Geometric topology | Four-dimensional geometry
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1a and S1b (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1a and S1b each exists in its own independent embedding space R2a and R2b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y. Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube. If S1a and S1b each has a radius of , their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4. The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface. (Wikipedia).
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
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Some minimal submanifolds generalizing the Clifford torus -Jaigyoung Choe
Workshop on Mean Curvature and Regularity Topic: Some minimal submanifolds generalizing the Clifford torus Speaker: Jaigyoung Choe Affiliation: KIAS Date: November 5, 2018 For more video please visit http://video.ias.edu
From playlist Workshop on Mean Curvature and Regularity
Spherical Earth: https://skfb.ly/NNrH Torus Earth: https://skfb.ly/MYpC Shapeways link: http://shpws.me/M9NI Joint work with Saul Schleimer.
From playlist 3D printing
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
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
Rob Kusner: Willmore stability and conformal rigidity of minimal surfaces in S^n
A minimal surface M in the round sphere S^n is critical for area, as well as for the Willmore bending energy W=∫∫(1+H^2)da. Willmore stability of M is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the W-stability of M persists in all higher dimensional
From playlist Geometry
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
Classification results for two-dimensional Lagrangian tori - Dimitroglou Rizell
Princeton/IAS Symplectic Geometry Seminar Topic: Classification results for two-dimensional Lagrangian tori Speaker: Georgios Dimitroglou-Rizell Date: Thursday, April 7 We present several classification results for Lagrangian tori, all proven using the splitting construction from symp
From playlist Mathematics
Twisted matrix factorizations and loop groups - Daniel Freed
Daniel Freed University of Texas, Austin; Member, School of Mathematics and Natural Sciences February 9, 2015 The data of a compact Lie group GG and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of
From playlist Mathematics
F. Andreatta - The height of CM points on orthogonal Shimura varieties and Colmez conjecture (part2)
We will first introduce Shimura varieties of orthogonal type, their Heegner divisors and some special points, called CM (Complex Multiplication) points. Secondly we will review conjectures of Bruinier-Yang and Buinier-Kudla-Yang which provide explicit formulas for the arithmetic intersecti
From playlist Ecole d'été 2017 - Géométrie d'Arakelov et applications diophantiennes
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
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Index Theory and Flexibility in Positive Scalar Curve Geometry -Bernhard Hanke
Emerging Topics Working Group Topic: Index Theory and Flexibility in Positive Scalar Curve Geometry Speaker: Bernhard Hanke Affilaion: Augsburg University Date: October 18, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics
Lifting Projective Galois Representations - Stefan Patrikis
Lifting Projective Galois Representations Stefan Patrikis Princeton University; Member, School of Mathematics October 1, 2012
From playlist Mathematics
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
Twisted real structures for spectral triples
Talk by Adam Magee in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on March 31, 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
In search of Lagrangians with non-trivial Floer cohomology by Sushmita Venugopalan
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Isadore Singer- 1. Index Theory Revisited [1996]
slides for this talk: http://www.math.stonybrook.edu/Videos/SimonsLectures/direct_download.php?file=PDFs/43-Singer.pdf Simons Lecture Series Stony Brook University Department of Mathematics and Institute for Mathematical Sciences October 1-10, 1996 Isadore Singer MIT http://www.math.st
From playlist Number Theory
necklace,two way,Torus by Villarceau circles,mobius ball
From playlist Handmade geometric toys