There are 5 great Geometry Masters in history: 欧高黎嘉陈
Euclid (300 BCE, Greece), Gauss (18CE, Germany), Riemann (19CE, Germany), Cartan (20CE, France), Chern (21CE, China).
Jim Simons (Hedge Fund Billionaire, Chern’s PhD Student) quoted Chern had said to him:
“If you do One Thing that is reallygood, that’s all you could really expect in a life time.” 一生作好一件事, 此生无悔矣!
Highlights:
1. Video below @82:00 mins, SS Chern criticised on Hardy’s famous statement: “Great Math is only discovered by young mathematicians before 30.” Chern’s response: “Don’t believe it ! 不要相信它”.
2. Chern’s Conjecture :“21世纪中国将是数学大国。 ” China will be a Math Kingdom in 21st century.
is a smooth manifold.![U_1=\{[z^1, z^2]\mid z^1\neq 0\}](https://s0.wp.com/latex.php?latex=U_1%3D%5C%7B%5Bz%5E1%2C+z%5E2%5D%5Cmid+z%5E1%5Cneq+0%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![U_2=\{[z^1, z^2]\mid z^2\neq 0\}](https://s0.wp.com/latex.php?latex=U_2%3D%5C%7B%5Bz%5E1%2C+z%5E2%5D%5Cmid+z%5E2%5Cneq+0%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Also define 
by ![g_1([z^1, z^2])=\frac{z^2}{z^1}](https://s0.wp.com/latex.php?latex=g_1%28%5Bz%5E1%2C+z%5E2%5D%29%3D%5Cfrac%7Bz%5E2%7D%7Bz%5E1%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![g_2([z^1, z^2])=\frac{\overline{z^1}}{\overline{z^2}}](https://s0.wp.com/latex.php?latex=g_2%28%5Bz%5E1%2C+z%5E2%5D%29%3D%5Cfrac%7B%5Coverline%7Bz%5E1%7D%7D%7B%5Coverline%7Bz%5E2%7D%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be the homeomorphism from 
to 
defined by 
and define 
by 
.
is an open cover of 
are well-defined homeomorphisms (from 
onto an open set in 
is an atlas of 
![\begin{aligned} \phi_2\phi_1^{-1}(x,y)&=\phi_2 g_1^{-1}(x+iy)\\ &=\phi_2([1,x+iy])\\ &=f(\frac{1}{x-iy})\\ &=f(\frac{x+iy}{x^2+y^2})\\ &=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}) \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++%5Cphi_2%5Cphi_1%5E%7B-1%7D%28x%2Cy%29%26%3D%5Cphi_2+g_1%5E%7B-1%7D%28x%2Biy%29%5C%5C++%26%3D%5Cphi_2%28%5B1%2Cx%2Biy%5D%29%5C%5C++%26%3Df%28%5Cfrac%7B1%7D%7Bx-iy%7D%29%5C%5C++%26%3Df%28%5Cfrac%7Bx%2Biy%7D%7Bx%5E2%2By%5E2%7D%29%5C%5C++%26%3D%28%5Cfrac%7Bx%7D%7Bx%5E2%2By%5E2%7D%2C%5Cfrac%7By%7D%7Bx%5E2%2By%5E2%7D%29++%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Similarly, 
is of class 
is a smooth manifold.
and 
. Then 
is an open cover of 
by 
and 
by 
.
. Hence 
is a homeomorphism from 
onto an open set in 
is a homeomorphism from 
onto an open set in 
is an atlas for 
is differentiable of class 
are of class 
is of class 
.
by 
and 
.
is well-defined since if 
then 
so that 
.
defined by 
and 
.
is smooth. Hence 
be a smooth manifold, and let 
. A linear map 
is called a derivation at 
if it satisfies 
, is defined as the set of all derivations of 
at 
and we wish to find its 
th homology 
. In each dimension the homology of complex 
becomes a vector space over a field, described fully by its rank 
. (Over a field 
, 
and 
) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module 
corresponding to 
). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.
-interval 
describes a basis element for the homology vector spaces starting at time 
until time 
. This element is a 
that is completed at time 
, at which time it joins the boundary group 
.
is a basis element for the persistent groups 
. Recall the equation 
Since 
for all 
, hence 
for 
. The three inequalities 
define a triangular region in the index-persistence plane, as shown in Figure below.
be the set of triangles defined by 
of 
.
), but in Singapore it is quite expensive probably due to import fees.
is a PID, then every finitely generated module 
such that 
where 
, and 
.
where 
are homogenous elements such that 
, and 
denotes an 
-shift upward in grading.![F[t]](https://s0.wp.com/latex.php?latex=F%5Bt%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
-modules by suitable objects.
.
of 
. We define ![\displaystyle Q(i,j)=\Sigma^i F[t]/(t^{j-i})](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+Q%28i%2Cj%29%3D%5CSigma%5Ei+F%5Bt%5D%2F%28t%5E%7Bj-i%7D%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
for a 
, we have ![Q(i,+\infty)=\Sigma^iF[t]](https://s0.wp.com/latex.php?latex=Q%28i%2C%2B%5Cinfty%29%3D%5CSigma%5EiF%5Bt%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
, we define 
defines a bijection between the finite sets of 
be a graded ring. An ideal 
is homogenous (also called graded) if for every element 
, its homogenous components also belong to 
.
are called the homogenous parts of 
, where 
is a smooth structure on 
-manifold 
, there are disjoint open subsets 
such that 
and 
.
. For each 
containing 
; and
.
is called a smooth atlas if 
and for any two charts 
, 
in 
), the transition map 
is a diffeomorphism.
by John Lee
by Lawrence Conlon
(a direct sum of abelian groups 
) such that 
for all 
for all 
. Elements of any factor 
of the decomposition are called homogenous elements of degree ![R[t]](https://s0.wp.com/latex.php?latex=R%5Bt%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
non-negatively with the standard grading 
for all 
.
and 
.
be a persistence module over 
where the 
, 
is given by 
is a family of 
, together with homomorphisms 
.
maps a homology class to the one that contains it.
(resp.\ persistence module 
) is of finite type if each component complex (resp.\ module) is a finitely generated 
(resp.\ 
for some integer 
.
of finite type, whose homology is a persistence module 
of finite type.
, 
, such that the associated map 
given by 
is continuous. Two maps 
are called homotopic, denoted 
, if there exists a homotopy 
connecting them.
is a family 
, 
, such that
and 
are independent of 
defined by 
is continuous.
and 
are connected in this way by a homotopy 
.
is a homomorphism and 
is a normal subgroup of 
contained in the kernel of 
, then 
uniquely.
between chain complexes 
and 
induces homomorphisms between the homology groups of the two complexes.
implies that 
implies 
. Also 
. Hence 
, by universal property of quotient groups.
, we have 
. Therefore 
.
with 
for each 
of 
with coefficients 
.
by writing 
if and only if 
. The quotient space 
is called projective 
.) The canonical projection 
is just 
. Define 
, 
, by setting 
.
covers 
.
is open in 
, so 
. Then since 
, so 
. Hence 
.
. Define 
for 
. If 
, then 
. Then 
is well-defined.
where 
. Both 
are continuous, so 
is a homeomorphism.
. Since CW-complexes are Hausdorff, so is 
in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance 
). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:
, is the abstract simplicial complex where a set of 
vertices spans a 
-ball neighborhoods have nonempty intersection.
, is the abstract simplicial complex where a set 
of 
) Bottom right: Vietoris-Rips complex has a different homotopy type (
). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.
is a 
is also a 
be two 
is also a 
is also a 
be 
are both 
is a diffeomorphism since both 
and 
are diffeomorphisms due to 
, 
implies 
so 
is also a 
. If 
, then 
, so that 
Mathtuition88 