MATH41101 - 2008/2009

 

Specification

Aims

1. To discuss the basic ideas associated to simple polytopes in all dimensions,
2. To explain the context and applications of toroidal symmetry in topology and geometry,
3. To explore the relationship between 1) and 2),
4. To introduce students to a subject which is currently under rapid development.

Brief Description of the unit

Problems connected with torus actions arise in different areas of mathematics and mathematical physics. So the theory is always fashionable, and is a constant source of novel applications; it also regularly contributes new ideas to topology. The course is intended as a systematic but elementary introduction to toric topology, with emphasis on the more accessible aspects related to discrete mathematics.

Learning Outcomes

On successful completion of this course, students should be able to

1. perform basic operations on, and straightforward calculations with, simple polytopes and simplicial complexes,
2. understand the significance and algebraic properties of their face rings, and carry out elementary manipulations,
3. describe moment angle complexes and associated toric constructions.
   

Future topics requiring this course unit

None.

Syllabus

1. Simplicial complexes and maps.
2. Basic operation with simplicial complexes.
3. Simplicial spheres and simple polytopes.
4. Stasheff polytopes and permutohedra.
5. Constructions of simple polytopes using Minkowski sum. Nestohedra and zonotopes.
6. Ring of simple polytopes and applications.
7. Generating functions of face polynomials.
8. Stanley – Reisner face rings.
9. The moment-angle functor and applications.
10. Moment-angle complexes corresponding to joins, connected sums and bistellar moves.
11. Quasitoric manifolds.
12. Stably complex structures on quasitoric manifolds and applications.
13. Constructions of toric manifolds.

Textbooks

[1] V. M. Buchstaber, T. E. Panov, Torus actions and their applications in topology and combinatirics., AMS, University Lecture Series, v. 24, Providence, RI, 2002, 152 pp.
[2] V. M. Buchstaber, T. E. Panov, N. Ray, Spaces of polytopes and cobordism of quasitoric manifolds., MoscowMath. J., v. 7, N 2, 2007, 219–242.
[3] V. M. Buchstaber, N. Ray, An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron., Toric Topology, Contemporary Mathematics, AMS, v. 460, 2008.

Manchester Institute for Mathematical Sciences Preprints (http://www.manchester.ac.uk/mims/eprints):

2007.195: V. Buchstaber, T. Panov and N. Ray (2007), Spaces of polytopes and cobordism of quasitoric manifolds. Article.
2007.232: Victor Buchstaber (2007), Toric Topology of Stasheff Polytopes. Conference or Workshop Item.
2008.31: Victor M Buchstaber and Nigel Ray (2008), An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron. Article.

Teaching and learning methods

Two lectures and an examples class each week. In addition students should expect to spend at least seven hours on private study on this course unit.

Assessment

End of semester examination: three hours weighting 100%

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Arrangements