Office Rm. Blocker 614A
E-mail: kuchment@math.tamu.edu
Home Page: /~kuchment
Algebraic topology tools have been used more and more recently in various applied area (numerical analysis,
imaging,
neuroscience, evolutionary biology, computer vision, complexity theory, statistics, machine learning, and what
not). The class will not be an in-depth and/or rigorous math course, but rather a pedestrian intuitive introduction for students with applied aspirations to main concepts,
with examples and applications that could entice users to a further study. Thus, the class does not substitute in any way our topology
and geometry graduate classes, which are needed for students going in geometry/topology related directions, although might be considered as fulfilling the geometry/topology breadth
requirement.
The topics listed below are (optimistically) planned to be addressed
(all being illustrated with examples from physics, data science, and other
applications). References are made to the Ghrist's book, while the additional reading listed further below is also helpful.
Intuitive introduction and a variety of examples from applications
Tentatively 1 class. Ref: Preface
Graphs, knots, links, braids
Tentatively 2 classes. Ref: Ch.1.
Surfaces. Euler characteristics.
Tentatively 3 classes. Ref: Ch. 1,3.
Vector fields, winding numbers.
Tentatively 2 classes. Ref: Sec. 1.4, 77.
Homotopy.
Tentatively 5 classes. Ref: Ch. 8.
Project 1.
Tentatively end of February.
Homology.
Tentatively 4 classes. Ref: Ch.2, 4.
Cohomology.
Tentatively 4 classes. Ref: Ch. 6
Project 2.
General position and transversality. Morse theory.
Tentatively 4-5 classes. Ref: Sec. 1.6, Ch. 7.
Time permitting: more stuff, e.g. sheaves. Ref: Ch. 9, 10.
Take home final exam
Linear algebra.
Basic notions of topology: open sets, continuous mappings, compactness, metrics. Basic notions of abstract algebra: group, field.
GRADING POLICY
Grading will be based upon attendance and class participation (30%) 2 home projects (40%) and a take-home final exam (30%).
Percentage of points |
Grade |
---|---|
80% and higher |
A |
70% and higher |
B |
60% and higher |
C |
50% and higher |
D |
Less than 50% |
F |
Herbert Edelsbrunner, A short course in computational geometry and topology Springer Verlag.
Herbert Edelsbrunner and John L. Harer,
Computational Topology: An Introduction. Amer. Math. Soc.
Parts A and B contain nicely written introduction to some topological notions and relevant algorithms.
Part C is concentrated on persistent homology and its applications to data science.
A. Zamorodian, Topology for computing, Cambridge Univ. Press. Another nice book somewhat similar to the previous one.
M. Monastyrsky, Riemann, topology, and physics, Birkhauser. While Part I contains scientific biography of B. Riemann, the Part II contains a nice introduction to topological notions and important examples of physics applications.
V.G. Boltyanskii, V.A. Efremovich, Intuitive Combinatorial Topology, Springer Verlag. Probably the nicest and enjoyable intuitive introduction to algebraic topology I know (accessible even to high school students).
V. V. Prasolov, Intuitive topology, AMS. A nice small indeed intuitive book on surfaces, knots, vector fields.
D. Farmer and T. Stanford, Knots and surfaces. A guide to discovering mathematics, AMS. Similar to the Prasolov's book above.
W. Chinn, N. Steenrod, First concepts of topology.
Hajime Saito, Algebraic topology: an intuitive approach, AMS.
C. Kosniowski, A first corse in algebraic topology.
AND PLENTY OF OTHER GOOD BOOKS