Topology of homogeneous spaces | Algebraic geometry
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. Even more generally, "Schubert calculus" is often understood to encompass the study of analogous questions in generalized cohomology theories. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety. The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring. In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (by block matrix). Putting Schubert's system on a rigorous footing is Hilbert's fifteenth problem. (Wikipedia).
11_4_1 The Derivative of the Composition of Functions
The composition of a multivariable function and a vector function and calculating its derivative.
From playlist Advanced Calculus / Multivariable Calculus
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11_3_1 The Gradient of a Multivariable Function
Using the partial derivatives of a multivariable function to construct its gradient vector.
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Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi Get 2 months of Curiosity Stream free by going to www.curiositystream.com/infinite and signing up with the promo code "infinite." It's said that Hermann Schubert perfor
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From playlist Seminar on Applied Geometry and Algebra (SIAM SAGA)
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PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t
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Lauren Williams: Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations
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[BOURBAKI 2017] 21/10/2017 - 2/4 - Simon RICHE
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