Events

Angus Gruen (CALTECH) to speak about "Large Colour R matrices and HOMFLY-PT polynomials" on February 21th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Sala de Juntes, 11:00h-13:30h. 

Since the discovery of the Jones polynomial and its subsequent reinterpretation by Reshetikhin and Turaev, Quantum Knot invariants have played a large role in Knot Theory. These invariants are associated to finite irreducible representations of Quantum Groups and, in certain cases, fit into a regular families parametrised by rank which can be described by higher parameter invariants such as the HOMFLY-PT polynomial. Recently work by Gukov, Manolescu and Park showed that, in the case of $\mathfrak{sl}_2$, it is possible to define a higher parameter invariant which combines all quantum invariants of a given rank. This talk will introduce the framework of Quantum Knot invariants and some recent results involving involving extensions of these ideas to $\mathfrak{sl}_N$ for both fixed and generic N. 

Angus Gruen (CALTECH) to speak about "Knot theory, from Gauss to Jones" on February 21th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Aula 101, 16:00h-17:00h.

Knot Theory in it's simplest form is the study of intertwined loops in three dimensional space. The field traces its roots back to 1833 with the Gauss linking integral and played a key role in the development of Low Dimensional Topology. More recently, since the discovery of the Jones Polynomial, the field has seen significant advances thanks to contributions from Quantum Algebra and Theoretical Physics. I will give an introduction to this topic, focusing on a mix of foundational results and invariants. 

Ángel González (UCM) to speak about "Character varieties of knots made easy" on February 21th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Sala de Juntes, 11:00h-13:30h. 

Knot invariants, such as the Jones polynomial or the Reshetikhin-Turaev invariants, are omnipresent in the geometry of 3-manifolds. Among them, one is especially notorious: the fundamental group of the complement of the knot, aka the knot group. Particularly interesting is to study the space of representations of these knot groups, the so-called character varieties of knots. Even the simplest properties of these character varieties had led to deep proofs in hyperbolic geometry. However, general methods to unravel the deep geometric features of these spaces are still missing.

In this talk, we will discuss how these character varieties can be studied through Topological Quantum Field Theories (TQFTs), a cutting-edge categorical construction coming from theoretical physics. From the striking reinterpretation of the Jones polynomial using Chern-Simmons theory to Khovanov homology and quantum groups, these TQFTs have been used over the years to quantize celebrated invariants in terms of skein relations. In this talk, we shall discuss a new application of TQFTs to provide unexpected generalized skein relations for the arithmetics of character varieties of knots. This construction is based on a novel approach that applies Fourier-Mukai transformation as quantization mechanism instead of the classical geometric quantization.

Ángel González (UCM) to speak about "Cobordisms in algebraic topology, category theory and physics" on February 21th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Aula 101, 17:00h-18:00h.

Daniel Álvarez-Gavela (MIT) to speak about "Cotangent bundles of exotic spheres" on January 11th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Sala de Juntes, 16:00h-17:00h. 

The cotangent bundle of a smooth closed manifold is one of the most basic examples of a symplectic manifold. The smooth topology of the base manifold determines the symplectic topology of its cotangent bundle, but whether the converse is true is a major open problem. In this talk we will focus on the special case where the base manifold is an exotic sphere, reviewing what is known as well as current efforts to go further. 

Jagna Wiśniewska  (ETHZ) to speak about " b-Contact Structures on Tentacular Hyperboloids" on January 11th, 2023. 

Universitat Politècnica de Catalunya, Facultat de Matemàtiques i Estadística, Sala de Juntes, 17:00h-18:00h. 

In recent years there have been independently developed a variety of techniques to deal with the issue of non-compactness of a contact manifold. One is the definition of a class of Hamiltonians called tentacular Hamiltonians and the extension of Rabinowitz Floor homology to the non-compact zero level sets thereof. Another is the extension of contact structures to manifolds with singularities called b-contact structures. That rises obvious questions: are any of those techniques related? In this talk I show that this question can be answered affirmatively and present a class of hyperboloids called tentacular hyperboloids for which those two techniques can be applied alternatively.