Lectures on Differential Topology

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

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Optimization of Utility Functions in an Admissible Space of Higher Dimension

Handbook of Research on Military, Aeronautical, and Maritime Logistics and Operations - Advances in Logistics, Operations, and Management Science ◽

10.4018/978-1-4666-9779-9.ch006 ◽

2016 ◽

pp. 102-113

Author(s):

German Almanza ◽

Victor M. Carrillo ◽

Cely C. Ronquillo

Keyword(s):

Graduate Students ◽

Linear Algebra ◽

Algebraic Topology ◽

Morse Theory ◽

Utility Functions ◽

Higher Dimension ◽

Differential Topology ◽

Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

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Differential topology of higher dimensional manifolds

Princeton University Press eBook Package 2014 ◽

10.1515/9781400865192-006 ◽

2000 ◽

pp. 41-72 ◽

Cited By ~ 1

Author(s):

William Browder

Keyword(s):

Differential Topology ◽

Higher Dimensional

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Preparations of Differentiable Manifolds and Differential Topology

Qualitative Theory of Dynamical Systems ◽

10.1142/9789814350983_0001 ◽

1993 ◽

pp. 1-63

Keyword(s):

Differential Topology ◽

Differentiable Manifolds

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A history of algebraic and differential topology, 1900–1960

Historia Mathematica ◽

10.1016/0315-0860(91)90381-7 ◽

1991 ◽

Vol 18 (4) ◽

pp. 374-377

Author(s):

Jeremy Gray

Keyword(s):

Differential Topology ◽

History Of

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Unknotting tori in codimension one and spheres in codimension two

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100039001 ◽

1965 ◽

Vol 61 (3) ◽

pp. 659-664 ◽

Cited By ~ 14

Author(s):

C. T. C. Wall

Keyword(s):

Euclidean Space ◽

Unit Sphere ◽

Piecewise Linear ◽

Differential Topology ◽

Standard Unit ◽

Codimension Two ◽

Codimension One

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.

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