Topology

AGT: W2

(2020-11-24)

Dato da fare la scorsa volta: Scrivere un notebook python/sagemath (dopo averlo correttamente installato?) che dato un grafo (assegnato come complesso simpliciale astratto) calcola i due gruppi di omologia $H_0$ e $H_1$. Opzionale: assegnare (in qualche modo razionale) delle coordinate ai vertici, trasformandolo in grafo possibilmente euclideo, e visualizzare quel che viene. # define a function for visualization: import matplotlib as mpl import matplotlib.pyplot as plt from matplotlib.collections import LineCollection import numpy as np mpl.

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AGT: W1

(2020-11-11)

Installare CAS, python, jupyter, sagemath? Per poter fare un po’ di matematica computazionale serve avere un calcolatore e degli strumenti di calcolo. Potrebbe bastare un abaco, una copia del Liber Abbaci di Fibonacci per imparare ad usare le cifre decimali (PDF oppure SpringerLink), un regolo calcolatore oppure qualcosa di più recente. Una lista (molto incompleta) di strumenti di calcolo utili per la matematica: maple Mathematica matlab (with a toolbox) sagemath per python: SymPy, SciLab, SciPy, e molti altri… jupyter Quali installare, e come?

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Winding Around and Mountains

(2017-10-26)

Just … …reading John Roe’s book Winding Around. Here: For an extended riff on the idea of mathematics as mountaineering see “Into Thin Air” by Colin Adams, on Mathematical Intelligencer. How much about “mathematics as mountaineering”?

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Fundamental group of SO(3) and Dirac's scissors

(2016-11-25)

Topology of $SO(3)$ and fixed-distance pairs of points in $S^2$. Consider the space $X\subset S^2\times S^2$ of all pairs $(A,B)$ of points such that $|A-B|=\sqrt 3$, where the distance is the euclidean distance of $S^2\subset \RR^3$, and $S^2$ is the unit sphere in $\RR^3$. (1) $X\approx SO(3) \approx \PP^3(\RR)$. Proof: The fact that $SO(3) \approx \PP^3(\RR)$ is a not-so-easy elementary exercise (cf. exercise (11.27) and exercise (7.30) of my Geometry Notes, protected by a single-character password: the number of non-zero digits appearing in the password itself; the problem might be ill-posed).

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Homology of (simplicial) spheres and applications

(2016-11-21)

Nullhomotopic odd maps, Borsuk, Lusternik and Schnirelmann Let $\Delta^n$ the standard simplex of dimension $n$ in $\RR^{n+1}$, which is homeomorphic to a $n$-dimensional disk, and $\Sigma^{n}\subset \Delta^{n+1}$ the simplicial $n$-sphere, i.e. the boundary of $\Delta^{n+1}$ (the union of its $n+2$ $n$-dimensional faces), which is homeomorphic to an $n$-dimensional sphere. The homeomorphisms can be more explicit: let $B\in \Delta^{n+1}$ be its barycenter, and $\varphi\from \Sigma^n\subset \Delta^{n+1} \to \RR^{n+2}$ the map defined as $\varphi(x) = \dfrac{x-B}{|x-B|}$.

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Mesh, Barycentric Subdivision and Simplicial Approximation Theorem

(2016-11-18)

Mesh of a euclidean simplicial complex. Let $|K| \subset \RR^k$ an euclidean simplicial complex (i.e., an abstract simplicial complex, where the set of vertices actually is a set of points in $\RR^k$, such that $|K|$ is the disjoint union of the interiors of the convex hulls of the sets of vertices ranging over the set of simplices). A vertex $P$ of $K$ is therefore a point of $\RR^k$. For each simplex $\sigma \subset K$, the diameter of $\sigma$ (or $|\sigma|$ more precisely) is the maximum distance $|x-y|$ as $x,y$ vary in $|\sigma|$.

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Joins and Products of polyhedra

(2016-11-14)

Join $K*L$ of two simplicial complexes Let $K = (X_ K, \Phi_ K)$ and $L = (X_ L, \Phi_ L)$ be two simplicial complexes ($X_ K$ denotes the set of vertices of $K$ and $\Phi_ K$ the set of simplices). Then the join $KL$ is the simplicial complex defined as follows. Its set of vertices $X=X_ K \cup X_ L$ is the union of the sets of vertices of $K$ and of $L$, and a non-empty subset of $X$ is a simplex of $KL$ if and only if it is of type $\sigma \cup \tau$, where $\sigma \in \Phi_ K \cup \{ \emptyset \}$ and $\tau \in \Phi_ L \cup \{\emptyset \}$.

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Ranks, Tensors and Abelian Groups

(2016-11-09)

$\require{AMScd}$ Abelian Groups Given $a\in A$, the homomirphism $i_ a \from \ZZ \to A$ defined by $i_ a(k) = ka$ is mono (injective) iff $a$ has infinite order. Of $S$ is a set, the free Abelian group generated by $S$ is $$ \ZZ[S] = \ZZ S = \bigoplus_ {s \in S} \ZZ, $$ and it is the group of all finite linear combinations (with integer coefficients) of elements in $S$. The elements of $S$ are a basis (base) of $\ZZ[S]$.

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Graphs, trees and simplicial complexes

(2016-11-04)

Graphs and simplicial complexes A one-dimensional simplicial complex is a graph. That is, $K=(X_ K,\Phi_ K)$ where $V(K) = X_ K$ is the set of vertices and $\Phi_ K$ is the set of simplexes, with $\Phi_ K = \Phi^1_ K \cup \Phi^0_ K$, where $E(K) = \Phi^1_ K$ are the 1-dimensional simplexes (termed edges of $K$) and $\Phi^0_ K = \{ \{P\} : P \in X_ K \}$ are the $0$-dimensional simplexes (sometimes termed again vertices of $K$).

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Computing the Smith Normal Form of a matrix, and homology groups?

(2016-10-27)

A simple algorithm for computing the Smith Normal Form of a matrix in $\ZZ$ The proof and the algorithm are the same. First, a few auxiliary functions. Given a matrix $M$, the follow two functions are self-explanatory. def dims(M): num_righe=len(M) num_colonne=len(M[0]) return (num_righe,num_colonne) def MinAij(M,s): num_righe, num_colonne=dims(M) ijmin=[s,s] valmin=max( max([abs(x) for x in M[j][s:]]) for j in range(s,num_righe) ) for i in (range(s,num_righe)): for j in (range(s,num_colonne)): if (M[i][j] != 0 ) and (abs(M[i][j]) <= valmin) : ijmin = [i,j] valmin = abs(M[i][j]) return ijmin def IdentityMatrix(n): res=[[0 for j in range(n)] for i in range(n)] for i in range(n): res[i][i] = 1 return res def display(M): r="" for x in M: r += "%s\n" % x return r +"" Then, one needs the elementary operations on rows and columns on the matrix $M$: swap (permute) two rows, add to a row an integer multiple of another row, and change sign of a row.

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Császár torus, python and sagemath

(2016-10-19)

The simplicial complex Consider the (abstract) simplicial $T$ defined as follows: vertices are $$ X=\{ 0,1,2,3,4,5,6 \} = \{ j \mod 7 \}. $$ Now, for each $x\in X$ let $\Phi_ x$ denote the following subsets of $2^X$ : $$ \begin{aligned} H_ x &= \{ \{j,j+1\} : j =0 \ldots 6 \} = \{ \{j,j-1\} : j =0 \ldots 6 \} \\\ V_ x &= \{ \{ j,j+x \} : j =0 \ldots 6 \} = \{ \{ j,j-x \} : j =0 \ldots 6 \} \\\ D_ x &= \{ \{ j, j + x+1 \} : j = 0 \ldots 6 \} = \{ \{ j, j - x-1 \} : j = 0 \ldots 6 \} \\\ \Phi_ x & = H_ x \cup V_ x \cup D_ x \end{aligned} $$ where all the integers $j$ are meant $\mod 7$.

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Categorical products and coproducts

(2016-10-12)

Universal property of coproducts Let $X_1, X_2$ be objects of the category $\vC$. The coproduct of $X_1$ and $X_2$ (if it exists) is a space $X_1 \coprod X_2$ and two morphisms $j_1 \from X_1 \to X_1 \coprod X_2$, $j_2 \from X_2 \to X_1 \coprod X_2$ such that the following diagram commutes (the first is with MathJax AMScd, the second is an embedded SVG produced with xypic): $\require{amscd}$ $$ \begin{CD} X_1 @>{j_1}» X_1 \coprod X_2 @<{j_2}« X_2 \\\ @| @VVhV @| \\\ X_1 @>{h_1}» Z @<{h_2}« X_2 \\\ \end{CD} $$

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A counter-example for the exponentiation / adjoint of a function

(2016-10-05)

The problem: Let $X$, $Y$, and $Z$ be topological spaces. Let $Y^X$ denote the space of all continuous functions $X\to Y$, with the compact-open topology generated by the elements of the sub-basis $W_{K,U} = \{\varphi \in Y^X : \varphi K \subset U\}$, where $K$ ranges over all compact subsets of $X$ and $U$ ranges over over all open subsets of $Y$. Finite intersections of $W_{K,U}$ are a basis for the C.

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