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
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
3D Rotations and Quaternion Exponentials: Special Case
In this video, we'll understand 3D rotations from the point of view of vector analysis and quaternions. We will solve the problem of rotating a vector which is perpendicular to the axis of rotation in this video which will help us solve the general case in the next video. We will especiall
From playlist Quaternions
Quaternion algebras via their Mat2x2(F) representations
In this video we talk about general quaternion algebras over a field, their most important properties and how to think about them. The exponential map into unitary groups are covered. I emphasize the Hamiltionion quaternions and motivate their relation to the complex numbers. I conclude wi
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
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 rotation problem and Hamilton's discovery of quaternions III | Famous Math Problems 13c
This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this rem
From playlist Famous Math Problems
Modular forms of half-integral weight on exceptional groups
Joint IAS/Princeton University Number Theory Seminar Topic: Modular forms of half-integral weight on exceptional groups Speaker: Spencer Leslie Affiliation: Duke University Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of
From playlist Joint IAS/PU Number Theory Seminar
Periods of Quaternionic Shimura Varieties - Kartik Prasanna
Kartik Prasanna University of Michigan, Ann Arbor March 3, 2011 In the early 80's, Shimura made a precise conjecture relating Petersson inner products of arithmetic automorphic forms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a conse
From playlist Mathematics
Talk by Rahul Dalal (University of California, Berkeley, USA)
Statistics of Automorphic Representations Through the Stable Trace Formula
From playlist Seminars: Representation Theory and Number Theory
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
Right-angled Coxeter groups and affine actions ( Lecture 01) by Francois Gueritaud
DISCUSSION MEETING SURFACE GROUP REPRESENTATIONS AND PROJECTIVE STRUCTURES ORGANIZERS: Krishnendu Gongopadhyay, Subhojoy Gupta, Francois Labourie, Mahan Mj and Pranab Sardar DATE: 10 December 2018 to 21 December 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore The study of spaces o
From playlist Surface group representations and Projective Structures (2018)
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
This lecture is part of an online graduate course on Lie groups. We give an introductory survey of Lie groups theory by describing some examples of Lie groups in low dimensions. Some recommended books: Lie algebras and Lie groups by Serre (anything by Serre is well worth reading) Repre
From playlist Lie groups
Groups with bounded generation: old and new - Andrei S. Rapinchuk
Joint IAS/Princeton University Number Theory Seminar Topic: Groups with bounded generation: old and new Speaker: Andrei S. Rapinchuk Date: May 06, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
GSE statistics without spin - Sebastian Mueller
Sebastian Mueller University of Bristol November 5, 2013 For more videos, please visit http://video.ias.edu
From playlist Mathematics
An explicit supercuspidal local Langlands correspondence - Tasho Kaletha
Joint IAS/Princeton University Number Theory Seminar Topic: An explicit supercuspidal local Langlands correspondence Speaker: Tasho Kaletha Affiliation: University of Michigan; von Neumann Fellow, School of Mathematics Date: October 29, 2020 For more video please visit http://video.ias.e
From playlist Mathematics
Intro to Fourier series and how to calculate them
Download the free PDF from http://tinyurl.com/EngMathYT This is a basic introduction to Fourier series and how to calculate them. An example is presented that illustrates the computations involved. Such ideas are seen in university mathematics.
From playlist Fourier