#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

I learned from Jon Brett about physicist David #Bohm 's colleague Basil Hiley https://en.wikipedia.org/wiki/Basil_Hiley Their distinction between implicate and explicate order seems to appear in my study of #orthogonal #Sheffer #polynomial s. The Sheffer constraint of exponentiality yields an implicate order of #partition of a set https://www.math4wisdom.com/wiki/Exposition/20221122SpaceBuilders whereas orthogonality yields 5 possible explicate orders upon #measurement. Also curious how they use #CliffordAlgebra ,real and #symplectic. #BottPeriodicity ?

'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

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

#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!

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.

#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! 🙂

#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

#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.

#ExplainingMyResearch 1
Broadly speaking, I study #Poisson and #symplectic geometry, which, at their core, is the geometry of mechanics and #dynamics. A physical system is described a number of positional and momentum degrees of freedom, which together form an even-dimensional phase space equipped with a Poisson bracket, an operation on functions/physical quantities. This Poisson bracket on the phase space encodes the dynamic behaviour of the system.