Daniel Tubbenhauer email

- Thomas Czempik email
- Michael Ehrig email
- Tina Kanstrup email
- Deniz Kus email
- Joanna Meinel email
- Olaf Schnürer email
- Catharina Stroppel email
- Jacinta Torres email
- Daniel Tubbenhauer email
- Arik Wilbert email

- Seminarraum, Max-Planck-Institut für Mathematik (MPIM), Vivatsgasse 7, 53111 Bonn, Germany
- Every Monday from 16:15-18:00
- First meeting: Monday 26.10.2015
- Preliminary meeting: Wednesday 14.10.2015
- There will be no meeting on Monday 16.11.2015
- The talk on Monday the 30.11.2015 is shifted to Tuesday 01.12.2015 (same time, but in Seminar room 0.007, Mathematik-Zentrum, Endenicher Allee 60)
- There will be only a short meeting (no long talk) on Monday 21.11.2015
- The talk on Monday the 11.01.2015 starts at 16:45. There will be only a long talk and no answering of questions
- On Monday the 25.01.2016 we will have two answering of questions and an extra 60 minutes talk by Aleksei Ilin (Moscow), but Michael's talk is moved to Monday the 01.02.2016
- There will be no meeting on Monday 08.02.2016 - instead we will meet Tuesday 09.02.2016 (same time, but in Seminar room 0.007, Mathematik-Zentrum, Endenicher Allee 60)

- Monday 26.10.2015,
**Speaker**: Daniel Tubbenhauer,**Title**: Web calculi in representation theory - I - Monday 02.11.2015,
**Speaker**: Daniel Tubbenhauer,**Title**: Web calculi in representation theory - II - Monday 02.11.2015,
**Speaker**: Arik Wilbert,**Title**: Topology of two-block Springer fibers for the even orthogonal and symplectic group - I - Monday 09.11.2015,
**Speaker**: Arik Wilbert,**Title**: Topology of two-block Springer fibers for the even orthogonal and symplectic group - II - Monday 09.11.2015,
**Speaker**: Thomas Czempik,**Title**: Schur-Weyl duality for partition algebras - I - Monday 23.11.2015,
**Speaker**: Thomas Czempik,**Title**: Schur-Weyl duality for partition algebras - II - Monday 23.11.2015,
**Speaker**: Joanna Meinel,**Title**: Combinatorics of particle configurations - I - Tuesday 01.12.2015,
**Speaker**: Joanna Meinel,**Title**: Combinatorics of particle configurations - II - Tuesday 01.12.2015,
**Speaker**: Tina Kanstrup,**Title**: Braid group actions on matrix factorizations - I - Monday 07.12.2015,
**Speaker**: Tina Kanstrup,**Title**: Braid group actions on matrix factorizations - II - Monday 07.12.2015,
**Speaker**: Deniz Kus,**Title**: Demazure flags and Macdonald polynomials - I - Monday 14.12.2015,
**Speaker**: Deniz Kus,**Title**: Demazure flags and Macdonald polynomials - II - Monday 14.12.2015,
**Speaker**: Jacinta Torres,**Title**: Branching rules and paths - I - Monday 21.12.2015,
**Speaker**: Jacinta Torres,**Title**: Branching rules and paths - II - Monday 11.01.2016,
**Speaker**: Catharina Stroppel,**Title**: Quiver Schur algebras and representations of the general $p$-adic group - I - Monday 25.01.2016,
**Speaker**: Catharina Stroppel,**Title**: Quiver Schur algebras and representations of the general $p$-adic group - II - Monday 18.01.2016,
**Speaker**: Olaf Schnürer,**Title**: Homological smoothness and enriched resolutions - I - Monday 25.01.2016,
**Speaker**: Olaf Schnürer,**Title**: Homological smoothness and enriched resolutions - II - Monday 25.01.2016,
**Speaker**: Aleksei Ilin (Moscow),**Title**: Degeneration of Bethe subalgebras in the Yangian - Monday 01.02.2016,
**Speaker**: Michael Ehrig,**Title**: Foams (Or: Web calculi now in $3D$) - I - Tuesday 09.02.2016,
**Speaker**: Michael Ehrig,**Title**: Foams (Or: Web calculi now in $3D$) - II

Daniel Tubbenhauer,

**Title**: Web calculi in representation theory,

**Abstract**:
A striking question is if one can present the
category $\boldsymbol{\mathrm{Mod}}(\mathfrak{g})$ of finite-dimensional
representations of some Lie algebra
\(\mathfrak{g}\),
via *generators and relations* -- or even better: via
*diagrammatic* generators and relations.

The question itself is very hard and only partial solutions are known:
it goes back to work of Schur and Brauer that the subcategory of
$\boldsymbol{\mathrm{Mod}}(\mathfrak{g})$ tensor
generated by the vector representation can be (almost) described if
\(\mathfrak{g}\) is of classical type and
certain dimensions are "big enough". Moreover,
Rumer, Teller and Weyl
showed (more or less) already in the 30ties that the Temperley-Lieb algebra can
be seen as a diagrammatic realization of
the representation category of \(\mathfrak{sl}_2\)-modules tensor generated
by the vector representation of \(\mathfrak{sl}_2\) --
providing a topological (and fun!) tool to study the latter.

In this talk I explain "howe" one can prove such a
(*diagrammatic*) realization.

Everything in this talk can be quantized
(roughly: put quantum brackets everywhere)
and everything is
amenable to categorification, but I will keep it easy
and explain the \(\mathfrak{sl}_2\)-case in details
and give some applications to knot theory of the
whole calculus.

Arik Wilbert,

**Title**: Topology of two-block Springer fibers
for the even orthogonal and symplectic group,

**Abstract**:
In this talk we construct an explicit topological model (similar to the
topological Springer fibers of type $A$ appearing in work of Khovanov and Russell)
for every two-block Springer fiber corresponding to the even orthogonal group (type $D$)
and explain how to prove that the respective topological model is indeed homeomorphic to
its corresponding Springer fiber (which confirms a conjecture by Ehrig and Stroppel).
Furthermore, we discuss how the two-block Springer fibers for the symplectic group (type $C$)
fit into this picture. If time permits, some applications in the context of Springer theory
(representations of Weyl groups on the (co)homology of (topological) Springer fibers) are provided.
Since the talk is supposed to be as self-contained as possible we do not assume any familiarity with Springer fibers.

Thomas Czempik,

**Title**: Schur-Weyl duality for partition algebras,

**Abstract**:
First we will repeat the classical case of Schur-Weyl duality.
Then we construct the partition algebra, where I will give some nice
visualization (referring to Ceccherini-Silberstein, Scarabotti and Tolli or the bachelor thesis of Heike Herr).
Then we define representations of the symmetrical group and partition
algebra on the tensor product $V^{\otimes k}$ of some complex vector space v.
The talk we will very basic, so everyone can follow.

Joanna Meinel,

**Title**: Combinatorics of particle configurations,

**Abstract**:
We consider bosonic and fermionic particle configurations: which relations are satisfied
by the operators that move the particles from one position to another along a line or a circle?
They come from the action of the negative half of (affine) $\mathfrak{sl}(n)$ on symmetric and alternating
products of the natural representation, giving rise to 'new' algebras such as the (affine) nilTemperley-Lieb
algebra and the partic algebra, for which we construct normal forms and embeddings.

Tina Kanstrup,

**Title**: Braid group actions on matrix factorizations,

**Abstract**:
Let $X$ be a smooth scheme with an action of a reductive algebraic group $G$
over an algebraically closed field $k$ of characteristic zero. We construct
an action of the extended affine Braid group on the $G$-equivariant absolute
derived category of matrix factorizations on the Grothendieck variety times $TX$ with potential given
by the Grothendieck-Springer resolution times the moment map composed with the natural pairing.

Deniz Kus,

**Title**: Demazure flags and Macdonald polynomials,

**Abstract**:
Demazure modules are finite-dimensional representations and
occur in a highest weight integrable representation of an affine
Lie algebra. In this talk, we introduce a family of representations which
are indexed by partitions and show that these representations admit a Demazure
flag, i.e. a flag where the successive quotients are Demazure modules. For special
partitions, this implies a positivity result: the specialized Macdonald polynomial can
be written as a linear combination
of Demazure characters (for any fixed level $m$) with non-negative integer coefficients.

Jacinta Torres,

**Title**: Branching rules and paths,

**Abstract**:
The study of restrictions of representations is basic in representation theory. I will
introduce the path model associated to an irreducible finite-dimensional representation
of a simple Lie algebra (focusing on examples) and I will explain how it can be used to
produce branching rules for Levi subalgebras. Then I will talk about branching rules for
non-Levi subalgebras (in particular, $\mathfrak{sp}_{2n}$ as a sub Lie algebra of
$\mathfrak{sl}_{2n}$), and I will explain
why in the first situation there is compatibility of the crystal structures, while in the
second one there is not
(I will provide examples of this). Then, if time allows, I will present a mysterious conjecture by Naito-Sagaki.

Catharina Stroppel,

**Title**: Quiver Schur algebras and representations of the general $p$-adic group,

**Abstract**:
The representation theory of the general linear group $\mathrm{GL}(n,\mathbb{Q}_p)$ over
the field $\mathbb{Q}_p$ of $p$-adic numbers is rather difficult and involved and part of the
so-called local Langlands program. Motivated by the Schur-Weyl duality over finite
fields, I will define an affine Schur algebra and explain its relation to representations
of $\mathrm{GL}(n,\mathbb{Q}_p)$ and to the affine Hecke algebra. At the end I will connect it with KLR algebras
and quiver Schur algebras, which can be
defined either diagrammatically or geometrically and play an important role in categorification.

Olaf Schnürer,

**Title**: Homological smoothness and enriched resolutions,

**Abstract**:
This talk is meant as an introduction to my talk on Friday. We discuss the
notions of smoothness in algebraic geometry and in dg category theory (non-commutative geometry)
in easy examples. Then we discuss the existence
of additive resolution functors and, possibly, of dg enriched resolution functors.

Aleksei Ilin (Moscow),

**Title**: Degeneration of Bethe subalgebras in the Yangian,

**Abstract**:
We study the family of Bethe subalgebras in Yangian for \(\mathfrak{gl}_n\), parametrized by
regular diagonal matrices. We prove that all limit subalgebras of this family are
free and maximal commutative and explicitly describe “simple” limits.

Michael Ehrig,

**Title**: Foams (Or: Web calculi now in $3D$),

**Abstract**:
In this talk I will explain how one can use certain types of
enhanced cobordisms to construct foam categories, i.e. categories of
cobordisms with singular seams. This is done by gluing together different
types of TQFTs, which I will explain and discuss what kind of freedom one
has in this construction. In all of this \(\mathfrak{gl}_2\)-webs will play a prominent
role.

After the topological construction I will explain how one can translate
this into an algebraic model similar, but more involved, than Khovanov's
original arc algebra. We will then discuss how this construction
generalizes/unifies the work of Khovanov, Bar-Natan, Blanchet and others
to categorify Khovanov homology and obtaining link/tangle invariants. I
will give a rough idea how to show that via this generalized construction
one shows that all of these theories in the end are more or less
equivalent. If time permits we will discuss the relations to category $\mathcal{O}$,
categorification of quantum groups and higher representation theory,
tangle invariants, and/or possible generalizations to \(\mathfrak{gl}_n\)-webs.