Representation theory of Lie groups | Lie groups

Maximal torus

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. Rn). The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. (Wikipedia).

Video thumbnail

Torus Magic with Ring 1

Buy at http://www.shapeways.com/shops/GeometricToy "Torus Magic" can eat another torus.This torus object is constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

Video thumbnail

Torus Magic with Ring 2

Buy at http://www.shapeways.com/shops/GeometricToy "Torus Magic" can eat another torus.This torus object is constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

Video thumbnail

Torus Magic 2

The torus magic is constructed with many rings. It transforms flat,spherical,etc. Farther more you can turn it inside out.

From playlist Handmade geometric toys

Video thumbnail

Torus Magic (50mm)

Buy at http://www.shapeways.com/shops/GeometricToy Torus Magic is a transformable torus. This torus object is constructed with 20 large rings(50mm diameter) and many small rings.It transforms flat,spherical etc. Also you can turn inside out the torus. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

Video thumbnail

Torus Magic

Buy at http://www.shapeways.com/shops/GeometricToy Torus Magic is a transformable torus. This torus object is constructed with many rings,and transforms flat,spherical etc. Also you can turn inside out the torus. Copyright (c) 2014,AkiraNishihara

From playlist 3D printed toys

Video thumbnail

Turn a Torus Inside Out

Buy at http://www.shapeways.com/shops/GeometricToy This object consists of two "Torus Magic".These torus objects are constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,Akira Nishihara

From playlist 3D printed toys

Video thumbnail

Torus Autologlyph

This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/KiL

From playlist 3D printing

Video thumbnail

Hinged flat torus

Hinged flat torus: http://shpws.me/I9py Polygonal torus: http://shpws.me/HJYZ Hinged flat square surface: http://shpws.me/HJYA Also see http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml

From playlist 3D printing

Video thumbnail

Parahoric Subgroups and Supercuspidal Representations of p-Adic groups - Dick Gross

Dick Gross Harvard University December 9, 2010 This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is th

From playlist Mathematics

Video thumbnail

Mirror symmetry and cluster algebras – Paul Hacking & Sean Keel – ICM2018

Algebraic and Complex Geometry Invited Lecture 4.15 Mirror symmetry and cluster algebras Paul Hacking & Sean Keel Abstract: We explain our proof, joint with Mark Gross and Maxim Kontsevich, of conjectures of Fomin–Zelevinsky and Fock–Goncharov on canonical bases of cluster algebras. We i

From playlist Algebraic & Complex Geometry

Video thumbnail

Supercuspidal L-packets - Tasho Kaletha

Computer Science/Discrete Mathematics Seminar I Topic: Supercuspidal L-packets Speaker: Tasho Kaletha Affiliation: Technion Date: March 5, 2018 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Regular supercuspidal representations - Tasho Kaletha

Beyond Endoscopy Topic: Regular supercuspidal representations Speaker: Tasho Kaletha, University of Toronto Date: Oct 01, 2016 Time/Room: 3:00pm-3:50pm/s-101 For more videos, visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Gopal Prasad: Descent in Bruhat-Tits theory

Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric grou

From playlist Algebraic and Complex Geometry

Video thumbnail

David Zywina, Computing Sato-Tate and monodromy groups.

VaNTAGe seminar on May 5, 2020. License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.

From playlist The Sato-Tate conjecture for abelian varieties

Video thumbnail

Towards a p-adic Deligne--Lusztig theory - Charlotte Chan

Joint IAS/Princeton University Number Theory Seminar Topic: Towards a p-adic Deligne--Lusztig theory Speaker: Charlotte Chan Affiliation: Princeton University Date: September 27, 2018 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Double covers of tori and the local Langlands correspondence - Tasho Kaletha

Workshop on Representation Theory and Geometry Topic: Double covers of tori and the local Langlands correspondence Speaker: Tasho Kaletha Affiliation: University of Michigan Date: April 02, 2021 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Gluing a Torus

Gluing is a good method to construct new topological spaces from known ones. Here a rectangles is glued along the edges to form a torus. Often the fundamental group of the glued object can be calculated from the pieces (here a rectangles) and the glue (here two intersecting circles). Th

From playlist Algebraic Topology

Video thumbnail

Introduction to Minimal surfaces by Rukmini Dey

SUMMER SCHOOL FOR WOMEN IN MATHEMATICS AND STATISTICS POPULAR TALKS (TITLE AND ABSTRACT) June 22, Wednesday, 15:45 - 16:45 hrs Rukmini Dey (ICTS, India) Title: Introduction to Minimal surfaces Abstract: In this talk I will introduce zero mean curvature surfaces, called minimal surface

From playlist Summer School for Women in Mathematics and Statistics - 2022

Related pages

Weyl character formula | Lie group | Dynkin diagram | Special unitary group | Unitary group | Exponential map (Lie theory) | Automorphism | Class function | Bruhat decomposition | Quaternion | Root system | Cartan subgroup | Identity component | Rotation | Connected space | Mathematics | Weyl group | Cartan subalgebra | Lie algebra | Compact group | Compact space | Symplectic group | Abelian group