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TALKS
Daniel Álvarez
ABELIANIZATION OF PGL_2 CONNECTIONS AND THE GOLDMAN FORMULA
We revisit the relationship between flat framed PGL_2 bundles over Riemann surfaces and twisted flat line bundles over suitable branched double covers, which give rise to Fock-Goncharov cluster coordinates. Using the language of quasi-Poisson manifolds, we obtain an intrinsic description of the underlying Poisson structures and the corresponding Goldman formula explains the fact that the abelianization map is Poisson. This observation enhances the recent categorified picture of abelianization proposed by N. Nikolaev to the Poisson setting.
Tom Ariel
OBSTRUCTIONS TO SOLVING THE MAURER-CARTAN EQUATIONS, OR - WHEN CAN SOMETHING NONINTEGRABLE BE DEFORMED INTO SOMETHING INTEGRABLE?
In geometry, most deformation problems (complex, symplectic, Poisson…) are governed by differential graded Lie algebras (DGLA’s). A curved DGLA is a generalization of a DGLA, which corresponds geometrically to deformation problems where the initial object is not integrable (almost complex, non-Poisson bivector, non-closed 2-form...).
I will present some obstructions to solving the Maurer-Cartan equation in curved DGLA's. As an application, we will consider the problem of finding a Dirac complement to a given Dirac structure.
Alessandro Arsie
Paula Balseiro
ON CONSERVED VOLUMES FOR NONHOLONOMIC SYSTEMS WITH SYMMETRIES
One of the principal differences between Hamiltonian systems—described on symplectic manifolds—and nonholonomic systems is the preservation of an invariant volume for the dynamics. However, certain examples of nonholonomic systems happen to admit an invariant measure. In this talk, we consider nonholonomic systems reduced by the action of a Lie group of symmetries and, under suitable assumptions, we explore the existence of an invariant volume for the underlying geometric structure.
Glauber Barbosa
DIRAC REDUCTION OF SPACES OF CONNECTIONS ON THE CIRCLE WITH SEVERAL MARKED POINTS
In this talk, we apply the infinite-dimensional Dirac reduction developed by Cabrera–Gualtieri–Meinrenken to the space of connections on the circle, considering the quotient by gauge transformations that are trivial at n marked points. This reduction yields a finite-dimensional Dirac structure E[n].
The Dirac structure E[n] is integrated by a quasi-symplectic groupoid obtained by reducing the space of connections on a cylinder with (n,n) marked points on its two boundary components, while the corresponding Morita equivalences are obtained by reducing one with (n,m) marked points. As a consequence, we establish a correspondence between the Hamiltonian spaces associated with the Dirac structures E[n] and E[m].
Finally, we discuss how the infinite-dimensional geometry of the loop group is recovered in the limit as the marked points increasingly cover the circle.
Francis Bischoff
DISCRIMINANT GROUPOIDS OF REFLECTION GROUPS
The discriminant of a finite reflection group is an early example of a Saito free divisor which is not normal crossing. Consequently, the vector fields which are tangent to the discriminant define a Lie algebroid. In this talk, I will discuss a family of algebraic Lie groupoids (and categories) which integrate these Lie algebroids.
Matias del Hoyo
EFFECTIVE STACKS AND THE GLOBAL QUOTIENT THEOREM
A classical theorem shows that an effective orbifold can always be presented as the global quotient of a manifold by a locally free action of a compact group. Within the framework of Lie groupoids and differentiable stacks, orbifolds correspond to proper étale groupoids, and the question of whether the theorem holds for non-étale groupoids naturally arises. In this talk, we will discuss effectiveness for Lie groupoids and differentiable stacks, building on the linearization of proper groupoids and representations up to homotopy, and present a global quotient theorem for effective stacks beyond the étale case.
Lino Grama
SYZ FOR ALMOST ABELIAN LIE GROUPS
In this talk, we discuss SYZ mirror symmetry in the non-Kähler setting, in the sense of Lau–Tseng–Yau. We construct SYZ dual pairs for a family of solvmanifolds arising from almost abelian Lie groups and explore applications to dualities between Bott–Chern and Tseng–Yau cohomologies. This is joint work with L. Cavenaghi, L. Katzarkov, and P. Muniz.
Aline Leite
MODULI SPACE OF (FLAT) MULTIPLICATIVE EHRESMANN CONNECTIONS
A classical result due to Atiyah and Bott establishes that the moduli space of flat principal bundle connections admits a symplectic structure under certain conditions. The notion of a principal connection was generalized to groupoids, first by Laurent-Gengoux, Stiénon, and Xu, and later by Fernandes and Mărcut, , through what they called multiplicative Ehresmann connections (MECs).
In this work, we take the first steps toward defining a Dirac structure on a moduli space of MECs such that, under a certain condition, the moduli space of flat MECs is one of the pre-symplectic leaves of the Dirac structure. This is a generalization of the classical symplectic structure on the moduli space of flat principal bundle connections.
In this talk, we will present the definition of this Dirac structure and see examples of it for certain groupoids.
Igor Mencattini
INTEGRABILITY AND POISSON QUASI-NIJENHUIS STRUCTURES
In this talk I will present the advances made during the last few years in the attempt to describe classical integrable systems in the framework of the Poisson quasi-Nijenhuis structures.
Andrei Mikhailov
SUPERSYMMETRY BICOMPLEX OF PURE SPINOR SUPERGRAVITY IN ADS BACKGROUND
Consider the linear superspace of infinitesimal deformations of the maximally supersymmetric Yang-Mills theory. It is a representation of the Lie superalgebra of supersymmetries. The structure of this representatiom is not well understood. We can obtain information about its cohomologies from its interpretation as cohomologies of Q-manifold, which arises from AdS/CFT correspondence and the pure spinor formalism.
Simone Noja
HOLOMORPHIC ODD SYMPLECTIC SUPERMANIFOLDS
A smooth odd symplectic supermanifold is a smooth supermanifold endowed with a closed, nondegenerate 2-form of odd parity; such spaces provide the geometric foundation of the Batalin–Vilkovisky formalism. While every smooth odd symplectic supermanifold is split, complex supermanifolds need not be, and it is therefore natural to ask whether the holomorphic symplectic condition still forces splitness. In this talk, we introduce holomorphic odd symplectic supermanifolds and study their classification. We show that their associated graded model is of the form ΠT∗X and we describe their global geometry in terms of Hamiltonian torsors. In dimensions 3∣3 and 4∣4, the classification reduces to H1(X,∧3TX), yielding explicit non-split examples, while higher dimensions exhibit a genuine obstruction theory. This is work in progress.
Andrés Rodríguez
NIJENHUIS STRUCTURES ON DIFFERENTIABLE STACKS
We frame quasi-Nijenhuis structures on a Lie groupoid as Maurer-Cartan elements of a graded Lie 2-algebra governed by the Frölicher-Nijenhuis bracket. We then outline how a Morita equivalence of Lie groupoids induces a quasi-isomorphism between their respective Frölicher-Nijenhuis 2-algebras. This quasi-isomorphism establishes a natural bijection between the gauge equivalence classes of their Maurer-Cartan elements. Consequently, quasi-Nijenhuis structures transfer naturally under Morita equivalence. Up to gauge equivalence, this transfer allows us to define a shifted Nijenhuis structure on the associated differentiable stack. Finally, we demonstrate that this framework realizes holomorphic stacks as the geometric integrations of generalized complex manifolds.
Nikolay Sheshko
DERIVED SYMPLECTIC REDUCTION VIA DG-GROUPOIDS
Symplectic reduction is a mechanism for reducing the dimension of the phase space of a physical system with symmetries. In its classical form it assumes that the level set of the moment map is a smooth submanifold on which the group acts freely. I will describe a derived version of symplectic reduction that works in the most general case, when reduction happens at a non-regular value and the group action has non-discrete stabilizers. In particular, I will model the symplectic quotient by the action dg-groupoid over the derived zero locus of the moment map, following Carchedi and Behrend–Liao–Xu. I will then show how to construct the reduced symplectic form inside the triple Bott–Shulman–Stasheff complex of the dg-groupoid and prove that this form is 0-shifted symplectic. Time permitting, I will discuss the relation to the classical BRST complex and to recent work of Cueca, Dorsch, Sjamaar and Zhu.
Alex Takeda
Fabricio Valencia
POISSON STRUCTURES ON QUADRATIC LIE ALGEBRA
A Lie algebra is called quadratic if it admits a symmetric, invariant, nondegenerate bilinear form. Classical examples include Lie algebras of compact Lie groups, semisimple Lie algebras, and cotangent Lie algebras of Lie groups. In this talk, I will discuss Poisson structures on quadratic Lie algebras that are compatible with the underlying quadratic structure. These structures can be characterized in terms of skew-symmetric solutions of the classical Yang-Baxter equation. I will introduce a generalized double extension of quadratic Lie algebras, which provides a systematic method for constructing such Poisson structures. As applications, I will describe examples on oscillator Lie algebras and on the quadratic Lie algebras associated with five-dimensional Wess-Zumino-Witten models. If time permits, I will also briefly discuss how the construction extends to Jacobi structures on quadratic Lie algebras.This is joint work with Alessandro Arsie.
POSTERS
Danilo Arcanjo
ON THE RELATION BETWEEN REGULAR COURANT AND QUADRATIC LIE ALGEBROIDS
This work investigates the relationship between regular Courant algebroids and quadratic Lie algebroids. A regular Courant algebroid $E$ naturally induces a quadratic Lie algebroid, called the associated ample Lie algebroid, obtained by taking the quotient of $E$ by the orthogonal complement of the kernel of its anchor. From this construction, the inverse problem arises naturally: given a quadratic Lie algebroid $A$, under what conditions does there exist a regular Courant algebroid whose associated ample Lie algebroid is isomorphic to $A$? It turns out that there is an obstruction given by the first Pontryagin class of a quadratic Lie algebroid and a quadratic Lie algebroid is isomorphic to the ample Lie algebroid of some regular Courant algebroid if and only if its first Pontryagin class vanishes. The results presented here were originally established by Chen, Stiénon, and Xu (2013) in the context of the classification of regular Courant algebroids.
Rafael Hernandez Bolaños
DIFFERENTIABLE STACKS VIA LOCALIZATION
Localization provides a way of constructing stacks by formally inverting weak equivalences between groupoids. In the usual approach, this leads to a category of stacks after collapsing isomorphic maps, for instance via generalized maps or bibundles. We propose instead to perform this construction directly at the level of bicategories, retaining the 2-morphisms between maps. Following Pronk’s bicategory of fractions, and subsequent simplifications by Roberts and Tommasini, we study the localization of Lie groupoids at weak equivalences. The differentiable setting allows for a particularly concrete form of the axioms and of the resulting calculus of fractions. The goal is to understand the bicategory of differentiable stacks from this point of view.
André Tanure
ON THE SPACE OF SIMPLICIAL SECTIONS OF A SIMPLICIAL VECTOR BUNDLE
In this work we define simplicial sections of simplicial vector bundles, and show that their set has a structure of simplicial vector space. An important fact obtained is that when the simplicial vector bundle is a higher VB n-groupoid, the space of sections is also a n-groupoid. We also introduce the simplicial vector space of projectable sections and show that it can be obtained as a pullback. Moreover, we start treating the Morita invariance of this space of sections, obtaining a partial result that in the case of linear hypercovers of higher VB-groupoids over the same base, the induced map between the spaces of sections is a hypercover of simplicial vector spaces. This work is part of the speaker's doctoral thesis, supervised by Prof. Cristian Ortiz.
Tianhao Ye
ONE-SHIFTED COISOTROPIC STRUCTURES
Abstract: We introduce a notion of 1-shifted coisotropic morphism from Lie groupoids to quasi-Poisson groupoids via Courant morphisms. This notion encompasses coisotropic subgroupoids, Hamiltonian G-spaces as well as 1-shifted coisotropic structures into quasi-symplectic groupoids in the sense of M.~Mayrand. We study several properties of these objects, including Morita invariance, symplectic realizations, transverse coisotropic intersections, Lie theory, and various other characterizations via the coisotropic condition, Maurer-Cartan elements, tangent complexes, and the vanishing ideal.
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