Mathematical principles | Partial differential equations
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric C1 embedding theorem and the Smale–Hirsch immersion theorem. (Wikipedia).
Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t
From playlist Introduction to Homotopy Theory
Introduction to Homotopy Theory- Part 5- Transition to Abstract Homotopy Theory
Credits: nLab: https://ncatlab.org/nlab/show/Introdu... Animation library: https://github.com/3b1b/manim Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • YouTube Track Link: https://bit.ly/31Ma5s0 • Spotify Track Link: https://spoti.fi/
From playlist Introduction to Homotopy Theory
Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie
Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Homotopy Group - (1)Dan Licata, (2)Guillaume Brunerie, (3)Peter Lumsdaine
(1)Carnegie Mellon Univ.; Member, School of Math, (2)School of Math., IAS, (3)Dalhousie Univ.; Member, School of Math April 11, 2013 In this general survey talk, we will describe an approach to doing homotopy theory within Univalent Foundations. Whereas classical homotopy theory may be des
From playlist Mathematics
Homomorphisms in abstract algebra
In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu
From playlist Abstract algebra
Homotopy elements in the homotopy group π₂(S²) ≅ ℤ. Roman Gassmann and Tabea Méndez suggested some improvements to my original ideas.
From playlist Algebraic Topology
Introduction to Homotopy Theory- PART 2: (TOPOLOGICAL) HOMOTOPY
We move on to the second section of nLab's introduction to homotopy theory, homotopy. Topics covered include left/right homotopy, topolocial path/cylinder objects, homotopy groups, and weak/standard homotopy equivalences. PLEASE leave any misconceptions I had or inaccuracies in my video i
From playlist Introduction to Homotopy Theory
Homomorphisms in abstract algebra examples
Yesterday we took a look at the definition of a homomorphism. In today's lecture I want to show you a couple of example of homomorphisms. One example gives us a group, but I take the time to prove that it is a group just to remind ourselves of the properties of a group. In this video th
From playlist Abstract algebra
Lie Groups and Lie Algebras: Lesson 34 -Introduction to Homotopy
Lie Groups and Lie Algebras: Introduction to Homotopy In order to proceed with Gilmore's study of Lie groups and Lie algebras we now need a concept from algebraic topology. That concept is the notion of homotopy and the Fundamental Group of a topological space. In this lecture we provide
From playlist Lie Groups and Lie Algebras
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
Existence of Symplectic and Contact forms by Mahuya Datta
DATE & TIME: 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 structure. The moduli space of these curves (
From playlist J-Holomorphic Curves and Gromov-Witten Invariants
Topological quantum phases - Alexei Kitaev
Special Seminar Topic: Topological quantum phases Speaker: Alexei Kitaev Affiliation: California Institute of Technology; Distinguished Visiting Professor, School of Natural Sciences Date: November 25, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Caustics of Lagrangian homotopy spheres with stably trivial Gauss map - Daniel Alvarez-Gavela
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Topic: Caustics of Lagrangian homotopy spheres with stably trivial Gauss map Speaker: Daniel Alvarez-Gavela Date: May 14, 2021 For more video please visit https://www.ias.edu/video
From playlist Mathematics
A geometric model for the bounded derived category of a gentle algebra, Sibylle Schroll, Lecture 1
Gentle algebras are quadratic monomial algebras whose representation theory is well understood. In recent years they have played a central role in several different subjects such as in cluster algebras where they occur as Jacobian algebras of quivers with potentials obtained from triangula
From playlist Winter School on “Connections between representation Winter School on “Connections between representation theory and geometry"
Homology cobordism and triangulations – Ciprian Manolescu – ICM2018
Geometry | Topology Invited Lecture 5.5 | 6.1 Homology cobordism and triangulations Ciprian Manolescu Abstract: The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with
From playlist Geometry
Mather-Thurston’s theory, non abelian Poincare duality and diffeomorphism groups - Sam Nariman
Workshop on the h-principle and beyond Topic: Mather-Thurston’s theory, non abelian Poincare duality and diffeomorphism groups Speaker: Sam Nariman Affiliation: Purdue University Date: November 1, 2021 Abstract: I will discuss a remarkable generalization of Mather’s theorem by Thurston
From playlist Mathematics
Computing with Univalence - Daniel Licata
Daniel Licata Carnegie Mellon University; Member, School of Mathematics September 28, 2012 For more videos, visit http://video.ias.edu
From playlist Mathematics
Foundations of Mathematics and Homotopy Theory - Vladimir Voevodsky
Vladimir Voevodsky Institute for Advanced Study March 22, 2006 More videos on http://video.ias.edu
From playlist Mathematics
Introduction to Homotopy Theory- Part 4: Fibrations
Wow! This one was a lot more detailed than usual, so I'd really recommend going through the proofs with the nLab in hand. I tried to elucidate some of their explanations, but it's still good to have both, so hopefully in between both of our presentations you can find understanding. And as
From playlist Introduction to Homotopy Theory