Source: https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/?W
The Strange Topology That Is Reshaping Physics
Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing
Charles Kane never thought he would be cavorting with topologists. “I don’t think like a mathematician,” admits Kane, a theoretical physicist who has tended to focus on tangible problems about solid materials. He is not alone. Physicists have typically paid little attention to topology—the mathematical study of shapes and their arrangement in space. But now Kane and other physicists are flocking to the field.
In the past decade, they have found that topology provides unique insight into the physics of materials, such as how some insulators can sneakily conduct electricity along a single-atom layer on their surfaces.
Some of these topological effects were uncovered in the 1980s, but only in the past few years have researchers begun to realize that they could be much more prevalent and bizarre than anyone expected. Topological materials have been “sitting in plain sight, and people didn’t think to look for them”, says Kane, who is at the University of Pennsylvania in Philadelphia.
Now, topological physics is truly exploding: it seems increasingly rare to see a paper on solid-state physics that doesn’t have the word topology in the title. And experimentalists are about to get even busier. A study on page 298 of this week’s Nature unveils an atlas of materials that might host topological effects, giving physicists many more places to go looking for bizarre states of matter such as Weyl fermions or quantum-spin liquids.
Read more at: https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/?WT.mc_id=SA_WR_20170726
.
, and take the nerve (Delaunay complex).
is alternating sum over all simplices 
in Delaunay complex.
partition into intervals (for acyclic case only)
components, 
loops, 
voids.
is born when it is not in image of previous homology group.
[Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.
.
. [E., Nikitenko, Reitones, 2016]
and 
. The diagram then has the following form:
is replaced by a subgroup 
. The differentials 
go two units to the right, and we replace the term 
by the term 
, where the 
‘s refer to the 
now go diagonally upward because of the formula ![j_2(i_1a)=[j_1a]](https://s0.wp.com/latex.php?latex=j_2%28i_1a%29%3D%5Bj_1a%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. The maps 
and 
still go vertically and horizontally, 
and 
.
and a subspace 
, define 
. Since the boundary map 
takes 
to 
, it induces a quotient boundary map 
.
where 
holds. The relative homology groups 
are the homology groups 
of this chain complex.
– chains 
such that 
.
for some 
and 
.
is as follows: If a class ![[\alpha]\in H_n(X,A)](https://s0.wp.com/latex.php?latex=%5B%5Calpha%5D%5Cin+H_n%28X%2CA%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is represented by a relative cycle ![\partial[\alpha]](https://s0.wp.com/latex.php?latex=%5Cpartial%5B%5Calpha%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is the class of the cycle 
in 
.
is a nonempty closed subspace that is a deformation retract of some neighborhood in 
is the inclusion 
and 
is the quotient map 
.
and 
for 
.
take 
so that 
. The terms 
in the long exact sequence are zero since 
is contractible.
are isomorphisms for 
and that 
. Starting with 
, 
for 
, the result follows by induction on 
to be the homology groups of the augmented chain complex 
where 
. We require 
and 
, 
vanishes on 
and hence induces a map 
with 
. So 
. Clearly, 
for 
such that 
, the exact MV sequence has the form
with 
the northern and southern hemispheres, so that 
. Then in the reduced Mayer-Vietoris sequence the terms 
are zero. So from the reduced Mayer-Vietoris sequence 
we get the exact sequence 
.
}
as the group of all rotations about the origin of 
as the unit 3-sphere 
with antipodal points identified.
in 
to the rotation about 
(and maps 
to the identity rotation). This mapping is clearly smooth and surjective. Its restriction to the interior of 
. On the boundary of 
and through 
are the same. Hence the mapping induces a smooth bijective map from 
, with antipodal points on the boundary identified, to 
is also smooth. (To see that the inverse is smooth, write 
. Then 
, and so 
exists and is continuous for all orders 
. Similar results hold for the variables 
and 
, and also mixed partials. By multivariable chain rule, one can see that all component functions are indeed smooth, so the inverse is smooth as claimed.)
, the unit ball 
is a diffeomorphism between 
(diffeomorphic)
, we may view 
is well-defined since if 
, then 
.
, it is clear that 
. So 
, where 
denotes the set of 2 by 2 complex matrices.
, it is clear that 
and 
are smooth maps since their component functions are of class 
. Since 
and 
) are also smooth.
, although the algorithm works for any field.![\mathbb{Z}_2[t]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D_2%5Bt%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
-module by the correspondence in previous Theorem. In this section we use 
and 
to denote homogeneous bases for 
and 
respectively.
since we are computing over 
is![\displaystyle M_1=\begin{bmatrix}[c|ccccc] &ab &bc &cd &ad &ac\\ \hline d & 0 & 0 & t & t & 0\\ c & 0 & 1 & t & 0 & t^2\\ b & t & t & 0 & 0 & 0\\ a &t &0 &0 &t^2 &t^3 \end{bmatrix}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+M_1%3D%5Cbegin%7Bbmatrix%7D%5Bc%7Cccccc%5D++%26ab+%26bc+%26cd+%26ad+%26ac%5C%5C+%5Chline++d+%26+0+%26+0+%26+t+%26+t+%26+0%5C%5C++c+%26+0+%26+1+%26+t+%26+0+%26+t%5E2%5C%5C++b+%26+t+%26+t+%26+0+%26+0+%26+0%5C%5C++a+%26t+%260+%260+%26t%5E2+%26t%5E3++%5Cend%7Bbmatrix%7D.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
of 
has the following basic property: 
provided 
.
relative to the standard basis for 
. We then reduce the matrix according to the reduction algorithm described previously.
, 
hence the standard basis may be used to represent 
.
and represent 
relative to 
and the homogeneous basis for 
in reverse degree order. Next, we make 
by Gaussian elimination on the columns, using elementary column operations. From linear algebra, we know that 
is the number of pivots in the echelon form. The basis elements corresponding to non-pivot columns form the desired basis for 
is said to be normal if 
is the splitting field of a family of polynomials in ![K[X]](https://s0.wp.com/latex.php?latex=K%5BX%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be an algebraic closure of 
containing 
of 
. In other words, ![L[X]](https://s0.wp.com/latex.php?latex=L%5BX%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. (One says that the polynomial splits in 
if and only if its minimal polynomial is a product of distinct linear factors over 
matrix. The characteristic polynomial of 
, is the polynomial defined by 
where 
.
for all 
a.e.\ and that 
, 
. If 
, we have 
for all 
. Note that the result is false if 
.
, where 
is the domain of integration).
, 
, 
otherwise the result is trivially true. Also, by Fatou’s Lemma, 
. Since 
and there exists 
such that for any measurable subset 
with 
, 
.
), by Egorov’s Theorem there exists closed 
such that 
and 
converge uniformly to 
such that for 
, 
for all 
.
.
). Error: See correction below.
, where 
is the ball with radius 
centered at the origin. Then 
, so there exists 
such that for 
, 
.
on 
such that for 
, 
.
,
.![f_k:=k\chi_{[0,\frac 1k]}](https://s0.wp.com/latex.php?latex=f_k%3A%3Dk%5Cchi_%7B%5B0%2C%5Cfrac+1k%5D%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Then 
. Note that 
, 
so that 
. Similarly, 
.
, 
for all 
.
on 
.
, by the finite measure case there exists 
such that for 
, 
.
is not Lipschitz on ![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
:
such that for all ![x,y\in [0,1]](https://s0.wp.com/latex.php?latex=x%2Cy%5Cin+%5B0%2C1%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, 
, 
for some 
between 
and 
is unbounded on 
is absolutely continuous on [0,1]. So Lipschitz is a stronger condition than absolutely continuous.
and 
, and let 
. If 
and 
, then 
and 
, then 
.
convergence and a.e. convergence:
, 
. If 
, then 
. Conversely, if 
, then 
.
).
for all 
.
.
and 
we see that 
.)
and 
so that 
as 
. Done.
for 
.
for 
gives 
with 
.
a.e.\ and 
a.e.\ which is integrable. Also, 
since 
. By Generalized Lebesgue’s DCT, we have 
thus 
![f_k=\chi_{[-k,k]}\in L^\infty(\mathbb{R})](https://s0.wp.com/latex.php?latex=f_k%3D%5Cchi_%7B%5B-k%2Ck%5D%7D%5Cin+L%5E%5Cinfty%28%5Cmathbb%7BR%7D%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Then 
, and 
. However 
.
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with 
, 
.
for 
. Each 
is measurable and 
. Note that 
. Then 
. By Monotone Convergence Theorem, 
, that is, 
be sufficiently large such that 
.
, and suppose 
.
for all 
.
. Let 
be a subsequence such that 
).
, there exists a further subsequence 
a.e. By Fatou’s Lemma, 
a contradiction.
be two sequences. Then,![\displaystyle \sum_{k=m}^n f_k(g_{k+1}-g_k)=[f_{n+1}g_{n+1}-f_mg_m]-\sum_{k=m}^n g_{k+1}(f_{k+1}-f_k).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bk%3Dm%7D%5En+f_k%28g_%7Bk%2B1%7D-g_k%29%3D%5Bf_%7Bn%2B1%7Dg_%7Bn%2B1%7D-f_mg_m%5D-%5Csum_%7Bk%3Dm%7D%5En+g_%7Bk%2B1%7D%28f_%7Bk%2B1%7D-f_k%29.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
on 
, then 
and 
.
be any subsequence of 
on 
a.e.\ in 
.
.
has a further subsequence that converges to 
, we have 
and 
.
, 
, etc.
.
bringing him great fame among the mathematical community. It is a beautiful equation; it is surprising that the constant 
(Parseval’s Identity)
![\displaystyle \frac 1\pi\cdot\left[\frac{x^3}{3}\right]_{-\pi}^\pi=\sum_{n=1}^\infty\frac{4}{n^2}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac+1%5Cpi%5Ccdot%5Cleft%5B%5Cfrac%7Bx%5E3%7D%7B3%7D%5Cright%5D_%7B-%5Cpi%7D%5E%5Cpi%3D%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B4%7D%7Bn%5E2%7D.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
(L’Hopital’s Rule Proof)
” is incorrect.
instead, and this implies 
.
.
where 
.
Mathtuition88 