Xah Math Blog Archive 2017-01

in part 2 will be lots goodies for me. bundles and connection, homology. ... chern classes in chapter 4, ... and more differential geometry goodies in Part 3.

2017-10-23

differential geometer Richard S Palais' Books and Papers http://vmm.math.uci.edu/PalaisPapers/

2017-10-23

The Life and Mathematics of Shiing-Shen Chern

one of the greatest differential geometer of the century.

The Life and Mathematics of Shiing-Shen Chern

Just read this. Soul touching.

to read bio of Chern, is to also have a glimpse of modern history of China, thru the tumultuous times of war.

(watch great movie Farewell My Concubine )

and, i learned, a towering figure of differential geometry is Élie Cartan

math things i've learned.

equivalence problem. For example, triangle are defined by 3 real numbers, length of 2 of which is less than the other. So, that's the condition. It “generates” all possible triangles. And, any 2 triangle can be decided if they are equivalent (by isometry here), by first reducing the 3 numbers to certain canonical form (such as by scaling so that shortest side is 1), then simply compare the numbers.

For plane curves, it's determined by curvature function. It generates all plane curves. And to determine if 2 plane curves are equivalent by an isometry, you just express a given curve by the curvature function. Then you simply just compare 2 functions, literally.

for space curves, it's 2 functions: curvature and torsion.

for surfaces, it's the 2 fundamental forms.

the question of “equivalence problem”, is to formulate a way, so any 2 geometric object can be so compared, and unify the cases for curves and surfaces.

and Cartan began it Cartan's equivalence method

and today the method is “g structure”. G-structure on a manifold

2017-10-22

see ~~https://plus.google.com/+johncbaez999/posts/DGYEUQ3WG4b~~

and http://math.ucr.edu/home/baez/diary/october_2017.html

2017-10-16

omg, how boring is topology?

you'd think you gonna see how coffee cup turns into a donut.

instead, you got a fiat “open set”. From there on, its set of sets, subset, empty set, union of set, intersection of set, complementary set, power set, super set, finer set, coarser set, and it's set all the way!

~~https://plus.google.com/+XahLee/posts/MqxVNtn572c~~

2017-10-15

The Method of Fluxions by Issac Newton https://archive.org/stream/methodoffluxions00newt#page/n3/mode/2up

2017-10-10

Hilbert's 16th problem. the relative positions of the branches of real algebraic curves of degree n. Hilbert's sixteenth problem

2017-10-10

Abuse of Math Notation

math notation idiocy ~~https://plus.google.com/+XahLee/posts/hzPVxNEWbe1~~

2017-10-07

Comprehensive Introduction to Differential Geometry By Michael Splvak. Classic. Comprehensive Introduction Differential Geometry

that's 5 volumes heavy expensive book. and, is graduate level.

you wonder, why's such book named “introduction”. It's actually graduate text, at 5 thick heavy volumes. Mathematicians not specialized in differential geometry have lots to learn from it, i assume.

but, “introduction” implies, this 5 volumes book only touches the surface. But then, its got the word “comprehensive” in it?

so, the proper interpretation seems to be, that the topic of differential geometry really is deep, and tied with lots other advanced math. Therefore, even at 5 volumes, and being a comprehensive tome, is, still just a intro.

On several occasions, most prominently in Volume 2, Spivak “translates” the classical language that Gauss or Riemann would be familiar with to the abstract language that a modern differential geometer might use.

2017-10-08 Wikipedia Michael Spivak

that'd be interesting to read.

2017-10-07