Show threadCharlotte Kirchhoff-LukatMar 3, 2023
#TodaysMath 2/2: The appropriate #Floer #cohomology theory to associate to (composable) Lagrangian correspondences is *quilted Floer cohomology*. I like this paper by Wehrheim and Woodward to learn about it:
https://arxiv.org/abs/0905.1368
Floer Cohomology and Geometric Composition of Lagrangian Correspondences

We prove an isomorphism of Floer cohomologies under geometric composition of Lagrangian correspondences in exact and monotone settings.

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
Show threadCharlotte Kirchhoff-LukatFeb 9, 2023
This is #TodaysMath, I guess. I've mostly been working on job applications this week, so no interesting #math developments to report...
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
Show threadCharlotte Kirchhoff-LukatDec 22, 2022
#TodaysMath
I am currently learning the hard way that the #FukayaCategory of even very simple objects (still thinking about these surfaces with singular symplectic structures) can have lots of higher \(A_{\infty}\)-operations that are very easy to overlook! Counting polygons is surprisingly hard when they have weird shapes. ๐Ÿ™ƒ
Charlotte Kirchhoff-LukatDec 22, 2022
For #TodaysMath, I am thinking about \(A_{\infty}\)-algebras on a small number of generators and \(A_{\infty}\)-categories with a small number of objects, and their equivalences.
Reading https://arxiv.org/pdf/1910.01096.pdf by Jack Smith to learn more.
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
#Todaysmath
This happens because there are holomorphic discs/polygons traversing the singularity, as long as Lagrangians share at least one intersection point inside the singular circle.
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
Charlotte Kirchhoff-LukatDec 16, 2022
Since I turned #ExplainingMyResearch into a fairly general rumination on my larger #research area, I'm starting another more specialist hashtag for what I'm actually doing on a more day-to-day basis: #TodaysMath
(I will not post for it every day, nor will I abandon the more general one.)
#math #geometry