Representation theory of Lie groups | Representation theory of Lie algebras | Homological algebra

In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space; in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general. (Wikipedia).

Affine polygon rendering (quads, not triangles)

In https://youtu.be/hxOw_p0kLfI I illustrated how affine texture mapping (non perspective-corrected) appears “wonky” when the shape is not an equilateral. Some of this was because it was constructed from triangles. So what happens when we render any convex polygons and not just triangles?

From playlist 3D Rendering Tutorial

What are affine transformations?

Algorithm Archive: https://www.algorithm-archive.org/contents/affine_transformations/affine_transformations.html Github sponsors (Patreon for code): https://github.com/sponsors/leios Patreon: https://www.patreon.com/leiosos Twitch: https://www.twitch.tv/leioslabs Discord: https://discor

From playlist Algorithm Archive

Novel Algebraic Operations for Affine Geometry | Algebraic Calculus One | Wild Egg

We introduce some novel conventions to help us set up the foundations of affine geometry. We learn about differences of points, sums of points and vectors, affine combinations and vector proportions. And then use these to state a number of important results from affine geometry, including

From playlist Algebraic Calculus One from Wild Egg

Affine Springer fibers and representation theory - Cheng-Chiang Tsai

Short talk by postdoctoral members Topic: Affine Springer fibers and representation theory Speaker: Cheng-Chiang Tsai, Member, School of Mathematics For more videos, visit http://video.ias.edu

From playlist Mathematics

Representations of Finite Groups | Definitions and simple examples.

We define the notion of a representation of a group on a finite dimensional complex vector space. We also explore one and two dimensional representations of the cyclic group Zn. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.mich

From playlist Representations of Finite Groups

Representation theory: Introduction

This lecture is an introduction to representation theory of finite groups. We define linear and permutation representations, and give some examples for the icosahedral group. We then discuss the problem of writing a representation as a sum of smaller ones, which leads to the concept of irr

From playlist Representation theory

MOR_012 - Linguistic Micro-Lectures: Affixes

What are affixes and how can they be defined and represented? Within less than two minutes Prof. Handke explains the main principles of affixation using examples from PDE and other languages.

From playlist Micro-Lectures - Morphology

Affine and mod-affine varieties in arithmetic geometry. - Charles - Workshop 2 - CEB T2 2019

François Charles (Université Paris-Sud) / 24.06.2019 Affine and mod-affine varieties in arithmetic geometry. We will explain how studying arithmetic versions of affine schemes and their bira- tional modifications leads to a generalization to arbitrary schemes of both Fekete’s theorem on

From playlist 2019 - T2 - Reinventing rational points

Concavity and Inflection Points for f(x) = ln(1 + x^2)

In this video I find the intervals on which the function f(x) = ln(1 + x^2) is concave up and concave down. I also find the inflection points. If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com Free Homewor

From playlist Concavity and Inflection Points

Laura Rider: Modular Perverse Sheaves on the affine Flag Variety

There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equiva

From playlist Workshop: Monoidal and 2-categories in representation theory and categorification

The Drinfeld-Sokolov reduction of admissible representations of affine Lie algebras - Gurbir Dhillon

Workshop on Representation Theory and Geometry Topic: The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras Speaker: Gurbir Dhillon Affiliation: Yale University Date: April 03, 2021 For more video please visit http://video.ias.edu

From playlist Mathematics

Modular Perverse Sheaves on the affine Flag Variety - Laura Rider

Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Modular Perverse Sheaves on the affine Flag Variety Speaker: Laura Rider Affiliation: University of Georgia Date: November 16, 2020 For more video please visit http://video.ias.edu

From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory

The Hecke category action on the principal block via Smith theory - Geordie Williamson

Geometric and Modular Representation Theory Seminar Topic: The Hecke category action on the principal block via Smith theory Speaker: Geordie Williamson Affiliation: University of Sydney; Distinguished Visiting Professor, School of Mathematics Date: January 27, 2021 For more video please

From playlist Geordie Williamson external seminars

Group Theory for Cryptology by Carlo Scoppola

PROGRAM GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION ORGANIZERS: Gurmeet Kaur Bakshi, Manoj Kumar and Pooja Singla DATE: 14 October 2019 to 23 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Determining explicit algebraic structures of semisimple group algebras is a fund

From playlist Group Algebras, Representations And Computation

Talk by Jonathan Wang (MIT, USA)

L-functions and Geometric Harmonic Analysis on Spherical Varieties

From playlist Seminars: Representation Theory and Number Theory

The center of the small quantum group - Pablo Boixeda Alvarez

Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: The center of the small quantum group Speaker: Pablo Boixeda Alvarez Affiliation: Member, School of Mathematics Date: November 17, 2020 For more video please visit http://video.ias.edu

From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory

The affine Hecke category is a monoidal colimit - James Tao

Geometric and Modular Representation Theory Seminar Topic: The affine Hecke category is a monoidal colimit Speaker: James Tao Affiliation: Massachusetts Institute of Technology Date: February 24, 2021 For more video please visit http://video.ias.edu

From playlist Seminar on Geometric and Modular Representation Theory

Monica Vazirani: Representations of the affine BMW category

The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. I will discuss Walker’s TQFT-motivated 1-handle construction of a family of

From playlist Workshop: Monoidal and 2-categories in representation theory and categorification

Using the property of exponents to multiply expressions

👉 Learn how to simplify expressions using the power rule of exponents. When several terms of an expression is raised to an exponent outside the parenthesis, the exponent is distributed over the individual terms in the expression and the exponent outside the parenthesis is multiplied to eac

From playlist Simplify Using the Rules of Exponents

Anthony Henderson: Hilbert Schemes Lecture 4

SMRI Seminar Series: 'Hilbert Schemes' Lecture 4 Kleinian singularities 1 Anthony Henderson (University of Sydney) This series of lectures aims to present parts of Nakajima’s book `Lectures on Hilbert schemes of points on surfaces’ in a way that is accessible to PhD students interested i

From playlist SMRI Course: Hilbert Schemes