# Seminar

## Spring 2022

#### When

Monday, April 25th 2022, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Juan Margalef (Memorial University)

#### Title

Symplectic formulation of the covariant phase space with boundaries

#### Abstract

There are two different standard ways of endowing a physical theory with a symplectic structure: the canonical and the covariant. The former is derived from the well-known symplectic structure of a certain cotangent bundle. The latter is based on the variational calculus. Including a boundary in the canonical formalism poses no problem, however, in the covariant formalism things break apart. In this talk, I will briefly introduce both formalisms without boundary and explain in detail a new framework that allows us to include boundaries in a straightforward way. I will also show briefly a new result where we proved that, actually, the canonical and covariant formalisms are equivalent in full generality.

#### When

Monday, April 4th 2022, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Alfonso Garmendia

#### Title

Basic ideas for pseudodifferential b-calculus and the holonomy groupoid of a singular foliation

#### Abstract

We will start reviewing the basics of pseudodifferential operators to find homotopic solutions to elliptic differential equations, and explain its generalization for manifolds with boundary (b-pseudodifferential calculus). This process uses a blow up of the pair groupoid. Then it will be introduced the Holonomy groupoid of a singular foliation and its relation with the blow up construction.

#### When

Monday, March 21st 2022, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

#### Title

Surgery on links in S³. A short story on Dehn surgery and Kirby calculus.

#### Abstract

Already in 1989, E. Witten showed in his famous paper "Quantum Field Theory and the Jones Polynomial" how the Chern-Simons TQFT recovers the skein relations defining the Jones polynomial, a well-known invariant of knots and links. This process is carried out by considering the expectation value of Wilson lines in S³ and generalized to any arbitrary closed oriented 3-manifold by means of Dehn surgery on links. Current work by S. Gukov and others searching for more powerful invariants relies on the use of this description of 3-manifolds in terms of integral surgery on links in S³, depicted by decorated plumbing graphs in a certain equivalence class given by the so-called Kirby moves. In this talk we present the description of this underlying framework enabling to better understand the foundations of the above mentioned invariants.

#### When

Monday, February 21st 2022, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Pablo Nicolás (UPC)

#### Title

Gauge models for Yang-Mills theory on manifolds with boundary and singular symplectic reduction

#### Abstract

Yang-Mills theories describe fields on a space-time *M* as connections on a principal *G*-bundle *π : P → M*. Wong's equations describe the parallel transport over an associated vector bundle to *P*; these, in turn, describe the movement of a point-mass particle interacting with the Yang-Mills field. In this talk we extend the formalism of vector bundles, principal bundles and principal connections to b-manifolds. Following the work of A. Weinstein, we show the existence of a universal model for the phase space of a particle interacting with a Yang-Mills field; moreover, these equations become hamiltonian. Also following R. Montgomery, we see that the universal b-symplectic spaces of Weinstein are symplectic leaves of a certain universal Poisson space. This is a joint work with P. Mir and E. Miranda.

#### When

Monday, February 7th 2022, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Martin Zika (Charles University)

#### Title

NQP Manifolds and dg-Lagrangian Relations

#### Abstract

Graded symplectic manifolds naturally arise in the context of Batalin-Vilkovisky quantization as one introduces fields of non-trivial ghost degrees. We will review the one-to-one correspondence between isomorphism classes of differential non-negatively graded symplectic manifolds (NQP manifolds) of degree 2 and Courant algebroids following Dmitry Roytenberg. Applying the construction of Lagrangian relations in the spirit of Weinstein’s symplectic category, we can extend the one-to-one correspondence on objects to an equivalence of suitable categories. We will comment on applications in the context of Ševera's quantum odd symplectic category and further promising outlooks.

#### When

Friday, January 28th 2022, at 10:30 at Barcelona/Paris/Berlin time.

#### Speaker

Ángel González Prieto (Universidad Complutense de Madrid)

#### Title

Topological Quantum Field Theories for dummies - Part IV

#### Abstract

In this introductory talk, we shall explain the fundamentals of one of the most groundbreaking developments in algebraic topology, the so-called Topological Quantum Field Theories (TQFTs). We will study their formulation as representations of the category of bordisms into the category of vector spaces. Additionally, we will explore some of the consequences of these axioms, including the classical result that identifies 1+1 TQFTs with Frobenius algebras. After that, we shall focus on the applications of TQFTs to knot theory. We will discuss Chern-Simmons theory as a fundamental example of a 2+1 decorated TQFT, reaching Witten's celebrated theorem that shows that SU(2) Chern-Simons theory recovers the Jones polynomial, and important knot invariant. Time permitting, we will explore some deeper connections between TQFTs and low dimensional topology, such as the application of $R$-matrices solving the Yang-Baxter equation to the construction of representations of the tangle category. No prerequisites will be needed to follow the talk, apart from basic algebraic topology.

## Autumn 2021

#### When

Wednesday, December 22nd 2021, at 16:30 at Barcelona/Paris/Berlin time.

#### Speaker

Ángel González Prieto (Universidad Complutense de Madrid)

#### Title

Topological Quantum Field Theories for dummies - Part III

#### Abstract

In this introductory talk, we shall explain the fundamentals of one of the most groundbreaking developments in algebraic topology, the so-called Topological Quantum Field Theories (TQFTs). We will study their formulation as representations of the category of bordisms into the category of vector spaces. Additionally, we will explore some of the consequences of these axioms, including the classical result that identifies 1+1 TQFTs with Frobenius algebras. After that, we shall focus on the applications of TQFTs to knot theory. We will discuss Chern-Simmons theory as a fundamental example of a 2+1 decorated TQFT, reaching Witten's celebrated theorem that shows that SU(2) Chern-Simons theory recovers the Jones polynomial, and important knot invariant. Time permitting, we will explore some deeper connections between TQFTs and low dimensional topology, such as the application of $R$-matrices solving the Yang-Baxter equation to the construction of representations of the tangle category. No prerequisites will be needed to follow the talk, apart from basic algebraic topology.

#### When

Monday, December 13th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Ood Shabtai (Tel Aviv University)

#### Title

Pairs of spectral projections of spin operators

#### Abstract

We study the semiclassical behavior of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, and contrast it with the behavior of the polynomial when evaluated on random pairs of projections.

#### When

Friday, December 10th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Ángel González Prieto (Universidad Complutense de Madrid)

#### Title

Topological Quantum Field Theories for dummies - Part II

#### Abstract

In this introductory talk, we shall explain the fundamentals of one of the most groundbreaking developments in algebraic topology, the so-called Topological Quantum Field Theories (TQFTs). We will study their formulation as representations of the category of bordisms into the category of vector spaces. Additionally, we will explore some of the consequences of these axioms, including the classical result that identifies 1+1 TQFTs with Frobenius algebras. After that, we shall focus on the applications of TQFTs to knot theory. We will discuss Chern-Simmons theory as a fundamental example of a 2+1 decorated TQFT, reaching Witten's celebrated theorem that shows that SU(2) Chern-Simons theory recovers the Jones polynomial, and important knot invariant. Time permitting, we will explore some deeper connections between TQFTs and low dimensional topology, such as the application of $R$-matrices solving the Yang-Baxter equation to the construction of representations of the tangle category. No prerequisites will be needed to follow the talk, apart from basic algebraic topology.

#### When

Monday, November 22nd 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Pau Mir (UPC)

#### Title

Geometric quantization of b-symplectic toric manifolds

#### Abstract

Geometric quantization is a tool which is being widely used in symplectic geometry, since it has applications in the relation between classical physics and quantum physics. When manifolds are b-symplectic, this quantization has to be redefined in order to obtain results in finite dimensions. In this talk, we will see a model for geometric quantization of b-symplectic toric manifolds and how it relates to other pre-existing models such as formal geometric quantization.

#### When

Monday, November 15th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Ángel González Prieto (Universidad Complutense de Madrid)

#### Title

Topological Quantum Field Theories for dummies

#### Abstract

#### When

**Tuesday**, November 9th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Robert Cardona (UPC / IRMA - Université de Strasbourg / CNRS)

#### Title

Dynamics and transversality via symplectic flexibility

#### Abstract

In this talk, we give an introduction to Hamiltonian structures and their relation to Hamiltonian dynamics. New results on the dynamical universality of such structures will be presented, as well as a very basic introduction to the main tool of the proof, the homotopy principle. Time permitting, we will state some flexibility theorems for transverse embeddings into contact manifolds, based on joint work with Francisco Presas.

#### When

Monday, October 18th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Romero Solha (UPV-EHU)

#### Title

Symplectic foliations induced by harmonic forms on 3-manifolds

#### Abstract

This talk aims to detail a construction of symplectic foliations on 3-dimensional orientable Riemannian manifolds from harmonic forms; and how it relates to the positiveness of gravitational mass in Newtonian gravity.

## Spring 2021

#### When

Monday, June 14th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Ryszard Nest (U. Copenhagen)

#### Title

Towards resolutions of singularities of Poisson structures

#### Abstract

In this joint work with Eva Miranda. We associate regular Poisson structures to linearizable Poisson structures.

Here you can find a recording of the talk.

#### When

Thursday, June 10th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Jean Gutt (Institut National Universitaire Champollion and the Institut de Mathématiques de Toulouse)

#### Title

Symplect'hearing the contact horizon and contacting the symplectic inner beauty

#### Abstract

This talk will be a wander through symplectic topology and contact geometry having in mind the question How much does the contact boundary know about the symplectic interior, and, reciprocally, how much does the symplectic interior know about its contact boundary? The aim of this talk is to review the main question I am interested in, explain my approach to one of them (the strong Viterbo conjecture) and ask further questions.

#### When

Monday, May 3rd 2021, at 15:30 at Barcelona/Paris/Berlin time.

#### Speaker

Konstantinos Kourliouros (ICMC-USP Säo Pao)

#### Title

Impasse Singularities of Generalised Hamiltonian Systems

#### Abstract

By a Generalised (or Constrained) Hamiltonian System we mean a pair consisting of a function (the generalised Hamiltonian) and an arbitrary closed 2-form (i.e. not necessarily symplectic). Such systems, considered first by P. A. M. Dirac, appear in abundance when considering variational and optimal control problems with constraints, in quantisation problems, but also in hydrodynamics (general vortex theory) e.t.c. In this talk we give local classification results for typical singularities of such systems at impasse points, i.e. at points belonging on the singular locus of the 2-form, where its rank is smaller than the maximum possible. In particular we show how to obtain exact local normal forms with functional invariants for a wide class of impasse singularities, and we give a geometric description of them in terms of other invariant objects (discriminants and period mappings) naturally associated to the system.

Here you can find a recording of the talk.

#### When

Monday, April 19th 2021, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Alessandro Tamai (Trieste)

#### Title

Singular solutions spaces of rolling balls problem

#### Abstract

"The "Rolling Balls Model", the model describing a pair of spheres of different ray rolling one on another without slipping or twisting, is a classical example of sub-Riemannian problem. The symmetries of the distribution associated with the system depend on the ratio of the rays and radically change when the ratio equals 3. Indeed, for this value of the ratio (and only for this value) it extends to the exceptional simple Lie group G2 which acts, still for this value of the ratio, also on the singular solutions related to the problem. In this talk we show how it is possible to describe the spaces of such singular solutions in a geometric way, as a family of 5-dimensional manifolds depending on the ratio. For rational values of the ratio such manifolds have a structure of SO(2)-principal bundles which are not topologically distinguished by their homology, homotopy and de Rham cohomology groups. In addition, we show that for integer values of the ratio the configuration manifold of the problem is a branched covering of each of such manifolds and how the covering maps associated allow to relate them with another known family of topological spaces, the lens spaces.This talk is based on the research works developed in my master thesis at University of Trieste, under the supervision of the professor Agrachev from SISSA (Trieste).

## Spring 2020

Here you can find the slides used in this talk.

#### When

Tuesday, June 30th 2020, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

Urs Frauenfelder (Augsburg University)

#### Title

Delayed Potentials and Weber's Electrodynamics

#### Abstract

This is joint work with Joa Weber. We first discuss work of Carl Neumann from 1868 about a delayed Coulomb potential and its relation to the electrodynamic potential of Wilhelm Weber. Then we explain how Weber's Electrodynamics can be interpreted as a Hamiltonian system and how it is related to Riemannian and Lorenzian geometry. Finally we quantize the system.

Here you can find the slides used in this talk.

#### When

Friday, June 26th 2020, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

Joaquim Brugués (UPC / UAntwerp)

#### Title

The Euler class of b-tangent bundles

#### Abstract

The Euler class is a basic tool to understand the topology of orientable vector bundles, in particular it aims to measure how "twisted" any given bundle is. In the particular case of the tangent bundle to a manifold, the Poincaré-Hopf theorem relates this construction to the topology of the base manifold, yielding an index theorem. In this talk we will review the definition of the Euler class and study a possible analogous to the Poincaré-Hopf theorem to b-tangent bundles.

#### When

Tuesday, June 23rd 2020, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

Anastasia Matveeva (UPC)

#### Title

b-Loop Hamiltonian spaces

#### Abstract

TBA

#### When

Tuesday, June 2nd 2020, at 15:30 at Barcelona/Paris/Berlin time.

#### Speaker

Robert Cardona (UPC)

#### Title

Integrable systems, Fomenko invariants and Fluid Dynamics

#### Abstract

TBA

#### When

Tuesday, May 26th 2020, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

Pau Mir (UPC)

#### Title

Integrable systems, cotangent lifts and rigidity of group actions

#### Abstract

TBA

#### When

Tuesday, May 19th 2020, at 15:30 at Barcelona/Paris/Berlin time.

#### Speaker

Baptiste Coquinot (ENS Paris)

#### Title

Helium and Hamiltonian delay equations

#### Abstract

TBA

Here you can find the slides used in the talk.

#### When

Friday, May 8th 2020, at 16:00 at Barcelona/Paris/Berlin time.

#### Speaker

Cédric Oms (UPC)

#### Title

Periodic orbits on the restricted three body problem: Different techniques and different results

#### Abstract

TBA

#### When

Tuesday, April 28th and Tuesday, May 5th 2020, at 17:00 at Barcelona/Paris/Berlin time.

#### Speaker

Daniel Peralta-Salas (CSIC)

#### Title

Contact structures and Beltrami fields on the torus and the sphere

#### Abstract

We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form. The talk will be tightly related to this article: https://arxiv.org/abs/2004.10185.