Complete Derivation: Universal Property of the Tensor Product
Previous tensor product video: https://youtu.be/KnSZBjnd_74 The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universa
From playlist Tensor Products
The TRUTH about TENSORS, Part 9: Vector Bundles
In this video we define vector bundles in full abstraction, of which tangent bundles are a special case.
From playlist The TRUTH about TENSORS
A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos. The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral. If a coveri
From playlist Algebraic Topology
What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will h
From playlist What is a Manifold?
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I
Lie Groups and Lie Algebras: Lesson 41: Elementary Representation Theory I I wanted to begin a more intricate example of the principle of a Universal Covering group, but I think I need to cover a little background material. We need to get a grip on what is meant by "Representation Theory"
From playlist Lie Groups and Lie Algebras
Parallels and the double triangle | Universal Hyperbolic Geometry 18 | NJ Wildberger
We discuss Euclid's parallel postulate and the confusion it led to in the history of hyperbolic geometry. In Universal Hyperbolic Geometry we define the parallel to a line through a point, NOT the notion of parallel lines. This leads us to the useful construction of the double triangle of
From playlist Universal Hyperbolic Geometry
Rational trigonometry: an overview | Universal Hyperbolic Geometry 39 | NJ Wildberger
This important video introduces Rational Trigonometry from first principles using a vector approach. The main notions of quadrance and spread replace distance and angle, and are introduced purely algebraically. The scalar/inner/dot product plays an important role, and allows us to introduc
From playlist Universal Hyperbolic Geometry
Marc Levine: Refined enumerative geometry (Lecture 4)
The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Lecture 4: Characteristic classes in Witt-cohomology Classical enumerative geometry relies heavily on the theory of Chern classes of vector bundles and the splitting principle, which
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Holomorphic Curves in Compact Quotients of SL(2,C) by Sorin Dumitrescu
DISCUSSION MEETING TOPICS IN HODGE THEORY (HYBRID) ORGANIZERS: Indranil Biswas (TIFR, Mumbai, India) and Mahan Mj (TIFR, Mumbai, India) DATE: 20 February 2023 to 25 February 2023 VENUE: Ramanujan Lecture Hall and Online This is a followup discussion meeting on complex and algebraic ge
From playlist Topics in Hodge Theory - 2023
Charles Rezk: Elliptic cohomology and elliptic curves (Part 4)
The lecture was held within the framework of the Felix Klein Lectures at Hausdorff Center for Mathematics on the 10. June 2015
From playlist HIM Lectures 2015
Geometry of vortices on Riemann surfaces (Lecture 2) by Oscar García-Prada
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
Quantization of Vortices (Lecture 2) by Nuno Romão
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
Ana Balibanu: The partial compactification of the universal centralizer
Abstract: Let G be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in G of regular elements in Lie(G), parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent
From playlist Algebra
Alina Marian - On the tautological cohomology of the moduli space of curves
Worldwide Center of Mathematics. Research lecture
From playlist Center of Math Research: the Worldwide Lecture Seminar Series
Transversality and super-rigidity in Gromov-Witten Theory (Lecture - 04) by Chris Wendl
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
Conformal Limits of Parabolic Higgs Bundles by Richard Wentworth
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
Eyal Markman: Hyperholomorphic sheaves and generalized deformations of K3 surfaces
This talk will elaborate on the role hyperholomorphic sheaves play in generalized deformations of K3 surfaces, described in the talk of Sukhendu Mehrotra. The lecture was held within the framework of the Junior Hausdorff Trimester Program Algebraic Geometry. (12.2.2014)
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Spherical and elliptic geometries: an introduction | Universal Hyperbolic Geometry 33
We introduce PART II of this course on universal hyperbolic geometry: Bringin geometries together. This lecture introduces the very basic definitions of spherical geometry; lines as great circles, antipodal points, spherical triangles, circles, and some related notions on points, lines and
From playlist Universal Hyperbolic Geometry