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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.

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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

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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

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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

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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

Aleksandar Mikovic, *Universidade Lusófona*

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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.

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

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06/11/2019, 14:30 — 15:00 — Room P3.10, Mathematics Building

Manuel Araújo, *Instituto Superior Técnico*

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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 $3$.

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

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06/11/2019, 14:00 — 14:30 — Room P3.10, Mathematics Building

John Huerta, *Instituto Superior Técnico*

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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).

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

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06/11/2019, 11:30 — 12:30 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

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$2$-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.

#### See also

Related talk in Norwich - 1

Related talk in Norwich - 2

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

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05/11/2019, 17:30 — 18:00 — Room P3.31, Mathematics Building

Marko Stosic, *Instituto Superior Técnico*

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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.

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

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05/11/2019, 17:00 — 17:30 — Room P3.31, Mathematics Building

Pedro Lopes, *Instituto Superior Técnico*

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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.

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

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05/11/2019, 16:00 — 16:30 — Room P3.31, Mathematics Building

João Esteves, *Instituto Superior Técnico*

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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.

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

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05/11/2019, 15:30 — 16:00 — Room P3.31, Mathematics Building

Roger Picken, *Instituto Superior Técnico*

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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.

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