# Oberseminar Darstellungstheorie, Sommer Term 2016

Contact:

Catharina Stroppel email

Daniel Tubbenhauer email

## Where and when?

- Seminar room 1.008, Mathematik-Zentrum, Endenicher Allee 60, 53115 Bonn, Germany
- Every second Friday from 14:15-15:15
- Exceptions:
- David Jordan's talk will take place Thursday 28.04.2016 from 13:00-14:00 in the Seminarraum of the MPIM (his talk is canceled due to strike at the airport Cologne/Bonn)
- There will be two talks on Friday the 10.06.2016. First, Beren Sanders from 12:15-13:15, then Marco Mackaay from 14:15-15:15. The room is seminar room 1.008

- First talk: Friday 15.04.2016

## Schedule

- Friday 15.04.2016,
**Speaker**: Paul Wedrich (Imperial College London),**Title**: Uniqueness and branching of knot homologies - Thursday 28.04.2016,
**Speaker**: David Jordan (University of Edinburgh) - canceled,**Title**: Difference operators and double affine Hecke algebras and from topological field theory - Friday 13.05.2016,
**Speaker**: Michael Brown (University of Bonn),**Title**: Topological K-theory of dg categories of graded matrix factorizations - Friday 27.05.2016,
**Speaker**: Travis Schedler (The University of Texas at Austin),**Title**: Special polynomials from symplectic singularities - Friday 03.06.2016,
**Speaker**: Gus Lehrer (The University of Sydney),**Title**: Invariants of the orthosymplectic super group - Friday 10.06.2016,
**Speaker**: Beren Sanders (University of Copenhagen),**Title**: Grothendieck-Neeman duality and the Wirthmüller isomorphism - Friday 10.06.2016,
**Speaker**: Marco Mackaay (Universidade do Algarve),**Title**: Simple transitive $2$-representations of Soergel bimodules - Friday 17.06.2016,
**Speaker**: Andrew Mathas (The University of Sydney),**Title**: Jantzen filtrations and graded Specht modules

## Abstracts

Paul Wedrich,

**Title**: Uniqueness and branching of knot homologies,

**Abstract**:
Knot homology theories are categorified versions of classical (and quantum) knot polynomials that are
(conjecturally/almost) functorial under knot cobordisms. The goal of this talk is to present
two examples of the role of higher representation theory in the study of categorifications of the
Reshetikhin-Turaev invariants of knots colored with $\mathfrak{sl}(N)$ representations.
As a first example, I will explain how categorical versions of skew Howe duality have been used to
prove that several, superficially very different constructions of $\mathfrak{sl}(N)$ knot homologies (via matrix
factorizations, category $\mathcal{O}$, coherent sheaves...) produce isomorphic invariants. The second example
concerns categorified branching rules, which provide relationships between these knot homologies in the
form of spectral sequences, which are also interesting from a topological perspective. If time permits,
I will talk about some open problems related to colored HOMFLY-PT homologies, whose underlying higher
representation theory is less well understood.

David Jordan,

**Title**: Difference operators and double affine Hecke algebras and from topological field theory ,

**Abstract**:
Cherednik introduced his double affine Hecke algebras (DAHAs) in the
90's, to solve some conjectures of Macdonald in the realm of algebraic
combinatorics. Since then, they have popped up in remarkably diverse branches
of representation theory and mathematical physics. Recently, works of many authors
have been devoted to understanding their role in various ``field theoretic" constructions,
i.e. to constructing invariants in low-dimensional topology.

In this talk, I'll survey some of these recent works, and I'll report
on some joint work with David Ben-Zvi and Adrien Brochier, which constructs
the representation theory of the DAHA in Type A in an intrinsically topological
fashion: namely, as the factorization homology of a marked torus. This leads
us more generally to the construction of a $4$-dimensional ``big sister" to the $3$-dimensional
Reshetikhin-Turaev topological field theory, in which the DAHA is expected to play a central role.

Michael Brown,

**Title**: Topological K-theory of dg categories of graded matrix factorizations,

**Abstract**:
This is a report on joint work with Tobias Dyckerhoff. We consider localizing
invariants of a certain dg category, the singularity category, associated to a
graded Gorenstein algebra A over a field k. If A is the coordinate ring of a weighted
projective hypersurface, the singularity category of A is quasi-equivalent to the dg category
of graded matrix factorizations associated to the hypersurface. When k is the complex numbers,
one such localizing invariant is topological K-theory, as constructed by A. Blanc. The main goal
of the talk will be to discuss a calculation of the topological K-theory of the singularity categories
of a large class of weighted projective hypersurfaces in terms of a classical topological invariant of a
hypersurface: the Milnor fiber and its monodromy.
I will also discuss some applications of this result, and, if time permits, some future directions.

Travis Schedler,

**Title**: Special polynomials from symplectic singularities,

**Abstract**:
I will explain how to obtain Kostka and Tutte polynomials from the nilpotent cone and
hypertoric varieties. The main tool (which I will recall) is a $D$-module on the variety expressing
invariance under Hamiltonian flow. This includes work with Bellamy, Etingof, and Proudfoot.

Gus Lehrer,

**Title**: Invariants of the orthosymplectic super group,

**Abstract**:
A combination of geometric and diagram theoretic methods have led to recent refinements of the
classical versions of the second fundamental theorem of invariant theory, and a new theorem in the
orthosymplectic case. I shall describe these results, as well as the circle of
ideas leading to them. This is joint work with Ruibin Zhang, and partly with Pierre Deligne.

Beren Sanders,

**Title**: Grothendieck-Neeman duality and the Wirthmüller isomorphism,

**Abstract**:
(Joint work with Paul Balmer and Ivo Dell'Ambrogio.)
In this talk, I will discuss an intimate relationship between Grothendieck
duality in algebraic geometry and the Wirthmüller isomorphism in equivariant
stable homotopy theory. To this end, we will make a general study of the
existence and properties of adjoints of an arbitrary coproduct-preserving
tensor-triangulated functor between rigidly-compactly generated tensor
triangulated categories. It turns out that the more adjoints exist, the more
strongly related they must be to each other, and the result is a surprising
trichotomy: There exist either exactly three adjoints, exactly five or
infinitely many. Moreover, this analysis will provide us with purely formal,
canonical constructions of Wirthmüller isomorphisms (when they exist) and
demonstrate that Grothendieck duality is in fact a necessary condition for the
existence of such an isomorphism. If time permits, I will mention some more recent
developments which show that the Adams isomorphism can also be constructed in
this way as a (suitably generalized) Wirthmüller isomorphism.

Marco Mackaay,

**Title**: Simple transitive $2$-representations of Soergel bimodules,

**Abstract**:
I will first recall Soergel bimodules, which categorify Hecke algebras, and
some bits of the theory of $2$-representations, which categorify the usual representations.

More specifically, I will explain cell $2$-representations and simple
transitive $2$-representations, both due to Mazorchuk and Miemietz. By definition,
every cell $2$-representation is simple transitive. Mazorchuk and Miemietz showed
that the converse also holds in finite type $A$.

However, this is not true in general, as Kildetoft, Mazorchuk,
Zimmermann and I showed in a joint paper (arXiv:1605.01373). In that paper,
we classified all transitive $2$-representations (up to equivalence) for the so-called
small quotient of the Soergel bimodules
in almost all finite Coxeter types. I will sketch our results in my talk.

Andrew Mathas,

**Title**: Jantzen filtrations and graded Specht modules,

**Abstract**:
I will explain how to give an easy proof of the Jantzen sum formula for
the Specht modules of the cyclotomic Hecke algebras of type $A$ using the KLR grading.
Then I will discuss some consequences and applications of this approach.

## Aktuelles

21./22.11.: Eröffnungsveranstaltung GlobalMathNetwork

Hausdorff-Kolloquium im WS 2017/18

Toeplitz Kolloquium zur "Didaktik und Geschichte der Mathematik" im WS 2017/18

Berufspraktisches Kolloquium im WS 2017/18

Prof. Daniel Huybrechts wurde zum Mitglied der Academia Europaea gewählt.

Bonner Mathematik im Shanghai-Ranking auf Platz 32 und bundesweit führend

Prof. Peter Scholze ist plenary speaker auf dem ICM 2018

Prof. Dr. Gerd Faltings erhält Cantor-Medaille der DMV

Prof. Peter Scholze in die Akademie der Wissenschaften und der Literatur aufgenommen

Prof. Peter Scholze erhält einen der zehn EMS-Preise

Humboldt-Forschungspreis für Daniel Tataru

Prof. Peter Scholze erhält den Gottfried Wilhelm Leibniz-Preis 2016