Watch back the talks by Arhtemy Kiselev and Giovanni von Marion below:
Title: Universal deformations of Poisson brackets and a mechanism of smooth structure for space-time. Speaker: Arthemy Kiselev (Bernoulli institute) Abstract: Poisson brackets --not necessarily "symplectic", i.e. not necessarily referring to the canonical conjugate "coordinates" and equally many "momenta"-- are ways to process energy into motion. The given store of energy is often fixed yet the ways to spend it can be negotiated, which results in the infinitesimal deformation problem: "How can Poisson brackets be changed a little bit, modulo their tensor behaviour under coordinate reparametrisations, in such a way that they stay Poisson?" Kontsevich's combinatorial language of (un)directed graphs allows the construction, itself related to many other domains of mathematics and quantum physics, of such deformations; they are universal for all Poisson brackets on all affine manifolds. For example, infinitely many such flows --on the spaces of Poisson bi- vectors, themselves hard to describe-- are obtained from the countable set of generators for the Lie algebra of Drinfeld's mysterious Grothendieck--Teichmueller Lie group. Whether a Kontsevich's universal deformation is Poisson-trivial in a given geometry, thus amounting to non-linear, smooth reparametrisations of local coordinates, is a hard open problem. In those Poisson geometries where the Kontsevich graph flows appear to be trivial, they instead start to generate a smooth structure on the underlying affine manifold (where coordinate changes were very simple at the `quantum' level). Independently, we learn that many natural classes of highly-nonlinear Poisson brackets are stable under Kontsevich's deformations, again leading to unexpected facts and nontrivial combinatorial, topological, and geometric open problems about the dynamics of Poisson structures. This talk, based on the joint work in progress with Ricardo Buring, will serve to introduce key notions and show basic examples.
Arthemy's extended lectures on this subject can be found here:
References:
[1] Buring R., Kiselev A. V. (2020) Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms, Physics of Particles and Nuclei Letters 17:5 Supersymmetry and Quantum Symmetries'2019, 707--713. arXiv:1912.12664 [math.SG] [2] Kiselev A. V., Buring R. (2021) The Kontsevich graph orientation morphism revisited, Banach Center Publ. 123 Homotopy algebras, deformation theory & quantization, 123--139. arXiv:1904.13293 [math.CO]
Speaker: Giovanni van Marion (Bernoulli/VSI)
Title: Applying Transition State Theory to Sphaleron Transitions
Abstract: Classical and quantum transition state theory (CTST and QTST) explains reactionrates in chemistry. The systems studied are those in which a molecule must pass through an intermediate excited state, related to a saddle point of the Hamiltonian in phase space, to be able to react. I present an approach to both kinds of TST based on expanding the Hamiltonian in normal form coordinates. On the other hand, classical and quantum field theories have similar kinds of processes to which these techniques might also be applicable. In our research, we study the dynamics related to so-called sphalerons using TST in a simplified model. In reality, the electroweak sector of the standard model of particlephysics contains such a sphaleron, which might be an ingredient in explaining the observable universe’s matter-antimatter asymmetry.