were:

Date

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

The speakers for the winter term 2020/21 were:

Date

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

Seminar Geometric Methods in Mathematics

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