1998 seminars

Marko Stošić, Instituto Superior Técnico
Knots-quivers correspondence and applications

In this talk I shall present the knots-quivers correspondence, as well as some surprising implications in combinatorics involving counting of lattice paths and number theory. The knots-quivers correspondence relates the colored HOMFLY-PT invariants of a knot with the motivic Donaldson-Thomas invariants of the corresponding quiver. This correspondence is made completely explicit at the level of generating series. The motivation for this relationship comes from topological string theory, BPS (LMOV) invariants, as well as categorification of HOMFLY-PT polynomial and A-polynomials. We compute quivers for various classes of knots, including twist knots, rational knots and torus knots.

One of the surprising outcomes of this correspondence is that from the information of the colored HOMFLY-PT polynomials of certain knots we get new expressions for the classical combinatorial problem of counting lattice paths, as well as new integrality/divisibility properties.

The main goal of this talk is to present basic ideas and to present numerous open questions and ramifications coming from knots-quivers correspondence.

(based on joint works with P. Sulkowski, M. Reineke, P. Kucharski, M. Panfil and P. Wedrich).

Ana Bela Cruzeiro, Department of Mathematics, Instituto Superior Técnico
Stochastic Clebsch variational principles

We derive the equations of motion associated with stochastic Clebsch action principles for mechanical systems whose configuration space is a manifold on which a Lie algebra acts transitively. These are stochastic differential equations (spde's in infinite dimensions).

We give the Hamiltonian version of the equations, as well as the corresponding Kolmogorov equations.

This is a joint work with D. D. Holm and T. S. Ratiu.

Marco Mackaay, Universidade do Algarve
2-representation theory

I will give an overview of 2-representation theory, following Mazorchuk and Miemietz' approach. After explaining the general setup, I will sketch the 2-representation theory of dihedral Soergel bimodules as an example.

After the seminar, for those interested we will continue with a discussion of approaches to 2-representation theory.

Ricardo Schiappa, Instituto Superior Técnico
Co-equational (i.e. Parametric) Resurgence and Topological Strings

I will briefly review the uses and applications of resurgence applied to topological string theory, with emphasis on nonperturbative completions and the large-order behaviour of enumerative invariants. Due to the nature of the holomorphic anomaly equations, there is a clear need to develop methods of co-equational (i.e. parametric) resurgence in order to achieve a complete description of the topological string transseries.

Björn Gohla, GFM Univ. Lisboa
A Categorical Model for the Hopf Fibration

We give a description up to homeomorphism of $S^3$ and $S^2$ as classifying spaces of small categories, such that the Hopf map $S^3\longrightarrow{}S^2$ is the realization of a functor.

Björn Gohla, GFM Univ. Lisboa
Monoids, Monads and Simplicial Objects

We will present some classical facts about the relationship between monoids and monads. We will use ordinal sums of categories and the join product of topological spaces to define the abstract and topological simplices. Along the way we show how the simplicial identities can be obtained. Time permitting we will indicate a 2-categorical generalization of this circle of ideas.

Leonardo Santilli, Grupo de Fisica Matematica, Univ. Lisboa
A Chern-Simons view on noncommutative scalar field theory

We present a recent result establishing a bridge between noncommutative scalar field theory in $2$ dimensions and topological field theory in $3$ dimensions.

The content of the seminar is split in two main parts, according to the twofold aspect of the result. In the first half, we show that a version of Abelian gauge theory on $\mathbb{R}^3 _{\lambda}$, when restricted to a single fuzzy sphere, reduces in the large $N$ limit to the Langmann-Szabo-Zarembo (LSZ) matrix model, which originally emerges in the study of scalar field theory on the Moyal plane. Then, throughout the second part, we prove that the LSZ matrix model is actually equivalent to the matrix model of $U(N)$ Chern-Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices of the LSZ model, the Chern-Simons matrix model either describes the Chern-Simons partition function, the unknot invariant, given by quantum dimensions, or the Hopf link invariant. Equivalently, the partition function of the LSZ model can be written in terms of the $S$ and $T$ modular matrices of the WZW model.

Based on: arXiv:1805.10543 [hep-th].

Björn Gohla, CAMGSD, Universidade de Lisboa
Internal Strictification

It is well known that ordinary bicategories can always be replaced by bi-equivalent strict 2-categories. Special cases of this are the strictification of monoidal categories and categorical groups. We give an abstract strictification construction for pseudo-monoids in a monoidal 2-category. It is easy to see that bicategories internal to an appropriate category are such pseudo-monoids, and can hence be strictified. (Joint work with Nelson Martins-Ferreira)

Miguel Tierz, Grupo de Fisica Matematica, Universidade de Lisboa
Random matrix theory in topological gauge theories

We present an overview of analytical tools in random matrix theory and related areas, involving Toeplitz/ Hankel determinants and symmetric functions, with an emphasis on their relevance in the study of topological gauge theory and focussing on some specific Chern-Simons theories and 2d Yang-Mills theories. We will also explain how these methods and results are intertwined with localization results in supersymmetric gauge theories.