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30/10/2020, 17:00 — 18:00 — Online

João Faria Martins, *University of Leeds*

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Crossed modules, homotopy 2-types, knotted surfaces and welded knots

I will review the construction of invariants of knots, loop braids and knotted surfaces derived from finite crossed modules. I will also show a method to calculate the algebraic homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere from a movie presentation of $\Sigma$. This will entail a categorified form of the Wirtinger relations for a knot group. Along the way I will also show applications to welded knots in terms of a biquandle related to the homotopy 2-type of the complement of the tube of a welded knots.

The last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: *Emergent Physics From Lattice Models of Higher Gauge Theory*.

#### See also

João Faria Martins slides

Projecto FCT `UIDB/04459/2020`.

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16/10/2020, 17:00 — 18:00 — Online

Miranda Cheng, *University of Amsterdam*

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Quantum modular forms and $3$-manifolds

Quantum modular forms are functions on rational numbers that have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upper-half plane which lead to quantum modular forms. Inspired by the $3d-3d$ correspondence in string theory, new topological invariants named homological blocks for (in particular plumbed) three-manifolds have been proposed a few years ago. My talk aims to explain the recent observations on the quantum modular properties of the homological blocks, as well as the relation to logarithmic vertex algebras.

The talk will be based on a series of work in collaboration with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah Harrison, and Gabriele Sgroi.

#### See also

Cheng slides.pdf

Projecto FCT `UIDB/04459/2020`.

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09/10/2020, 17:00 — 18:00 — Online

Alexander Shapiro, *University of Notre Dame*

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Cluster realization of quantum groups and higher Teichmüller theory

Quantum higher Teichmüller theory, as described by Fock and Goncharov, endows a quantum character variety on a surface $S$ with a cluster structure. The latter allows one to construct a canonical representation of the character variety, which happens to be equivariant with respect to an action of the mapping class group of $S$. It was conjectured by the authors that these representations behave well with respect to cutting and gluing of surfaces, which in turn yields an analogue of a modular functor. In this talk, I will show how the quantum group and its positive representations arise in this context. I will also explain how the modular functor conjecture is related to the closedness of positive representations under tensor products as well as to the non-compact analogue of the Peter-Weyl theorem. If time permits, I will say a few words about the proof of the conjecture.

This talk is based on joint works with Gus Schrader.

#### See also

Shapiro talk notes.pdf

Projecto FCT `UIDB/04459/2020`.

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02/10/2020, 17:00 — 18:00 — Online

Davide Masoero, *Group of Mathematical Physics, University of Lisbon*

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Counting Monster Potentials

The monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that

- The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
- The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of $N$ a unique monster potential with $N$ roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer $N$, the number of monster potentials with $N$ roots coincide with the number of integer partitions of $N$, which is the dimension of the level $N$ subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

The talk is based on the preprint https://arxiv.org/abs/2009.14638, written in collaboration with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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25/09/2020, 17:00 — 18:00 — Online

André Henriques, *University of Oxford*

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Reps of relative mapping class groups via conformal nets

Given a surface with boundary $\Sigma$, its relative mapping class group is the quotient of $\operatorname{Diff}(\Sigma)$ by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If $\Sigma$ has no boundary, then that's the usual mapping class group; if $\Sigma$ is a disc, then that's the group $\operatorname{Diff}(S^1)$ of diffeomorphisms of $S^1$.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

Projecto FCT `UIDB/04459/2020`.

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11/09/2020, 17:00 — 18:00 — Online

Alexis Virelizier, *Université de Lille*

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Homotopy Quantum Field Theories

Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs) by replacing manifolds by maps from manifolds to a fixed target space $X$. For example, any cohomology class in $H^3(X)$ defines a 3-dimensional HQFT with target $X$. If $X$ is aspherical, that is $X = K(G, 1)$ for some group $G$, then this cohomological HQFT is related to the Dijkgraaf-Witten invariant and is computed as a Turaev-Viro state sum via the category of $G$-graded vector spaces. More generally, the state sum Turaev-Viro TQFT and the surgery Reshetikhin-Turaev TQFT extend to HQFTs (using graded fusion categories) which are related via the graded categorical center.

This is joint work with V. Turaev.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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24/07/2020, 17:00 — 18:00 — Online

Ezra Getzler, *Northwestern University*

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Gluing local gauge conditions in BV quantum field theory

In supersymmetric sigma models, there is frequently no global choice of Lagrangian submanifold for BV quantization. I will take the superparticle, a toy version of the Green Schwarz superstring, as my example, and show how to extend the light-cone gauge to the physically relevant part of phase space. This involves extending a formula of Mikhalkov and A. Schwarz that generalizes the prescription of Batalin and Vilkovisky for the construction of the functional integral.

This is joint work with S. Pohorence.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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17/07/2020, 17:00 — 18:00 — Online

Pedro Boavida de Brito, *Instituto Superior Técnico and CAMGSD*

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Galois symmetries of knot spaces

I’ll describe how the absolute Galois group of the rationals acts on a space which is closely related to the space of all knots. The path components of this space form a finitely generated abelian group which is, conjecturally, a universal receptacle for integral finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteristic zero. I will then discuss some work in progress concerning higher-dimensional variants.

This is joint work with Geoffroy Horel.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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10/07/2020, 17:00 — 18:00 — Online

Ricardo Campos, *CNRS - University of Montpellier*

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The homotopy type of associative and commutative algebras

Given a topological space, how much of its homotopy type is captured by its algebra of singular cochains? The experienced rational homotopy theorist will argue that one should consider instead a commutative algebra of forms. This raises the more algebraic question

Given a dg commutative algebra, how much of its homotopy type (quasi-isomorphism type) is contained in its associative part?

Despite its elementary formulation, this question turns out to be surprisingly subtle and has important consequences.

In this talk, I will show how one can use operadic deformation theory to give an affirmative answer in characteristic zero.

We will also see how the Koszul duality between Lie algebras and commutative algebras allows us to use similar arguments to deduce that under good conditions Lie algebras are determined by the (associative algebra structure of) their universal enveloping algebras.

Joint with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra and based on arXiv:1904.03585.

Projecto FCT `UIDB/04459/2020`.

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03/07/2020, 17:00 — 18:00 — Online

Tom Sutherland, *Group of Mathematical Physics, University of Lisbon*

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Mirror symmetry for Painlevé surfaces

This talk will survey aspects of mirror symmetry for ten families of non-compact hyperkähler manifolds on which the dynamics of one of the Painlevé equations is naturally defined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the complement of a triangle of lines in a (singular) cubic surface. The two realisations relate closely to a space of stability conditions and a cluster variety of a quiver respectively, providing a perspective on SYZ mirror symmetry for these manifolds. I will discuss joint work in progress with Helge Ruddat studying the canonical basis of theta functions on these cubic surfaces.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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26/06/2020, 17:00 — 18:00 — Online

Marko Stošić, *Instituto Superior Técnico and CAMGSD*

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Rational and algebraic links and knots-quivers correspondence

I will start with a brief overview of knots-quivers correspondence, where colored HOMFLY-PT (or BPS) invariants of the knot are expressed as motivic Donaldson-Thomas invariants of a corresponding quiver.

This deep conjectural relationship already had some surprising applications.

In this talk I will focus on showing that the knots-quivers correspondence holds for rational links, as well as much larger class of arborescent links (algebraic links in the sense of Conway). This is done by extending the correspondence to tangles, and showing that the set of tangles satisfying tangles-quivers correspondence is closed under the tangle addition operation.

This talk is based on joint work with Paul Wedrich.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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19/06/2020, 17:00 — 18:00 — Online

Mikhail Khovanov, *Columbia University*

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Introduction to foam evaluation

Foam evaluation was discovered by Louis-Hardrien Robert and Emmanuel Wagner slightly over three years ago. It's a remarkable formula assigning a symmetric function to a foam, that is, to a decorated 2-dimensional CW-complex embedded in three-space. We'll explain their formula in the 3-color case in the context of unoriented foams and discuss its relation to Kronheimer-Mrowka homology of graphs and the four-color theorem.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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12/06/2020, 17:00 — 18:00 — Online

Antti Kupiainen, *University of Helsinki*

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Integrability of Liouville Conformal Field Theory

A. Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a natural measure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry.

In the context of 2D quantum gravity models, Polyakov’s approach is conjecturally equivalent to the scaling limit of Random Planar Maps and through the Alday-Gaiotto-Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn, Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable.

I will review a probabilistic construction of LCFT developed together with David, Rhodes and Vargas and recent proofs of the integrability of LCFT:

- The proof in a joint work with Rhodes and Vargas of the DOZZ formula (Annals of Mathematics, 81-166,191 (2020))
- The proof in a joint work with Guillarmou, Rhodes and Vargas of the bootstrap conjecture for LCFT (arXiv:2005.11530).

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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05/06/2020, 17:00 — 18:00 — Online

John Huerta, *Instituto Superior Técnico and CAMGSD*

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Bundle Gerbes on Supermanifolds

Bundle gerbes are a generalization of line bundles that play an important role in constructing WZW models with boundary. With an eye to applications for WZW models with superspace target, we describe the classification of bundle gerbes on supermanifolds, and sketch a proof of their existence for large families of super Lie groups.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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29/05/2020, 17:00 — 18:00 — Online

Danica Kosanović, *Max-Planck Institut für Mathematik*

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Knot invariants from homotopy theory

The embedding calculus of Goodwillie and Weiss is a certain homotopy theoretic technique for studying spaces of embeddings. When applied to the space of knots this method gives a sequence of knot invariants which are conjectured to be universal Vassiliev invariants. This is remarkable since such invariants have been constructed only rationally so far and many questions about possible torsion remain open. In this talk I will present a geometric viewpoint on the embedding calculus, which enables explicit computations. In particular, we prove that these knot invariants are surjective maps, confirming a part of the universality conjecture, and we also confirm the full conjecture rationally, using some recent results in the field. Hence, these invariants are at least as good as configuration space integrals.

#### See also

Slides of the talk

Projecto FCT `UIDB/04459/2020`.

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22/05/2020, 17:00 — 18:00 — Online

Sergei Gukov, *California Institute of Technology*

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Hidden Algebraic Structures in Topology

Which 4-manifold invariants can detect the Gluck twist? And, which 3-manifold invariants can detect the difference between surgeries on mutant knots? What is the most powerful topological quantum field theory (TQFT)? Guided by questions like these, we will look for new invariants of 3-manifolds and smooth 4-manifolds. Traditionally, a construction of many such invariants and TQFTs involves a choice of certain algebraic structure, so that one can talk about "invariants for SU(2)" or a "TQFT defined by a given Frobenius algebra." Surprisingly, recent developments lead to an opposite phenomenon, where algebraic structures are labeled by 3-manifolds and 4-manifolds, so that one can speak of VOA-valued invariants of 4-manifolds or MTC-valued invariants of 3-manifolds. Explaining these intriguing connections between topology and algebra will be the main goal of this talk.

#### See also

Gukov_slides.pdf

Projecto FCT `UIDB/04459/2020`.

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11/12/2019, 11:30 — 12:30 — Room P3.10, Mathematics Building

Paul Wedrich, *Max Planck Institute and University of Bonn*

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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.

Projecto FCT `UID/MAT/04459/2019`.

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27/11/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Alex Bullivant, *University of Leeds*

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$3+1D$ 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.

Please note that there is also a TQFT Club talk in the morning on the same day starting at 11h15.

Projecto FCT `UID/MAT/04459/2019`.

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27/11/2019, 11:15 — 12:15 — Room P3.10, Mathematics Building

Ángel González-Prieto, *ICMAT (Madrid)*

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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.

Please note the earlier starting time 11h15!

Also please note that there is a second TQFT Club talk on the same day starting at 15h.

Projecto FCT `UID/MAT/04459/2019`.

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06/11/2019, 16:30 — 17:00 — Room P3.10, Mathematics Building

Pedro Brito, *Instituto Superior Técnico*

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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.

Part of the Higher Structures and Applications mini-meeting, 5-6 Nov. 2019.

Projecto FCT `UID/MAT/04459/2019`.