TORIC TOPOLOGY - Malaga Spring School 2010

Last updated: 06 April 2010

Aims and objectives

I am not enthusiastic about mathematics lectures that consist mainly of reading notes, pdf files, or powerpoint presentations out loud, however well-written they may be. In this situation, the audience often gains little more than by private study of the supporting material. I prefer to hang lectures on a less rigid framework, by providing signposts, statements, references, examples, (and warnings!) for listeners to develop detailed proofs and interconnections for themselves. Hopefully, this happens through discussion and debate, in which I also like to participate.

Things may, of course, work out less well than I would like!

Prerequisites

I shall assume that participants are aware of (this certainly does not mean ‘have read’!) the following, and hope that they will have access to them during the week.

1. Michele Audin. Torus Actions on Symplectic Manifolds. 2nd revised edition. Progress in Mathematics 93, Birkhauser (2004).
2. Victor Buchstaber and Taras Panov. Torus Actions and Their Applications in Topology and Combinatorics}. University Lecture Series 24, AMS, Providence RI (2002).
3. Victor Buchstaber and Nigel Ray. An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron. In Proceedings of the International Conference in Toric Topology; Osaka City University 2006}, 1--27. Edited by Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov. Contemporary Mathematics 460, AMS, Providence RI (2008).
4. William Fulton. Introduction to Toric Varieties. Annals of Mathematics Studies 131, Princeton University Press, Priceton NJ (1993).
5. Richard Stanley. Combinatorics and Commutative Algebra. 2nd edition. Progress in Mathematics 41, Birkhauser (1996).

So far as more detailed knowledge is concerned, the following are basic, and will be assumed.

• Abelian groups such as n-dimensional lattices and tori, how they act on sets and topological spaces, and the concept of equivariance;
• finite simplicial complexes, both abstract and geometric;
• elementary homotopy theory, such as the idea of the fundamental group and homotopy equivalence;
• the notion of cohomology ring of a topological space, particularly with integer coefficients;
• a feeling for the concepts of smooth manifold and vector bundle, and familiarity with examples such as n-dimensional complex projective space and its tautological complex line bundle;
• graded polynomial algebras, and their quotients by graded ideals.

Everyone has their own favourite sources for this material. I often suggest that graduate students should consult the following.

1. Allen Hatcher. Algebraic Topology, and Vector Bundles and K-theory; the former is published by Cambridge University Press (2002). Both are available for download here.
2. J Peter May. A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics, University of Chicago (1999). This is available for download here.
3. John Milnor. Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ (1997) (reprint of the 1965 original).

More advanced background material, and an overview of activity around the planet, should be displayed on the Manchester Toric Topology page (if only I can find time to keep it up to date!)

An Apology

I would prefer to give these lectures in Spanish, rather than assume that listeners understand English. Unfortunately, my vocabulary is about 20 words, including ‘hola’! However, if I set myself the task of learning a few additional words and phrases each day, maybe the audience can help me to improve during my visit.

The Lectures

As I complete my planning for each lecture, I shall add a corresponding pdf file below. This process is unlikely to be in chronological order, and the presentation may turn out to be rather unorthodox!

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Bibliography