Inversive geometry | Geometric algebra | Conformal geometry | Computational geometry
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations. The effect of the mapping is that generalized (i.e. including zero curvature) k-spheres in the base space map onto (k + 2)-blades, and so that the effect of a translation (or any conformal mapping) of the base space corresponds to a rotation in the higher-dimensional space. In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. A geometric object (a k-sphere) can be synthesized as the wedge product of k + 2 linearly independent vectors representing points on the object; conversely, the object can be decomposed as the repeated wedge product of vectors representing k + 2 distinct points in its surface. Some intersection operations also acquire a tidy algebraic form: for example, for the Euclidean base space R3, applying the wedge product to the dual of the tetravectors representing two spheres produces the dual of the trivector representation of their circle of intersection. As this algebraic structure lends itself directly to effective computation, it facilitates exploration of the classical methods of projective geometry and inversive geometry in a concrete, easy-to-manipulate setting. It has also been used as an efficient structure to represent and facilitate calculations in screw theory. CGA has particularly been applied in connection with the projective mapping of the everyday Euclidean space R3 into a five-dimensional vector space R4,1, which has been investigated for applications in robotics and computer vision. It can be applied generally to any pseudo-Euclidean space - for example, Minkowski space R3,1 to the space R4,2. (Wikipedia).
Conformal Field Theory (CFT) | Infinitesimal Conformal Transformations
Conformal field theories are used in many areas of physics, from condensed matter physics, to statistical physics to string theory. They are defined as quantum field theories that are invariant under so-called conformal transformations. In this video, we will investigate these conformal tr
From playlist Particle Physics
Symposium on Geometry Processing 2017 Graduate School Lecture by Keenan Crane https://www.cs.cmu.edu/~kmcrane/ http://geometry.cs.ucl.ac.uk/SGP2017/?p=gradschool#abs_conformal_geometry Digital geometry processing is the natural extension of traditional signal processing to three-dimensi
From playlist Tutorials and Lectures
Geometric Algebra - The Matrix Representation of a Linear Transformation
In this video, we will show how matrices as computational tools may conveniently represent the action of a linear transformation upon a given basis. We will prove that conventional matrix operations, particularly matrix multiplication, conform to the composition of linear transformations.
From playlist Geometric Algebra
Geometric Algebra - Duality and the Cross Product
In this video, we will introduce the concept of duality, involving a multiplication by the pseudoscalar. We will observe the geometric meaning of duality and also see that the cross product and wedge product are dual to one another, which means that the cross product is already contained w
From playlist Geometric Algebra
Geometric Algebra - Rotors and Quaternions
In this video, we will take note of the even subalgebra of G(3), see that it is isomorphic to the quaternions and, in particular, the set of rotors, themselves in the even subalgebra, correspond to the set of unit quaternions. This brings the entire subject of quaternions under the heading
From playlist Math
Conformal Field Theory (CFT) | More on Infinitesimal Conformal Transformations
Conformal field theories are quantum field theories that are invariant under so-called conformal transformations. In this video, we will investigate these conformal transformations in three or more dimensions. More information and details can be found in the excellent book "Introduction
From playlist Particle Physics
Rod Gover - An introduction to conformal geometry and tractor calculus (Part 1)
After recalling some features (and the value of) the invariant « Ricci calculus » of pseudo‐Riemannian geometry, we look at conformal rescaling from an elementary perspective. The idea of conformal covariance is visited and some covariant/invariant equations from physics are recovered in
From playlist Ecole d'été 2014 - Analyse asymptotique en relativité générale
SIGGRAPH 2022 - Geometric Algebra
The SIGGRAPH 2022 course on Geometric Algebra. by Alyn Rockwood and Dietmar Hildenbrand
From playlist Introductory
Analytic Geometric Langlands-correspondence: Relations to Conformal (Lecture 1) by Joerg Teschner
Program Quantum Fields, Geometry and Representation Theory 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pandi
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Analytic Geometric Langlands-correspondence: Relations to Conformal (Lecture 2) by Joerg Teschner
Program Quantum Fields, Geometry and Representation Theory 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pandi
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Knot Categorification From Mirror Symmetry (Lecture- 1) by Mina Aganagic
PROGRAM QUANTUM FIELDS, GEOMETRY AND REPRESENTATION THEORY 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pan
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Boundaries of quasi-Fuchsian spaces and continuous/discontinuous (Lecture -1) by Ken'ichi Ohshika
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
Analytic Geometric Langlands-correspondence: Relations to Conformal ..(Lecture 3) by Joerg Teschner
Program Quantum Fields, Geometry and Representation Theory 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pandi
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Katrin Wendland: How do quarter BPS states cease being BPS?
CONFERENCE Recorded during the meeting "Vertex Algebras and Representation Theory" the June 09, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Luca Récanzone Find this video and other talks given by worldwide mathematicians on CIRM's Audiovi
From playlist Mathematical Physics
Rod Gover - An introduction to conformal geometry and tractor calculus (Part 4)
After recalling some features (and the value of) the invariant « Ricci calculus » of pseudo‐Riemannian geometry, we look at conformal rescaling from an elementary perspective. The idea of conformal covariance is visited and some covariant/invariant equations from physics are recovered in
From playlist Ecole d'été 2014 - Analyse asymptotique en relativité générale
Quasi-topological gauged sigma models, the geometric Langlands program, and knots by Meng-Chwan Tan
Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries
From playlist Quantum Fields, Geometry and Representation Theory
GAME2020 2. Hugo Hadfield, Eric Wieser. Robots, Ganja & Screw Theory (new audio!)
(* this version has an updated filtered audio track *) Hugo Hadfield and Eric Wieser explore how Conformal Geometric Algebra can be used to simplify robot kinematics. (slides : https://slides.com/hugohadfield/game2020). More information at https://bivector.net Chapters: 0:00 Introduction
From playlist Bivector.net
SGP2018 Graduate School | July 7-11 | Paris, France Speaker: Keenan Crane, Carnegie Mellon University Abstract: Digital geometry processing is the natural extension of traditional signal processing to three-dimensional geometric data. In recent years, methods based on so-called conformal
From playlist Tutorials and Lectures
On The Work Of Narasimhan and Seshadri (Lecture 3) by Edward Witten
Program Quantum Fields, Geometry and Representation Theory 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pandi
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Rod Gover - An introduction to conformal geometry and tractor calculus (Part 2)
After recalling some features (and the value of) the invariant « Ricci calculus » of pseudo‐Riemannian geometry, we look at conformal rescaling from an elementary perspective. The idea of conformal covariance is visited and some covariant/invariant equations from physics are recovered in
From playlist Ecole d'été 2014 - Analyse asymptotique en relativité générale