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The aim of the [course_title] is to show how basic geometric structures may be studied by transforming them into algebraic questions that are then subject to computations, thus measuring geometric and topological complexity. These methods are often used in other parts of mathematics, and also in biology, physics and other areas of application. The course is meant to give a basis for studies in topology, geometry, algebra, and applications.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: Open Culture 

Course Curriculum

Module: 01
AlgTop0: Introduction to Algebraic Topology 00:30:00
AlgTop1: One-dimensional objects 00:32:00
AlgTop2: Homeomorphism and the group structure on a circle 00:52:00
AlgTop3: Two-dimensional surfaces: the sphere 00:42:00
AlgTop4: More on the sphere 00:41:00
AlgTop5: Two-dimensional objects- the torus and genus 00:49:00
AlgTop6: Non-orientable surfaces—the Mobius band 00:42:00
AlgTop7: The Klein bottle and projective plane 00:40:00
AlgTop8: Polyhedra and Euler’s formula 00:45:00
AlgTop9: Applications of Euler’s formula and graphs 00:42:00
Module: 02
AlgTop10: More on graphs and Euler’s formula 00:48:00
AlgTop11: Rational curvature, winding and turning 00:48:00
AlgTop12: Duality for polygons and the Fundamental Theorem of Algebra 00:45:00
AlgTop13: More applications of winding numbers 00:27:00
AlgTop14: The Ham Sandwich theorem and the continuum 00:36:00
AlgTop15: Rational curvature of a polytope 00:50:00
AlgTop16: Rational curvature of polytopes and the Euler number 00:35:00
AlgTop17: Classification of combinatorial surfaces I 00:50:00
AlgTop18: Classification of combinatorial surfaces II 00:59:00
AlgTop19: An algebraic ZIP proof 00:42:00
Module: 03
AlgTop20: The geometry of surfaces 00:44:00
AlgTop21: The two-holed torus and 3-crosscaps surface 00:39:00
AlgTop22: Knots and surfaces I 00:52:00
AlgTop23: Knots and surfaces II 00:38:00
AlgTop24: The fundamental group 00:43:00
AlgTop25: More on the fundamental group 00:35:00
AlgTop26: Covering spaces 00:54:00
AlgTop27: Covering spaces and 2-oriented graphs 00:31:00
AlgTop28: Covering spaces and fundamental groups 00:47:00
AlgTop29: Universal covering spaces 00:48:00
Module: 04
AlgTop30: An introduction to homology 00:47:00
AlgTop31: An introduction to homology (cont.) 00:41:00
AlgTop32: Simplices and simplicial complexes 00:49:00
AlgTop33: Computing homology groups 00:41:00
AlgTop34: More homology computations 00:43:00
AlgTop35: Delta complexes, Betti numbers and torsion 00:48:00
Module: 05
AlgTopReview: An informal introduction to abstract algebra 00:49:00
AlgTopReview2: Introduction to group theory 00:47:00
AlgTopReview3: More on commutative groups—isomorphisms, homomorphisms, cosets and quotient groups 00:32:00
AlgTopReview4: Free abelian groups and non-commutative groups 00:51:00
Assessment
Submit Your Assignment 00:00:00
Certification 00:00:00

Course Reviews

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