Charlotte Kirchhoff-Lukat<p><span class="h-card" translate="no"><a href="https://mathstodon.xyz/@MotivicKyle" class="u-url mention">@<span>MotivicKyle</span></a></span> The history of the <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> <a href="https://mathstodon.xyz/tags/category" class="mention hashtag" rel="tag">#<span>category</span></a> in this form, the Wehrheim-Woodward category, is roughly as follows:<br />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.")<br />This, of course, does not work due to smooth Lagrangian correspondences not always composing smoothly. <br />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. <br />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!</p>
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Charlotte Kirchhoff-Lukat<p><span class="h-card" translate="no"><a href="https://mathstodon.xyz/@MotivicKyle" class="u-url mention">@<span>MotivicKyle</span></a></span> Correspondences as morphisms are a thing in <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> geometry; Lagrangian correspondences. You can use these to construct a category of smooth symplectic manifolds (of arbitrary dimension). <br />However, they wouldn't be submersions on each side, merely so-called reductions -- so may be more general than what you want. <br />I like <a href="http://arxiv.org/abs/1012.0105" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">http://</span><span class="">arxiv.org/abs/1012.0105</span><span class="invisible"></span></a> as an introduction; it includes a little treatment of the smooth case without symplectic structure, too.</p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/TodaysMath" class="mention hashtag" rel="tag">#<span>TodaysMath</span></a> 1/2: I am thinking and reading about the <a href="https://mathstodon.xyz/tags/category" class="mention hashtag" rel="tag">#<span>category</span></a> of <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> manifolds with (equivalence classes of sequences of) Lagrangian relations as morphisms; originally proposed by Weinstein and formalised by Wehrheim and Woodward.<br />Here's a nice paper by Weinstein that both explains the WW construction and gives a nice description of the morphisms: <br /><a href="https://arxiv.org/abs/1012.0105" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/1012.0105</span><span class="invisible"></span></a> <br /><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a></p>
Andrius (Math 4 Wisdom)<p>I learned from Jon Brett about physicist David <a href="https://mathstodon.xyz/tags/Bohm" class="mention hashtag" rel="tag">#<span>Bohm</span></a> 's colleague Basil Hiley <a href="https://en.wikipedia.org/wiki/Basil_Hiley" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Basil_Hi</span><span class="invisible">ley</span></a> Their distinction between implicate and explicate order seems to appear in my study of <a href="https://mathstodon.xyz/tags/orthogonal" class="mention hashtag" rel="tag">#<span>orthogonal</span></a> <a href="https://mathstodon.xyz/tags/Sheffer" class="mention hashtag" rel="tag">#<span>Sheffer</span></a> <a href="https://mathstodon.xyz/tags/polynomial" class="mention hashtag" rel="tag">#<span>polynomial</span></a> s. The Sheffer constraint of exponentiality yields an implicate order of <a href="https://mathstodon.xyz/tags/partition" class="mention hashtag" rel="tag">#<span>partition</span></a> of a set <a href="https://www.math4wisdom.com/wiki/Exposition/20221122SpaceBuilders" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">math4wisdom.com/wiki/Expositio</span><span class="invisible">n/20221122SpaceBuilders</span></a> whereas orthogonality yields 5 possible explicate orders upon <a href="https://mathstodon.xyz/tags/measurement" class="mention hashtag" rel="tag">#<span>measurement</span></a>. Also curious how they use <a href="https://mathstodon.xyz/tags/CliffordAlgebra" class="mention hashtag" rel="tag">#<span>CliffordAlgebra</span></a> ,real and <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a>. <a href="https://mathstodon.xyz/tags/BottPeriodicity" class="mention hashtag" rel="tag">#<span>BottPeriodicity</span></a> ?</p>
JMLR<p>'Discrete Variational Calculus for Accelerated Optimization', by Cédric M. Campos, Alejandro Mahillo, David Martín de Diego.</p><p><a href="http://jmlr.org/papers/v24/21-1323.html" rel="nofollow noopener noreferrer" target="_blank"><span class="invisible">http://</span><span class="ellipsis">jmlr.org/papers/v24/21-1323.ht</span><span class="invisible">ml</span></a> <br> <br><a href="https://sigmoid.social/tags/variational" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>variational</span></a> <a href="https://sigmoid.social/tags/symplectic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>symplectic</span></a> <a href="https://sigmoid.social/tags/optimization" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>optimization</span></a></p>
Charlotte Kirchhoff-Lukat<p>I am returned from my winter holiday (including a holiday away from social media)! <br /><a href="https://mathstodon.xyz/tags/TodaysMath" class="mention hashtag" rel="tag">#<span>TodaysMath</span></a> 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. <br />They are learning about Hamiltonian mechanics and the underlying geometric structures - symplectic structures/ Poisson brackets - for the first time. <br /><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="tag">#<span>geometry</span></a> <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> <a href="https://mathstodon.xyz/tags/Mechanics" class="mention hashtag" rel="tag">#<span>Mechanics</span></a></p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/TodaysMath" class="mention hashtag" rel="tag">#<span>TodaysMath</span></a> is still the <a href="https://mathstodon.xyz/tags/FukayaCategory" class="mention hashtag" rel="tag">#<span>FukayaCategory</span></a> for log <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> surfaces.<br />Log symplectic structures on oriented closed surfaces are one of the relatively few classes of <a href="https://mathstodon.xyz/tags/Poisson" class="mention hashtag" rel="tag">#<span>Poisson</span></a> <a href="https://mathstodon.xyz/tags/manifold" class="mention hashtag" rel="tag">#<span>manifold</span></a> that are fully classified: <br /><a href="https://arxiv.org/abs/math/0110304" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/math/0110304</span><span class="invisible"></span></a><br />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).<br />The paper is self-contained and readable with few prerequisites, have a look!</p>
Charlotte Kirchhoff-Lukat<p>So, <a href="https://mathstodon.xyz/tags/TodaysMath" class="mention hashtag" rel="tag">#<span>TodaysMath</span></a> is figuring out the higher operation in the <a href="https://mathstodon.xyz/tags/FukayaCategory" class="mention hashtag" rel="tag">#<span>FukayaCategory</span></a> for a real log <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> 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.<br /><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="tag">#<span>geometry</span></a></p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/ExplainingMyResearch" class="mention hashtag" rel="tag">#<span>ExplainingMyResearch</span></a> 23<br />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.<br />My <a href="https://mathstodon.xyz/tags/preprint" class="mention hashtag" rel="tag">#<span>preprint</span></a> <br /><a href="https://arxiv.org/abs/2207.06894" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/2207.06894</span><span class="invisible"></span></a><br />describes how to define <a href="https://mathstodon.xyz/tags/Floer" class="mention hashtag" rel="tag">#<span>Floer</span></a> <a href="https://mathstodon.xyz/tags/cohomology" class="mention hashtag" rel="tag">#<span>cohomology</span></a> for so-called log-<a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> surfaces. I am currently finishing off the full description of the category of branes in this setting, so keep your eyes open! 🙂</p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/Symplectic" class="mention hashtag" rel="tag">#<span>Symplectic</span></a>/ <a href="https://mathstodon.xyz/tags/Fukaya" class="mention hashtag" rel="tag">#<span>Fukaya</span></a> <a href="https://mathstodon.xyz/tags/category" class="mention hashtag" rel="tag">#<span>category</span></a> /#MirrorSymmetry people, if you are out there: What is your favourite introduction/overview of the <a href="https://mathstodon.xyz/tags/HomologicalAlgebra" class="mention hashtag" rel="tag">#<span>HomologicalAlgebra</span></a> of \(A_{\infty}\) -categories, specifically with regards to embedding them in Twisted Complexes, sets of (split) generators etc? <br />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.<br /><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="tag">#<span>CategoryTheory</span></a></p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/ExplainingMyResearch" class="mention hashtag" rel="tag">#<span>ExplainingMyResearch</span></a> 2<br />But <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> and <a href="https://mathstodon.xyz/tags/Poisson" class="mention hashtag" rel="tag">#<span>Poisson</span></a> structures appear in many contexts beyond classical mechanics: <a href="https://mathstodon.xyz/tags/Quantization" class="mention hashtag" rel="tag">#<span>Quantization</span></a>, <a href="https://mathstodon.xyz/tags/stringtheory" class="mention hashtag" rel="tag">#<span>stringtheory</span></a> and the study of symmetries (via <a href="https://mathstodon.xyz/tags/Lie" class="mention hashtag" rel="tag">#<span>Lie</span></a> groups) come to mind.</p>
Charlotte Kirchhoff-Lukat<p><a href="https://mathstodon.xyz/tags/ExplainingMyResearch" class="mention hashtag" rel="tag">#<span>ExplainingMyResearch</span></a> 1 <br />Broadly speaking, I study <a href="https://mathstodon.xyz/tags/Poisson" class="mention hashtag" rel="tag">#<span>Poisson</span></a> and <a href="https://mathstodon.xyz/tags/symplectic" class="mention hashtag" rel="tag">#<span>symplectic</span></a> geometry, which, at their core, is the geometry of mechanics and <a href="https://mathstodon.xyz/tags/dynamics" class="mention hashtag" rel="tag">#<span>dynamics</span></a>. 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.</p>
Refurio Anachro<p>Before we get back to <a href="https://mastodon.cloud/tags/symplectic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>symplectic</span></a> geometry let's have some fun with alternate ways to do classical mechanics!</p><p><a href="https://mastodon.cloud/tags/altmechanics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>altMechanics</span></a> – I mentioned Lagrangian and Hamiltonian mechanics earlier. Lagrange's is often simpler, but it cannot handle cyclic coordinates. Meet Routhian mechanics! Routh found out that you can cherry-pick momenta or velocities as your generalized coordinates to your delight.</p><p><a href="https://en.wikipedia.org/wiki/Routhian_mechanics" rel="nofollow noopener noreferrer" target="_blank"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Routhian</span><span class="invisible">_mechanics</span></a></p><p>1/</p>
Refurio Anachro<p>SYMPLECTIC GEOMETRY? – Over the next few days, I'll be developing an introduction to <a href="https://mastodon.cloud/tags/symplectic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>symplectic</span></a> <a href="https://mastodon.cloud/tags/geometry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>geometry</span></a> here. Don't be afraid, we will be taking the easy route...</p><p><a href="https://mastodon.cloud/tags/symplecticgeometry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>symplecticGeometry</span></a> is the geometry of <a href="https://mastodon.cloud/tags/phasespace" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>phaseSpace</span></a>!</p><p>1/</p>
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