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.

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.

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.

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).

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.

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.

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.

In this informal talk, I will look at some considerations that show up when extending gauge-theoretic construction of Topological Quantum Field Theory to connections on gerbes and higher structures. In particular, I will mention some contexts where higher categories with cubical or other more complex shapes of higher morphisms seem to recur, and suggest a few questions this raises.

Starting from the data of an $(n+1)$-dimensional state-sum TQFT, to each $(n-1)$-manifold we can associate a so-called tube algebra whose modules admit a physical interpretation as topological excitations in the theory. In this talk I will first introduce the tube algebra before describing an appropriate categorification. Such categorified tube algebras associate to each $(n-2)$-manifold a $2$-algebra and we propose an interpretation of the module theory in terms of excitations in the theory pinned to gapped boundaries.

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.

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 introduce a new class of states inspired by topological symmetries of links.

(joint work with Antonio Michele Miti)

In this talk we survey the symplectic geometric approach to knot framing via Lagrangian submanifold theory and geometric quantization developed in Besana & S. (2006) and extend it, taking inspiration from Liu & Ricca (2012, 2015), to achieve a geometric interpretation of the HOMFLYPT polynomial based on helicity only.

• Besana A. and Spera M., On some symplectic aspects of knots framings, J. Knot Theory Ram. 15 (2006), 883-912.
• Liu X. and Ricca R.L., The Jones polynomial for fluid knots from helicity, J. Phys A: Math. Theor. 45 (2012), 205501 (14pp).
• Liu X. and Ricca R.L., On the derivation of the HOMFLYPT polynomial invariant for fluid knots, J. Fluid Mechanics 773 (2015), 34-48.

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.

I will explain how to reduce the classification of ‘simple’ 2-representations of the 2-category of Soergel bimodules in many (most) cases to the known problem of the same classification for certain fusion categories.

In this talk I plan to present and discuss a rather weak bicategorical setup in which one can talk about genuine adjoint 1-morphisms. I will describe the main motivation from representation theory of finitary 2-categories (or bicategories) and make some parallells with the stucture theory of finite semigroups. I will also try to show how this approach simplifies formulation of some results from 2-representation theory, but also how it makes some other “classical” results much more difficult.

Joint work with Hankyung Ko and Xiaoting Zhang.

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.

We introduce a notion of enriched infinity categories over a given monoidal category, in analogy with quasi-categories over the category of sets. We make use of certain colax monoidal functors, which we call templicial objects, as a variant of simplicial objects that respects the monoidal structure. We relate the resulting enriched quasi-categories to nonassociative Frobenius monoidal functors, allowing us to prove that the free templicial module over an ordinary quasi-category is a linear quasi-category. To any dg category we associate a linear quasi-category, the linear dg nerve, which enhances the classical dg nerve, and we argue that linear quasi-categories can be seen as relaxations of dg-categories.

Joint work with Arne Mertens.

I will describe an approach to implementing classical topological field theories in the framework of extended TQFTs using higher categories of iterated spans. In the simplest version, which will be the main focus of the talk, these are arbitrary spans in some (infinity-)category, but I will also mention variants using derived stacks and either symplectic structures and iterated Lagrangian correspondences, or Poisson structures and iterated coisotropic correspondences. The latter was set up in joint work with V. Melani and P. Safronov, and the former is developed in an ongoing project with D. Calaque and C. Scheimbauer.

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.

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.

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

1. The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
2. 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).

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.

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.

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.

Finitary birepresentation theory of finitary bicategories is a categorical analog of finite-dimensional representation theory of finite-dimensional algebras. The role of the simples is played by the so-called simple transitive birepresentations and the classification of the latter, for any given finitary bicategory, is a fundamental problem in finitary birepresentation theory (the classification problem).

After briefly reviewing the basics of birepresentation theory, I will explain an analog of the double centralizer theorem for finitary bicategories, which was inspired by Etingof and Ostrik's double centralizer theorem for tensor categories. As an application, I will show how it can be used to (almost completely) solve the classification problem for Soergel bimodules in any finite Coxeter type.

A comprehensive study of periodic trajectories of the billiards within ellipsoids in the $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of $d$ intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods $T$, $d \leq T \leq 2d$, is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d + 1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d = 3$.

The talk is based on the following papers:

• V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications Mathematical Physics, 2019, Vol. 372, p. 183-211.
• G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

I will discuss some current work (with Garner, Hilburn, Oblomkov, and Rozansky) on new and old constructions of HOMFLY-PT link homology in physics and mathematics, and new connections among them. In particular, we relate the classic proposal of Gukov-Schwarz-Vafa, involving M-theory on a resolved conifold, to constructions in $3d$ TQFT's. In the talk, I will focus mainly on the $3d$ part of the story. I'll review general aspects of $3d$ TQFT's, in particular the "$3d$ A and B models" that play a role here, and how link homology appears in them.

This talk is based on joint work with M. Khovanov and Y. Kononov. By evaluating a topological field theory in dimension $2$ on surfaces of genus $0,1,2$, etc., we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

Newton’s interpolation is a method to reconstruct a function from its values at different points. In the talk I will explain how one can use this method to construct an explicit basis for the center of quantum $gl_N$ and to show that the universal $gl_N$ knot invariant expands in this basis. This will lead us to an explicit construction of the so-called unified invariants for integral homology 3-spheres, that dominate all Witten-Reshetikhin-Turaev invariants. This is a joint work with Eugene Gorsky, that generalizes famous results of Habiro for $sl_2$.

In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. The latter theory is furthermore semisimple which allows for complete computation from the knowledge of a few basic elements which can be computed explicitly. As a consequence of the explicit expressions we find in the Calabi-Yau case, we obtain the expected Gopukumar-Vafa formula and relation to SO/Sp Chern-Simons theory.