# Conference

The GQT colloquium takes place on the final two days, Nov 29 and 30, and consists of nine research talks. These are attended by Ph.D. students, members of the GQT staff, and visitors from abroad. We aim at organizing research talks which feature topics covered during the school, though some of the talks are unrelated.

The program is as follows:

**Thursday 29 Nov**

9:00-10:00: arrival

10:00-11:00: Bohmann

11:30 -12:30: Heuts

lunch

14:00-15:00: Tam

15:30-16:30: Hausmann

18:30: dinner

**Friday 30 Nov**

9:00-10:00: Sibilla

10:15 – 11:15: Goncalves de Martino * *

11:30 – 12:30: Nadimpalli

lunch

14:00-15:00: Sauvaget

15:30 – 16:30: Wong

**Abstracts**

Anna Marie Bohmann (Vanderbilt)

*Loop spaces, coTHH, and a new spectral sequence*

Given a space with its diagonal map, we construct a topological invariant called “topological coHochschild homology.” This is in fact a special case of an invariant for a coalgebral. It generalizes a classical algebraic invariant of coalgebras due to Doi. We show this invariant is related to algebraic K- theory and Waldhausen’s A-theory. In the case of a space with the diagonal map, we obtain the free loop space, which indicates connections to string topology as well. Additionally, we build a new spectral sequence to calculate topological coHochschild homology, which allows for new computational techniques. This it’s joint work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen.

Markus Hausmann (Copenhagen)

*Commuting Matrices and Atiyah’s Real K-theory*

Let G be a Lie group and n a natural number. The space C_n(G) of n commuting elements of G is a classical object of study, with connections to mathematical physics and interesting homotopy-theoretic properties. This talk will concern the spaces of commuting elements in the stable unitary group U and the stable orthogonal group O. I will explain how the homotopy groups of C_n(U) can be computed via stable homotopy theory and those of C_n(O) via equivariant stable homotopy theory, the main tool being Atiyah’s Real K-theory spectrum. This is joint work with Simon Gritschacher.

Marcelo Goncalves de Martino (Oxford)

*On the Dunkl version of a Howe duality*

In this talk, I will report on a joint work (in progress) with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe duality (O(r),spo(2|2)). Similar deformed versions of Howe dualities, specially in the (Pin(r),spo(2|1))-case, were discussed before in, for example, the works of Ben-Said, De Bie, Oste, Orsted, Soucek, Somberg and Van der Jeugt. Our work builds on the notion of a Dirac operator for Drinfeld algebras recently introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras (and the work of Vogan, Huang and Pandzic), as well as the work of Cheng and Wang on classical Howe dualities.

Gijs Heuts (Utrecht)

*Lie algebras in stable and unstable homotopy theory*

I will survey a homotopical version of the theory of Lie algebras, which has only recently begun to be developed. The underlying objects of these Lie algebras, rather than being vector spaces or modules, belong to the world of stable homotopy theory. One of the uses of these Lie algebras is in describing the homotopy theory of spaces. Classically, differential graded Lie algebras are used as models for spaces in rational homotopy theory. These new homotopical Lie algebras play a similar role, not in rational homotopy theory, but in localizations of homotopy theory at other cohomology theories (e.g. K-theory).

Santosh Nadimpalli (Nijmegen)

*Whittaker models of cuspidal representations of p-adic groups*

In this talk we explain about Whittaker models of representations of p-adic groups. We will briefly discuss their importance in Langlands program. For groups other than general linear groups, Whittaker models do not necessarily exist for cuspidal representations. We will explain the existence and non-existence of Whittaker models for small rank p-adic groups.

Adrien Sauvaget (Utrecht)

*Mazur-Veech volumes and intersection theory in the Hodge Bundle*

In the 80’s Mazur and Veech defined a volume form on moduli spaces Riemann surfaces endowed with a differential form with prescribed singularities. This volume form is a powerful tool to compute dynamical invariants of these surfaces. We will discuss the problem of expressing this volume form as product of Chern forms of “tautological” line bundles.

Nicolò Sibilla (Kent)

*Topological Fukaya category and homological mirror symmetry*

Kam-Fai Tam (Nijmegen)

*A crash course on Bruhat-Tits buildings*

In this semester, Santosh Nadimpalli and I are teaching a

course on Bruhat-Tits buildings. I present a one-hour summary on this

subject, based on the textbook ‘Buildings, by Kenneth Brown’. This

talk should be understandable by master students.

Michael Wong (UCL)

*Dimer models and Hochschild cohomology*

A dimer model is a type of quiver embedded in a Riemann surface. It gives rise to an associative, generally noncommutative algebra called the Jacobi algebra. In the version of mirror symmetry proved by Bocklandt, the wrapped Fukaya category of a punctured surface is equivalent to the category of matrix factorizations of the Jacobi algebra of a dimer, equipped with its canonical potential. For the purposes of deformation theory, we explicitly describe the Hochschild cohomologies of the Jacobi algebra and the associated matrix factorization category in terms of dimer combinatorics.