1998 seminars

Marco Mackaay, Universidade do Algarve
The 2-representation theory of Soergel bimodules of finite Coxeter type: a road map to the complete classification of all simple transitive 2-representations

I will first recall Lusztig's asymptotic Hecke algebra and its categorification, a fusion category obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is a 2-colored version of the semisimplified quotient of the module category of quantum $\operatorname{sl}(2)$ at a root of unity, which Reshetikhin-Turaev and Turaev-Viro used for the construction of 3-dimensional Topological Quantum Field Theories.

In the second part of my talk, I will recall the basics of 2-representation theory and indicate how the fusion categories above can conjecturally be used to study the 2-representation theory of Soergel bimodules of finite Coxeter type.

This is joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

Gonçalo Quinta & Rui André, Physics of Information and Quantum Technologies Group - IST (GQ); Center for Astrophysics and Gravitation - IST (RA)
Topological Links and Quantum Entanglement

We present a classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for any given number of rings. We demonstrate the use of this new classification scheme for three and four qubits and its potential in the context of qubit networks.

Federico Cantero, University of Barcelona, Spain
Higher Steenrod squares for Khovanov homology

We describe stable cup-$i$ products on the cochain complex with $\mathbb{F}_2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

Brian Hall, University of Notre Dame
Eigenvalues of random matrices in the general linear group

I will begin by discussing the two standard prototype random matrix models, one for Hermitian matrices and one for general matrices. For large matrices, the eigenvalues follow the "semicircular law" in the first case and the "circular law" in the second case. Furthermore, there is a simple relationship between these two laws.

I will then discuss two "multiplicative" analogs of these models, in which the random matrices are chosen from the unitary group and the general linear group, respectively. In the unitary case, the limiting eigenvalue distribution was computed by Biane and exhibits an interesting phase transition when a certain scaling parameter equals 4. I will then describe recent results of mine with Driver and Kemp on the general linear case. The limiting distribution again undergoes a phase transition and turns out to have a remarkably simple structure. The talk will be self-contained with lots of pictures and possibly even a few jokes.

Roger Picken, Departamento de Matemática, Instituto Superior Técnico
Quantum theory via (higher) groupoids and quantum measures

The idea of this informal seminar is to present, in cherry-picking fashion, some fascinating work by Ciaglia, Ibort and Marmo, who formulate the Schwinger approach to quantum theory using the contemporary language of groupoids and $2$-groupoids. By bringing in the notion of quantum measures, due to Sorkin, they achieve an elegant description of examples like the qubit or the two-slit experiment.

The aim is for the seminar to be comprehensible also to students with some awareness of quantum theory, and hopefully will be followed up by future seminars around quantum maths topics such as topological quantum computation and entanglement.

Main references

Ciaglia, Ibort, Marmo: https://arxiv.org/abs/1905.12274, https://arxiv.org/abs/1907.03883, https://arxiv.org/abs/1909.07265

Manuel Araújo, Instituto Superior Técnico
Topological Field Theory in dimension $3$

I will give an overview of some aspects of $3d$ TFT, from the Turaev-Viro and Reshetikin-Turaev invariants of oriented $3$-manifolds, to the more recent classifications of fully extended theories in terms of fusion categories and once extended theories in terms of Modular Tensor Categories.

Roger Picken, Instituto Superior Técnico
Research topics in higher gauge theory, knot theory, and anyons

Under the heading higher gauge theory I will briefy discuss:

  • gerbes on supergroups (to be described by John Huerta in his talk).
  • the relation between 2-group (discretized) connections and transports (with Jeff Morton).
  • surface transport in 3D quantum gravity (with Jeanette Nelson).

I will then focus on two projects:

  • development of the Eisermann invariant of knots (based on previous work with João Faria Martins).
  • (higher) gauge theoretical models for anyons in topological quantum computation.

João Esteves, Instituto Superior Técnico
A quantization of the Loday-Ronco Hopf algebra

In previous works we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by identifying pairs of nearest neighbour leaves and thus producing graphs with loops we obtain the full recursion formula of Eynard and Orantin. We also discussed the algebraic structure of the spaces of correlation functions in $g=0$ and in $g\gt 0$. By taking a classical and a quantum product respectively we endowed both spaces with a ring structure. Here we will show that the extended algebra of graphs is in fact a Hopf algebra and can be seen as a sort of quantization of the Loday-Ronco Hopf algebra. This is work in progress.

Pedro Lopes, Instituto Superior Técnico
Research topics in persistent tangles and hyperfinite knots

We look into persistent tangles i.e., tangles such that its presence in a diagram implies that diagram is knotted and remark their prevalence. We also look into hyperfinite knots i.e., limits of sequences of classes of knots and relate them to wild knots.

Marko Stosic, Instituto Superior Técnico
On Colored HOMFLY homology

I’ll try to give a brief description of the categorification of colored HOMFLY polynomial for knots, and discuss various open problems related with these invariants, including their definition, properties, relationship with string theory, as well as surprising combinatorial identities and properties of these invariants in special cases of torus knots and rational knots.

Marco Mackaay, Universidade do Algarve
$2$-Representation theory

The $2$-representation theory of $2$-categories is a categorical analogue of the representation theory of algebras. In my talk, I will recall its origins, its purposes, its basic features and explain some important examples. This talk is based on joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

Additional file

Related talk in Norwich - 1

John Huerta, Instituto Superior Técnico
Higher structures on supermanifolds

My recent work has concerned two projects, both concerned with higher structures on supermanifolds. I will probably focus on my work extending the classification theorems for gerbes to supermanifolds, with an eye to examples on super Lie groups. My other project, that I will probably not discuss, concerns applying cyclic cohomology to super-Yang-Mills theory in the so-called superspace formalism (i.e., formulated as a theory on a supermanifold).

Manuel Araújo, Instituto Superior Técnico
Coherence for $3$-dualizable objects

I will explain the notion of coherence for duals and adjoints in higher categories. I will discuss a strategy for using knowledge of coherence data and the cobordism hypothesis to give presentations of fully extended bordism categories and mention some progress in dimension $3$.

Aleksandar Mikovic, Universidade Lusófona
Categorification of LQG spin-network basis

We describe an approach of how to find a $2$-group generalization of the spin-network basis from Loop Quantum Gravity. This gives a basis of spin-foam functions which are generalizations of the Wilson surface holonomy invariant for the $3$-dimensional Euclidean $2$-group.

Pedro Brito, Instituto Superior Técnico
Galois symmetries in geometry

I’ll say a few words about some homotopical (“higher”) methods to study knot spaces and diffeomorphism groups. A fascinating appearance, and one of my current obsessions, is made by the absolute Galois group of the rationals.

Ángel González-Prieto, ICMAT (Madrid)
Topological recursion in the motivic theory of character varieties

The algebraic structure of the moduli spaces of representations of surface groups (aka character varieties) has been widely studied due to their tight relation with moduli spaces of Higgs bundles. In particular, Hodge-type invariants, like the so-called E-polynomial, has been objective of intense research over the past decades. However, subtler algebraic invariants as their motivic classes in the Grothendieck ring of algebraic varieties remain unknown in the general case.

In this talk, we will construct a Topological Quantum Field Theory that computes the motivic classes of representation varieties. This tool gives rise to an effective computational method based on topological recursion on the genus of the surface. As application, we will use it to compute the motivic classes of parabolic $\operatorname{SL}(2,\mathbb{C})$-character varieties over any compact orientable surface.

Alex Bullivant, University of Leeds
$3+1D$ Dijkgraaf-Witten theory and the Categorified Quantum Double

The quantum double is a quasi-triangular Hopf algebra whose category of representations can be interpreted physically as describing the processes of fusion and braiding of anyons in the $2+1D$ Dijkgraaf-Witten TQFT. Motivated by the possibilities of topological quantum computing in $3+1D$, in this talk I will give an informal overview of my ongoing research towards understanding the categorified quantum double and its bicategory of $2$-representations. In particular, I will focus on the relation between such constructions and the Hamiltonian formulation of $3+1D$ Dijkgraaf-Witten TQFT in order to describe the braiding and fusion of extended excitations such as loops.

Paul Wedrich, Max Planck Institute and University of Bonn
Invariants of $4$-manifolds from Khovanov-Rozansky link homology

Ribbon categories are $3$-dimensional algebraic structures that control quantum link polynomials and that give rise to $3$-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the $\operatorname{\mathfrak{gl}}(N)$ quantum link polynomial, to obtain a $4$-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth $4$-manifolds. The technical heart of this construction is the newly established functoriality of Khovanov-Rozansky homology in the $3$-sphere. Based on joint work with Scott Morrison and Kevin Walker https://arxiv.org/abs/1907.12194.