Show threadCharlotte Kirchhoff-LukatMar 7, 2023
@MotivicKyle The history of the #symplectic #category in this form, the Wehrheim-Woodward category, is roughly as follows:
People wanted to construct a symplectic category with more general morphisms than symplectomorphisms (which only exist between diffeomorphic manifolds). The idea of using Lagrangian correspondences is due to Weinstein and his observation that symplectomorphisms are examples. (Weinstein's philosophy: "Everything is a Lagrangian.")
This, of course, does not work due to smooth Lagrangian correspondences not always composing smoothly.
The Wehrheim-Woodward construction avoids this transversality issue entirely by taking as morphisms paths of relations modulo composing individual relations in the sequence that can be smoothly composed -- as far as I am aware, nobody has found a way of making a nice category with only smoothly composable relations.
But now this note by Weinstein shows that, actually, every morphism in the WW category has a representative path consisting of *at most* two non-composable Lagrangian relations, resulting in the span picture. I was somehow surprised by that!
Show threadCharlotte Kirchhoff-LukatMar 6, 2023
@MotivicKyle Correspondences as morphisms are a thing in #symplectic geometry; Lagrangian correspondences. You can use these to construct a category of smooth symplectic manifolds (of arbitrary dimension).
However, they wouldn't be submersions on each side, merely so-called reductions -- so may be more general than what you want.
I like http://arxiv.org/abs/1012.0105 as an introduction; it includes a little treatment of the smooth case without symplectic structure, too.
A note on the Wehrheim-Woodward category

Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.

arXiv.org
Charlotte Kirchhoff-LukatMar 3, 2023
#TodaysMath 1/2: I am thinking and reading about the #category of #symplectic manifolds with (equivalence classes of sequences of) Lagrangian relations as morphisms; originally proposed by Weinstein and formalised by Wehrheim and Woodward.
Here's a nice paper by Weinstein that both explains the WW construction and gives a nice description of the morphisms:
https://arxiv.org/abs/1012.0105
#math
A note on the Wehrheim-Woodward category

Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.

arXiv.org
JMLRFeb 1, 2023

'Discrete Variational Calculus for Accelerated Optimization', by Cédric M. Campos, Alejandro Mahillo, David Martín de Diego.

http://jmlr.org/papers/v24/21-1323.html

#variational #symplectic #optimization

Discrete Variational Calculus for Accelerated Optimization

Charlotte Kirchhoff-LukatJan 13, 2023
I am returned from my winter holiday (including a holiday away from social media)!
#TodaysMath is mostly very elemental stuff: I am helping two undergraduates read Marsden & Ratiu's "Introduction to Mechanics and Symmetry", accompanied by the lecture notes "Geometry and Mechanics by Mehta.
They are learning about Hamiltonian mechanics and the underlying geometric structures - symplectic structures/ Poisson brackets - for the first time.
#math #geometry #symplectic #Mechanics
Charlotte Kirchhoff-LukatDec 21, 2022
#TodaysMath is still the #FukayaCategory for log #symplectic surfaces.
Log symplectic structures on oriented closed surfaces are one of the relatively few classes of #Poisson #manifold that are fully classified:
https://arxiv.org/abs/math/0110304
The classification was done by Olga Radko in this paper, where they are called topologically stable Poisson structures (since their degeneracy locus is stable under small perturbation).
The paper is self-contained and readable with few prerequisites, have a look!
A classification of topologically stable Poisson structures on a compact oriented surface

Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. We construct a complete set of invariants classifying these structures up to an orientation-preserving Poisson isomorphism. We show that there is a set of non-trivial infinitesimal deformations which generate the second Poisson cohomology and such that each of the deformations changes exactly one of the classifying invariants. As an example, we consider Poisson structures on the sphere which vanish linearly on a set of smooth closed disjoint curves.

arXiv.org
Show threadCharlotte Kirchhoff-LukatDec 16, 2022
So, #TodaysMath is figuring out the higher operation in the #FukayaCategory for a real log #symplectic surface, meaning a surface with a particular "nice" singularity on a collection of embedded circles. These circles divide the surface into multiple symplectic components, but the components are not all separate! They interact with each other in the Fukaya category.
#math #geometry
Show threadCharlotte Kirchhoff-LukatDec 9, 2022
#ExplainingMyResearch 23
To start, I am working on 2-dimensional "universes" like the disc with boundary, which are not strictly speaking GC, but similar enough to be useful.
My #preprint
https://arxiv.org/abs/2207.06894
describes how to define #Floer #cohomology for so-called log-#symplectic surfaces. I am currently finishing off the full description of the category of branes in this setting, so keep your eyes open! 🙂
Log Floer cohomology for oriented log symplectic surfaces

This article provides the first extension of Lagrangian Intersection Floer cohomology to Poisson structures which are almost everywhere symplectic, but degenerate on a lowerdimensional submanifold. The main result of the article is the definition of Lagrangian intersection Floer cohomology, referred to as log Floer cohomology, for orientable surfaces equipped with log symplectic structures. We show that this cohomology is invariant under suitable isotopies and that it is isomorphic to the log de Rham cohomology when computed for a single Lagrangian.

arXiv.org
Charlotte Kirchhoff-LukatDec 9, 2022
#Symplectic/ #Fukaya #category /#MirrorSymmetry people, if you are out there: What is your favourite introduction/overview of the #HomologicalAlgebra of \(A_{\infty}\) -categories, specifically with regards to embedding them in Twisted Complexes, sets of (split) generators etc?
I've been working with all of these concepts for a while, but bee muddling through a lot. Sources are either fairly inexplicit or a 300-page book on homological algebra.
#math #CategoryTheory
Show threadCharlotte Kirchhoff-LukatNov 30, 2022
#ExplainingMyResearch 2
But #symplectic and #Poisson structures appear in many contexts beyond classical mechanics: #Quantization, #stringtheory and the study of symmetries (via #Lie groups) come to mind.