In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points: 1. * the operations can be studied by combinatorial means; and 2. * the effect of the operations is to yield an interesting bicommutant theory. The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group. In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is hard to come by. This connection established the deep interest of the cohomology operations for homotopy theory, and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints. (Wikipedia).
http://www.teachastronomy.com/ Cosmology is the study of the universe, its history, and everything in it. It comes from the Greek root of the word cosmos for order and harmony which reflected the Greek belief that the universe was a harmonious entity where everything worked in concert to
From playlist 22. The Big Bang, Inflation, and General Cosmology
Trigonometry 5 The Cosine Relationship
A geometrical explanation of the law of cosines.
From playlist Trigonometry
From playlist Courses and Series
Trigonometry 7 The Cosine of the Sum and Difference of Two Angles
A geometric proof of the cosine of the sum and difference of two angles identity.
From playlist Trigonometry
Using the law of cosines for a triangle with SAS
Learn how to solve for the lengths of the sides and the measures of the angles of a triangle using the law of cosines. The law of cosines is used in determining the lengths of the sides or the measures of the angles of a triangle when no angle measure and the length of the side opposite th
From playlist Law of Cosines
Applying the law of cosines to solve a word problem
Learn how to solve for the lengths of the sides and the measures of the angles of a triangle using the law of cosines. The law of cosines is used in determining the lengths of the sides or the measures of the angles of a triangle when no angle measure and the length of the side opposite th
From playlist Solve Law of Cosines (Word Problem) #ObliqueTriangles
How to use law of cosines for SSS
Learn how to solve for the lengths of the sides and the measures of the angles of a triangle using the law of cosines. The law of cosines is used in determining the lengths of the sides or the measures of the angles of a triangle when no angle measure and the length of the side opposite th
From playlist Law of Cosines
A Riemann-Roch theorem in Bott-Chern cohomology - Jean-Michel Bismut
Jean-Michel Bismut Université Paris-Sud April 21, 2014 If MM is a complex manifold, the Bott-Chern cohomology H(⋅,⋅)BC(M,C)HBC(⋅,⋅)(M,C) of MM is a refinement of de Rham cohomology, that takes into account the p,q p,q grading of smooth differential forms. By results of Bott and Chern, vect
From playlist Mathematics
Coulomb's Law (3 of 7) Force Between Two One Coulomb Charges
Using Coulomb's law shows how to calculate the electric force between two 1 Coulombs charges that are separated by a distance of 1 meter. Coulomb's law states that the magnitude of the force between two point charges is directly proportional to the product of the magnitudes of charges and
From playlist Coulomb's Law and the Electric Force
Cong Xue - 2/2 Cohomology Sheaves of Stacks of Shtukas
Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the Eichler-Shimura relations imply the finiteness property of the cohomology groups, (2) how the finiteness and Drinfeld's lemma imply the action of
From playlist 2022 Summer School on the Langlands program
Applying the law of cosines when given SAS
Learn how to solve for the lengths of the sides and the measures of the angles of a triangle using the law of cosines. The law of cosines is used in determining the lengths of the sides or the measures of the angles of a triangle when no angle measure and the length of the side opposite th
From playlist Law of Cosines
Stable Homotopy without Homotopy - Toni Mikael Annala
IAS/Princeton Arithmetic Geometry Seminar Topic: Stable Homotopy without Homotopy Speaker: Toni Mikael Annala Affiliation: Member, School of Mathematics Date: January 30, 2023 Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy in
From playlist Mathematics
Duality for Rabinowitz-Floer homology - Alex Oancea
IAS/PU-Montreal-Paris-Tel-Aviv Symplectic Geometry Topic: Duality for Rabinowitz-Floer homology Speaker: Alex Oancea Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche Date: May 27, 2020 For more video please visit http://video.ias.edu
From playlist PU/IAS Symplectic Geometry Seminar
Categorical aspects of vortices (Lecture 2) by Niklas Garner
PROGRAM: VORTEX MODULI ORGANIZERS: Nuno Romão (University of Augsburg, Germany) and Sushmita Venugopalan (IMSc, India) DATE & TIME: 06 February 2023 to 17 February 2023 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru For a long time, the vortex equations and their associated self-dual fie
From playlist Vortex Moduli - 2023
Cong Xue - 1/2 Cohomology Sheaves of Stacks of Shtukas
Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the Eichler-Shimura relations imply the finiteness property of the cohomology groups, (2) how the finiteness and Drinfeld's lemma imply the action of
From playlist 2022 Summer School on the Langlands program
Tim Perutz: From categories to curve-counts in mirror symmetry
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Jean-Morlet Chair - Lalonde/Teleman
The cup product operation [Ling Zhou]
In this tutorial, you will learn about the cup product operation in the simplicial setting, and go through an example of computing it in the 2-torus. The cup product induces a ring structure on cohomology, making it more informative than homology. In the TDA community, many attentions have
From playlist Tutorial-a-thon 2021 Fall
Categorical aspects of vortices (Lecture 1) by Niklas Garner
PROGRAM: VORTEX MODULI ORGANIZERS: Nuno Romão (University of Augsburg, Germany) and Sushmita Venugopalan (IMSc, India) DATE & TIME: 06 February 2023 to 17 February 2023 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru For a long time, the vortex equations and their associated self-dual fie
From playlist Vortex Moduli - 2023
Sylvia Serfaty: Microscopic description of Coulomb gases
We are interested in the statistical mechanics of systems of N points with Coulomb interactions in general dimension for a broad temperature range. We discuss local laws characterizing the rigidity of the system at the microscopic level, as well as free energy expansion and Central Limit T
From playlist Analysis and its Applications
Lars Hesselholt: Around topological Hochschild homology (Lecture 8)
The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "Workshop: Hermitian K-theory and trace methods" Introduced by Bökstedt in the late eighties, topological Hochschild homology is a manifestation of the dual visions of Connes and Waldhausen to
From playlist HIM Lectures: Junior Trimester Program "Topology"