# Trimester Seminar

## Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall

## Tuesday, October, 24^{th}, 10:00

**Noncommutative crepant resolutions of quotient singularities for reductive groups **

### Michel Van den Bergh (Universiteit Hasselt)

## Blackboard photos

## Wednesday, October, 25th, 10:00

**Arc algebras**

### Michael Ehrig (University of Sydney)

** Abstract**

Originally defined by Khovanov to categorify invariants of tangles, arc algebras have been generalized and modified in different ways. The talk will start with a basic introduction into what arc algebras are, followed by an overview on how they are related to different topics. This will range from Springer fibres and perverse sheaves on Grassmannians on the geometry side, to webs and foams on the side of link and tangle invariant, to the representation theory of Lie algebras and superalgebras, and to the categorification of quantum group representations.

## Slides

## Blackboard photos

## Thursday, October, 26^{th}, 10:00

**Convolution algebras arising from Springer fibers**

### Arik Wilbert (University of Bonn)

** Abstract**

We will explain how to use the irreducible components of Springer fibers to construct interesting convolution algebras. More precisely, the goal is to understand a geometric construction of the arc algebra* obtained by Stroppel-Webster. Along the way we will highlight some intriguing connections between the geometry of Springer fibers, cup diagram combinatorics, 2d TQFTs and Fukaya categories.

*Depending on prior knowledge it might be helpful to attend Michael’s talk to fully appreciate the contents of this talk.

## Blackboard photos

## Monday, October, 30^{th}, 14:45

**A universal source of braid group actions**

### David Jordan (University of Edinburgh)

** Abstract**

One of the reasons we love quantum groups is that they are the best-understood examples of braided tensor categories, and these are a systematic way to build representations of braid groups, which can then be studied using Lie theory (and its deformations). We can then ask, what about braid groups of other surfaces, is there a similar Lie theoretic construction of representations of these? In this talk I'll explain a positive answer to this question, and we'll consider lots of examples. It will turn out that the categories we find in this way are connected in a surprising way to important examples in (classical) geometric representation theory, like Harish-Chandra D-modules, adjoint orbits, character sheaves, etc. I'll discuss a bit of this as well.

## Blackboard Photos

## Thursday, November, 2^{nd}, 14:45

**Representation Theory of Tensor Categories**

### Tim Weelinck (University of Edinburgh)

** Abstract**

Tensor categories are categories with a multiplication structure and in this sense categorify algebras. There is a natural notion of module over a tensor category and so the fundamental question looms: 'can we classify the modules of our favourite tensor category?'. As in the case of algebras, this is very difficult in general. However, if the algebra happens to be a Frobenius algebra we have powerful tools available to study its representation theory. We will recall those tools and then explain that rigid tensor categories (where objects have duals) are categorified Frobenius algebras. This analogy proves very fruitful and provides us with powerful tools to study the representation theory of rigid tensor categories. We conclude by giving a detailed example of a module over the tensor category of representations of a quantum group, and explain how the reflection equation algebra arises naturally there.

## Blackboard photos

## Monday, November, 6^{th}, 14:45

**From A-infinity algebras to bocses via Koszul duality**

### Julian Külshammer (Universität Stuttgart)

** Abstract**

Exceptional collections frequently appear in algebraic and symplectic geometry as well as in representation theory. In representation theory, algebras having a full exceptional collection in their module category are called quasi-hereditary. In this talk, I will explain how to use a variant of Koszul duality for A-infinity algebras to construct a bocs (= a coring in a category of bimodules over an algebra) given an exceptional collection. On this level, also Ringel duality can be seen as a special case of Koszul duality. If time permits, we will show in an example, how the language of bocses can help to understand the category of modules filtered by the exceptional collection. This reports on joint work with Steffen Koenig and Sergiy Ovsienko as well as Agnieszka Bodzenta.

## Blackboard photos

## Tuesday, November, 14^{th}, 10:00

**On symmetric spaces for Kac-Moody groups**

### Guido Pezzini (La Sapienza University in Rome)

** Abstract**

In the talk I will report on a research project aimed at studying symmetric spaces for Kac-Moody groups, joint with Bart Van Steirteghem. Our work is motivated by the relevant connections that symmetric spaces, and more in general spherical varieties, have with other areas of mathematics such as representation theory and symplectic geometry.

We expect that similar interactions will be possible in the infinite-dimensional setting of Kac-Moody groups too, in particular with the representation theory of such groups and also with multiplicity free symplectic (Fréchet) manifolds under the action of loop groups.

Our goals include defining a structure of infinite-dimensional algebraic variety (ind-variety) on such symmetric spaces, studying functions defined by matrix coefficients, and define compactifications. We will also highlight some peculiarities that arise with Kac-Moody groups of affine type, and that show quite a different behavior from the classical finite-dimensional case.

## Blackboard photos

## Wednesday, November, 15^{th}, 10:00

**On equations defining the affine Grassmannian of SL**_{n}

_{n}

### Oded Yacobi (University of Sydney)

**Abstract**

The affine Grassmannian Gr of a semisimple Lie group G is an important infinite dimensional variety that appears in geometric representation theory. This talk concerns the projective geometry of Gr when G=SL_{n}. More precisely, in this case Gr naturally embeds into the Sato Grassmannian, which is a limit of finite dimensional Grassmannians Gr(n,2n) as n → ∞. We are interested in the equations defining the embedding Gr ⊂ SGr.

Kreiman, Lakshmibai, Magyar and Weyman constructed linear equations on SGr which vanish on Gr and conjectured that these equations suffice to cut out the affine Grassmannian. We recently proved this conjecture by reducing it to a question about finite dimensional Grassmannians. I'll describe our method of proof and mention some conjectures that arise from our work. I'll motivate this discussion by relating our work to the problem of describing the equations of an interesting class of singular varieties: the nilpotent orbit closures in positive characteristic.

This is joint work with Dinakar Muthiah and Alex Weekes.

## Blackboard photos

## Friday, November, 17^{th}, 16:15

**Perverse sheaves and knot contact homology **

### Wai-kit Yeung (Indiana University)

**Abstract**

In this talk, I will present a universal construction, called homotopy braid closure, that produces invariants of links in R^{3} starting with a braid group action on objects of a (model) category. Applying this construction to the natural action of the braid group B_{n} on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that extends knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R^{3} with singularities at most along the link.

This is joint work with Yu. Berest and A. Eshmatov.

## Slides

## Tuesday, December, 12^{th}, 14:45

**Mukai flops and P-twists via non-commutative crepant resolutionsy**

### Wahei Hara (Waseda University)

**Abstract**

In this talk I would like to discuss three derived equivalences, mutation equivalence for NCCRs of minimal nilpotent orbit closure of type A, Kawamata-Namikawa equivalence for Mukai flop and P-twists on the cotangent bundle of a projective space. We will observe correspondences among these equivalences, which can be regarded as a higher-dimensional analog of results of Donovan and Wemyss.