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day 4

\(\def\Real{\mathbb{R}}\)

Homology.

Simplicial complexes; definition of simplicial homology. Basic properties. Functoriality.
Geometric realizations; simplicial approximations.
Homotopy invariance.

Euler calculus.

Additivity; Euler characteristic as measure. Integrals with respect to Euler characteristics.
Properties: linearity, Fubini theorem.
Classes of admissible sets.
Applications in …

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day 3

\(\def\Real{\mathbb{R}}\)

Thom’s strong transversality theorem. Jets.

Examples: codimension of set of points of given corank.Typical singularities of visible contours.

Taken’s embedding theorem.

Short survey of geometric dimensionality reduction tools in data analysis.

  • Model-based dimensionality reduction tools.
    Setup: a point cloud
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day 2

We will address essential tools in differential topology:

Partition of unity, using it to embed a manifold into a (high-dimensional) Euclidean space.

First applications of these tools to data analysis will be discussed (mapping of the terrain using local beacon …

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day 1

We will be covering the basic notions: topological spaces, compactness, homeomorphisms, homotopy,
and introduce the most important class of models for data analysis: manifolds.

Homework:

  1. Three cyclists start and end in the same formation, A>B>C. During the race, they never
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problems for review

  1. Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function
    $$
    f(x)=\sum |L_n| z^n,
    $$
    where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\#
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midterm 2

\(\def\pd{\partial}
\def\Real{\mathbb{R}}
\)

  1. Consider the closed cycle \(\gamma:[0,1]\to\Real^2\) shown on the left (we assume that \(\gamma(0)=\gamma(1)\) is the dot on the positive \(x\)-axis). Define the functions
    \[
    I_-(t)=\int_0^t \left(\frac{(x+1)dy -ydx}{(x+1)^2+y^2}.\dot{\gamma}(t)\right)dt,\quad I_+(t)=\int_0^t \left(\frac{(x-1)dy -ydx}{(x-1)^2+y^2}.\dot{\gamma}(t)\right)dt,
    \]
    given as partial integrals over the
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Stokes and relatives

  • The (general) Stokes Theorem: this unique theorem puts
    Green’s/Gauss/Stokes Theorems (from intermediate calculus)
    under one umbrella. Let \(C\) be a polyhedron with boundary \(\partial C\). Then
    $$ {\int_{\partial C} \omega = \int_C d\omega }$$
  • Example: integrate the form \(\omega =
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    solutions for problems on vector analysis and differential forms

    Solutions

    Exercise:
    Assume that 4 vectors in \(\mathbb{R}^3\) satisfy \(A+B+C+D\)=0.
    Simplify
    $$ A\times B-B\times C+C \times D-D\times A$$

    Solution: By the anti-symmetry \(-B\times C = C\times B\)
    and \(-D\times A = A\times D\) we can rewrite the expression as
    $$ …

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    March 28

  • Consider the \(k\)-form \(\omega \) defined at \(x\in \Omega\subset \mathbb{R}^n\), given by the sum
    $$ \omega(x)= \sum c_I(x) dx_I $$
    where the multi-index
    \( I=\{1\leq i_1\leq i_2 \leq, \ldots, \leq i_k \leq n\} \).
    Each basis element
    \(dx_I:= dx_{i_1} \wedge
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    Exterior Differential forms

    \(\def\pd{\partial}
    \def\Real{\mathbb{R}}
    \)

    • We define a differential \(k\)-form as
      $$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
      an exterior form with coefficients depending on the positions.

      (We will be using sometimes simplifying notation \(\xi_i = dx_i\) or \(\eta_i=dy_i\).)

    • Pullback maps: Let
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