Hamiltonian mechanics | Smooth manifolds | Symplectic geometry | Differential topology
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. (Wikipedia).
Which manifolds are symplectic? - Yakov Eliashberg
Members’ Colloquium Topic: Which manifolds are symplectic? Speaker: Yakov Eliashberg Affiliation: Stanford University; Member, School of Mathematics Date: November 08, 2021 The question in the title was one of the founding questions in symplectic topology 40 years ago, and despite a lot
From playlist Mathematics
Uniqueness aspects of symplectic fillings - Zhengyi Zhou
Short talks by postdoctoral members Topic: Uniqueness aspects of symplectic fillings Speaker: Zhengyi Zhou Affiliation: Member, School of Mathematics Date: October 4, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Exploring Symplectic Embeddings and Symplectic Capacities
Speakers o Alex Gajewski o Eli Goldin o Jakwanul Safin o Junhui Zhang Project Leader: Kyler Siegel Abstract: Given a domain (e.g. a ball) in Euclidean space, we can ask what is its volume. We can also ask when one domain can be embedded into another one without distorting volumes. These
From playlist 2019 Summer REU Presentations
Symplectic topology of open manifolds - Laurent Côté
Short Talks by Postdoctoral Members Topic: Symplectic topology of open manifolds Speaker: Laurent Côté Affiliation: Member, School of Mathematics Date: September 22, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
What is a Manifold? Lesson 6: Topological Manifolds
Topological manifolds! Finally! I had two false starts with this lesson, but now it is fine, I think.
From playlist What is a Manifold?
Microlocal category for a closed symplectic manifold II - Dmitry Tamarkin
Dmitry Tamarkin Northwestern May 11, 2011 For more videos, visit http://video.ias.edu
From playlist Mathematics
Symplectic topology and the loop space - Jingyu Zhao
Topic: Symplectic topology and the loop space Speaker: Jingyu Zhao, Member, School of Mathematics Time/Room: 4:45pm - 5:00pm/S-101 More videos on http://video.ias.edu
From playlist Mathematics
What is a Manifold? Lesson 2: Elementary Definitions
This lesson covers the basic definitions used in topology to describe subsets of topological spaces.
From playlist What is a Manifold?
Brent Pym: Holomorphic Poisson structures - lecture 3
The notion of a Poisson manifold originated in mathematical physics, where it is used to describe the equations of motion of classical mechanical systems, but it is nowadays connected with many different parts of mathematics. A key feature of any Poisson manifold is that it carries a cano
From playlist Virtual Conference
Symplectic Dynamics of Integrable Hamiltonian Systems - Alvaro Pelayo
Alvaro Pelayo Member, School of Mathematics April 4, 2011 I will start with a review the basic notions of Hamiltonian/symplectic vector field and of Hamiltonian/symplectic group action, and the classical structure theorems of Kostant, Atiyah, Guillemin-Sternberg and Delzant on Hamiltonian
From playlist Mathematics
Symplectic fillings and star surgery - Laura Starkston
Laura Starkston University of Texas, Austin September 25, 2014 Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theor
From playlist Mathematics
Stability conditions in symplectic topology – Ivan Smith – ICM2018
Geometry Invited Lecture 5.8 Stability conditions in symplectic topology Ivan Smith Abstract: We discuss potential (largely speculative) applications of Bridgeland’s theory of stability conditions to symplectic mapping class groups. ICM 2018 – International Congress of Mathematicians
From playlist Geometry
Lectures on Homological Mirror Symmetry II - Sheridan Nick
Lectures on Homological Mirror Symmetry Sheridan Nick Institute for Advanced Study; Member, School of Mathematics November 4, 2013
From playlist Mathematics
Isocontact and isosymplectic immersions and embeddings by Mahuya Datta
J-Holomorphic Curves and Gromov-Witten Invariants DATE:25 December 2017 to 04 January 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex stru
From playlist J-Holomorphic Curves and Gromov-Witten Invariants
Act globally, compute...points and localization - Tara Holm
Tara Holm Cornell University; von Neumann Fellow, School of Mathematics October 20, 2014 Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing inte
From playlist Mathematics
How to Construct Topological Invariants via Decompositions and the Symplectic Category - Wehrheim
Katrin Wehrheim Massachusetts Institute of Technology; Institute for Advanced Study October 17, 2011 A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was introduced by Weinstein in an
From playlist Mathematics
Symplectic geometry of hyperbolic cylinders and their homoclinic intersections - Jean-Pierre Marco
Emerging Topics Working Group Topic: Symplectic geometry of hyperbolic cylinders and their homoclinic intersections Speaker: Jean-Pierre Marco Affiliation: Pierre and Marie Curie University Date: April 9, 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Frédéric BOURGEOIS - A symplectic invariant for contact manifolds
The construction of S^1-equivariant symplectic homology with Alexandru Oancea can be used to define an invariant for a wide class of contact manifolds. This is a substitute for cylindrical contact homology, which often has transversality issues. This symplectic invariant can then be applie
From playlist 2015 Summer School on Moduli Problems in Symplectic Geometry
How to Find Periodic Orbits and Exotic Symplectic Manifolds - Mark McLean
Mark McLean Massachusetts Institute of Technology; Member, School of Mathematics October 15, 2012 I will give an introduction to symplectic geometry and Hamiltonian systems and then introduce an invariant called symplectic cohomology. This has many applications in symplectic geometry and
From playlist Mathematics