Split-quaternion

Quaternions EXPLAINED Briefly

This is a video I have been wanting to make for some time, in which I discuss what the quaternions are, as mathematical objects, and how we do calculations with them. In particular, we will see how the fundamental equation of the quaternions i^2=j^2=k^2=ijk=-1 easily generates the rule for

From playlist Quaternions

Quaternions as 4x4 Matrices - Connections to Linear Algebra

In math, it's usually possible to view an object or concept from many different (but equivalent) angles. In this video, we will see that the quaternions may be viewed as 4x4 real-valued matrices of a special form. What is interesting here is that if you know how to multiply matrices, you a

From playlist Quaternions

Quaternions: Extracting the Dot and Cross Products

The most important operations upon vectors include the dot and cross products and are indispensable for doing physics and vector calculus. The dot product gives a quick way to check whether vectors are orthogonal and the cross product calculates a new vector orthogonal to both its inputs.

From playlist Quaternions

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

The Lie-algebra of Quaternion algebras and their Lie-subalgebras

In this video we discuss the Lie-algebras of general quaternion algebras over general fields, especially as the Lie-algebra is naturally given for 2x2 representations. The video follows a longer video I previously did on quaternions, but this time I focus on the Lie-algebra operation. I st

From playlist Algebra

The rotation problem and Hamilton's discovery of quaternions IV | Famous Math Problems 13d

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connec

From playlist Famous Math Problems

Set Theory (Part 14c): More on the Quaternions

No background in sets required for this video. In this video, we will learn how the quaternions can be thought of as pairings of complex numbers. We also will show how the quaternions can be written as a 2x2 complex matrix as opposed to a 4x4 real matrix and how the unit quaternions form t

From playlist Set Theory

Euler's Formula for the Quaternions

In this video, we will derive Euler's formula using a quaternion power, instead of a complex power, which will allow us to calculate quaternion exponentials such as e^(i+j+k). If you like quaternions, this is a pretty neat formula and a simple generalization of Euler's formula for complex

From playlist Math

Set Theory (Part 14b): Quaternions and 3D Rotations

No background in sets needed for this video - learn about the foundations of quaternions, derivation of the Hamilton product, and their application to 3D rotations. We will also see how dot and cross products are related to quaternion math. This video will be of particular interest to comp

From playlist Set Theory by Mathoma

Example of Quaternions

Matrix Theory: For the quaternion alpha = 1 - i + j - k, find the norm N(alpha) and alpha^{-1}. Then write alpha as a product of a length and a direction.

From playlist Matrix Theory

Simultaneous equidistribution of supersingular reductions of CM-curves by Manuel Luethi

PROGRAM SMOOTH AND HOMOGENEOUS DYNAMICS ORGANIZERS: Anish Ghosh, Stefano Luzzatto and Marcelo Viana DATE: 23 September 2019 to 04 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Ergodic theory has its origins in the the work of L. Boltzmann on the kinetic theory of gases.

From playlist Smooth And Homogeneous Dynamics

QED Prerequisites Geometric Algebra 24 Paravectors

In this lesson we discover yet another way to partition the components of a general multivector. In this method, the partitioning is entirely dependent on a choice of reference frame. That is \gamma_0 must be chosen and it represents the 4-velocity of an observer who is stationary in that

From playlist QED- Prerequisite Topics

R. Gutierrez - Quaternionic monodromies of the Kontsevich–Zorich cocycle

The monodromy group of a translation surface M is the Lie group spanned by all symplectic matrices arising from the homological action of closed loops at M (inside its embodying orbit closure). In the presence of zero Lyapunov exponents, Filip showed that these groups are —up to compact fa

From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications

Overview - Jacob Lurie

Milnor Conjecture Learning Seminar 1:00pm – 3:30pm Rubenstein Commons | Meeting Room 5 [REC but DO NOT PUBLISH] Topic: Overview Speaker: Jacob Lurie Affiliation: Faculty, Frank C. and Florence S. Ogg Professor, School of Mathematics Date: February 3, 2023

From playlist Mathematics

From PSL2 representation rigidity to profinite rigidity - Alan Reid and Ben McReynolds

Arithmetic Groups Topic: From PSL2 representation rigidity to profinite rigidity Speakers: Alan Reid and Ben McReynolds Affiliations: Rice University; Purdue University Date: February 9, 2022 In the first part of this talk, we take the ideas of the second talk and focus on the case of (a

From playlist Mathematics

Finding cocompact Fuchsian groups of given trace field and quaternian algebra - Jeremy Kahn

Jeremy Kahn, IAS October 7, 2015 http://www.math.ias.edu/wgso3m/agenda 2015-2016 Monday, October 5, 2015 - 08:00 to Friday, October 9, 2015 - 12:00 This workshop is part of the topical program "Geometric Structures on 3-Manifolds" which will take place during the 2015-2016 academic year

From playlist Workshop on Geometric Structures on 3-Manifolds

G. Molino - The Horizontal Einstein Property for H-Type sub-Riemannian Manifolds

We generalize the notion of H-type sub-Riemannian manifolds introduced by Baudoin and Kim, and then introduce a notion of parallel Clifford structure related to a recent work of Moroianu and Semmelmann. On those structures, we prove an Einstein property for the horizontal distribution usin

From playlist Journées Sous-Riemanniennes 2018

Lie Groups and Lie Algebras: Lesson 12 - The Classical Groups Part X (redux)

Lie Groups and Lie Algebras: Lesson 12 - The Classical Groups Part X (redux) We name the classical groups, finally! This video ended a bit short, I added the missing part in the "redux" version of this lesson. Please consider supporting this channel via Patreon: https://www.patreon.com/

From playlist Lie Groups and Lie Algebras

On the notion of genus for division algebras and algebraic groups - Andrei Rapinchu

Joint IAS/Princeton University Number Theory Seminar Topic: On the notion of genus for division algebras and algebraic groups Speaker: Andrei Rapinchu Affiliation: University of Virginia Date: November 2, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Folding the Klein Quartic

https://github.com/timhutton/klein-quartic

From playlist Geometry