JUNIOR
GLOBAL POISSON
WORKSHOP
II

Non-algebraicity of hypercomplex nilmanifolds

Anna Abasheva

Columbia University; Higher School of Economics

This is a joint work with Misha Verbitsky, arXiv:2103.05528

A hypercomplex manifold $X$ is a manifold equipped with an action of the quaternion algebra on its tangent bundle satisfying an integrability condition. Every hypercomplex manifold has a whole 2-sphere of complex structures; in this way it makes sense to talk about a generic complex structure $L$ on a $X$. It turns out that if $X$ is a compact hyperkähler manifold then the complex manifold $X_L$ is non-algebraic for a generic complex structure (Fujiki, 87). Furthermore, $X_L$ admits no rational non-trivial morphisms onto an algebraic variety ( = “algebraic dimension of $X_L$ vanishes”). By a later result by Misha Verbitsky (1995) all the subvarieties of $X_L$ for a generic $L$ are trianalytic, namely, they are complex analytic with respect to every complex structure. Consequently, $X_L$ doesn’t contain even-dimensional subvarieties (f.e. curves and divisors).

It might be tempting to conjecture that similar assertions hold for hypercomplex manifolds; this is, however, false in general. Nevertheless, the first assertion turns out to hold for so called hypercomplex nilmanifolds. A nilmanifold is a quotient of a nilpotent Lie group by a lattice. A left-invariant (hyper)complex structure on a Lie group is inherited by the quotient; in this way it makes sense to talk about (hyper)complex nilmanifolds. Complex nilmanifolds are non-Kähler, except for complex tori. Under an additional assumption on a hypercomplex nilmanifold (the existence of an HKT-structure) we are able to prove the assertion about subvarieties. Moreover, we provide a classification of trianalytic subvarieties in this case. My talk will be dedicated to the explanation of these results.

Slides

Equivariant Cohomology Models for Differentiable Stacks

Luis Alejandro Barbosa Torres

University of São Paulo (USP)

We introduce the concept of equivariant cohomology in the smooth manifold case and the notion of differentiable stacks. Then we consider an action of a Lie group on a differentiable stack in the sense of Romagny and consider the stacky quotient associated to this action. Consequently, we construct an atlas that makes these stacky quotient a differentiable stack. Using that the nerve of the associated Lie groupoid of that stack gives its the homotopy type, we provide a Borel model for equivariant cohomology in this context. In order to follow the classical approach for equivariant cohomology, we build a Cartan model for differentiable stacks and we prove that both models compute the same cohomology as the proposed by the Borel model.

Slides (Plenary), Slides (Parallel)

On some structures of the second Painlevé equation and related hierarchies

Irina Bobrova

Higher School of Economics

This talk is divided into two parts. The first part is general: we will discuss what the Painlevé equations are and what structures they have, namely integrability, confluences, Hamiltonian structures, sigma-coordinates and symmetries. In the second part, we will consider some integrable hierarchies associated with the second Painlevé equation, their sigma-coordinates, symmetries, and further generalizations to non-commutative cases.

arXiv:2010.10617, arXiv:2012.11010

Slides

The Poisson saturation of regular submanifolds

Stephane Geudens

KU Leuven

I will talk about a class of submanifolds in Poisson geometry, which are defined in terms of a constant rank condition. Their main feature is the fact that their local saturation is an embedded Poisson submanifold. I will give a normal form for the induced Poisson structure on the local saturation, and discuss some consequences. Time permitting, I will show how these results generalize to the setting of Dirac geometry.

Slides

Local gauge field theory from the perspective of non-linear PDE geometry

Jacob Kryczka

LAREMA, University of Angers

Higher structures and derived geometry have become ubiquitous tools when studying the mathematics of quantum field theory. Specifically, shifted Poisson structures and their quantization have found application in quantum field and string theory with derived symplectic geometry providing a powerful reinterpretation of the AKSZ formalism.

In the most `basic' setting, these notions appear when describing the homotopical space of critical points of an action functional. Rather than start with the critical locus, we would like to study the corresponding space of solutions of the equation of motion and the natural geometric structures it possesses. The upshot of this type of an approach is that we can study non-linear PDEs which are not necessarily of Euler-Lagrange form.

In my talk I will describe a functorial approach to non-linear PDEs in the presence of symmetries. We will pay special attention to describing gauge field theories and the derived covariant phase space, equipped with its canonical shifted symplectic form.

Slides

Hamiltonian $S^1$-spaces, semitoric integrable systems, and hyperbolic singularities

Joseph Palmer

University of Illinois, Urbana-Champaign

A Hamiltonian action of $S^1$ on a symplectic 4-manifold comes with a real valued Hamiltonian function $J$. When we can we find a smooth map $H$ such that $(J,H)$ is an integrable system? Moreover, what can we say about the properties of the resulting system $(J,H)$ in different situations? We explore these questions and how their answers relates to toric integrable systems, semitoric integrable systems, and a class of integrable systems with hyperbolic singularities which generalize semitoric systems. This is joint work with S. Hohloch.

Slides

Quantum R-matrix identities and Interacting Integrable Tops

Ivan Sechin

Skoltech

Integrability of classical integrable systems, for example, multi-particle Calogero–Moser system, is based on some functional identities on rational, trigonometric, or elliptic functions, which ensure the existence of Lax pair and the Poisson commutativity of integrals of motion. It appears that some quantum R-matrices satisfy the matrix analogues of the relations, known as associative Yang–Baxter equation and its degenerations. This fact allows us to use such quantum R-matrices in Lax pairs instead of scalar functions and construct new classical integrable systems.

I will describe the example of the application of quantum R-matrices relations in classical integrability, introducing the system of interacting integrable tops, generalizing both Calogero–Moser systems of particles and Euler tops. I will also show how the resulting integrable structures simultaneously contain the properties of particle and top systems. If time permits, I briefly discuss the quantization of these structures, in the elliptic case it leads to quadratic quantum algebras which generalize both Sklyanin algebra and Felder elliptic quantum group.

Slides

Symplectic excision

Xiudi Tang

University of Toronto

We consider closed subsets of a noncompact symplectic manifold and determine when they can be removed by a symplectomorphism, in which case we say the subsets are symplectically excisable. We prove that, in the case of a ray and more generally, the embedding of the epigraph of a lower semi-continuous function, there is a time-independent Hamiltonian flow that excises it from a noncompact symplectic manifold.

arXiv:2101.03534

Slides

On a generalisation of Lie and Courant algebroids, and its application to exceptional generalised geometry

Fridrich Valach

Imperial College London

I will introduce and discuss the notion of G-algebroids. These objects provide a common generalisation for Lie, Courant, and a special class of Leibniz algebroids used in exceptional generalised geometry (the 3 classes correspond to taking G to be the A, D, E simple groups, respectively). I will present a classification result in the exact case and provide an algebroid formulation for the recently introduced Poisson–Lie U-duality. This is a joint work with M. Bugden, O. Hulik, and D. Waldram.

arXiv:2103.01139

Slides (Plenary), Slides (Parallel)

Frobenius manifolds, mirror symmetry and integrable systems

Karoline van Gemst

University of Sheffield

Frobenius manifolds were introduced by Boris Dubrovin in the early 90’s as a means to describe 2-dimensional topological field theories in a coordinate-free way. Now, however, they arise in seemingly very distant mathematical areas and provide a bridge between them. Examples of such topics are enumerative geometry, singularity theory and integrable systems. In fact, mirror symmetry can be phrased as an isomorphism of Frobenius manifolds.

In this talk I will give a brief overview of what a Frobenius manifold is and how they are useful in the context of mirror symmetry. I will then present recent results obtained together with Andrea Brini. Lastly, I will highlight the connection between Frobenius manifolds and integrable systems, and an application of our mirror theorem in this context.

arXiv:2103.12673

Slides

Invariant generalized complex structures on flag manifolds

Carlos Augusto Bassani Varea

University of São Paulo (USP)

The aim of this talk is to describe the invariant generalized complex structures look on a maximal flag manifold in terms of a fixed root system associated to the complex semisimple Lie algebra determining the flag manifold. This is a joint work with Luiz A. B. San Martin.

Slides

Polynomial multivector fields (cancelled)

Aldo Witte

Utrecht University

We will discuss vector fields on a vector bundle $E \rightarrow M$ which are (homogeneous) polynomial with respect to the scalar multiplication. We will show that the space of such vector fields $X^p(E)_{pol}$ carries a very interesting algebraic structure, which will allow us, among others to describe elements of $X^p(E)_{pol}$ as sections of a very explicit vector bundle over $M$.

This will give us a strategy of studying normal form results in Poisson geometry: first we (abstractly) prove that a given Poisson structure is Poisson diffeomorphic to a polynomial Poisson structure, and then we give an explicit formula for the polynomial Poisson structure using the algebraic structure on $X^p(E)_{pol}$. If time permits we will discuss how to apply this to Lagrangian neighbourhood theorems, and log and elliptic symplectic structures.

Poly-Jacobi Geometry: a serendipitous discovery

Carlos Zapata-Carratala

University of Edinburgh

With motivations in the implementation of physical dimension into geometric mechanics, I will introduce the formalism of dimensioned algebra and the unit-free approach to Jacobi geometry. These will be shown to lead to a natural generalization of Jacobi/Poisson geometry where many constructions, such as products and quotients, become much more natural. Finally, I will present a breadth of structures, tentatively called Poly-Jacobi, which are natural to define within this formalism but don't seem to have been identified in the Poisson literature.

Slides