BMS/IMPAN Block Course: Convex Geometry Related to Hamiltonian Group Actions

This is a block course held jointly by the Institute for Mathematics of the Polish Academy of Science (IMPAN) and Berlin Mathematical School (BMS), organized by Tadeusz Januszkiewicz and Michael Joswig.

The course will be held one week in Berlin (Nov 27 - Dec 1, 2017) and one week in Bedlewo (Mar 19 - Mar 23, 2018). There will be lectures and exercises during these two weeks. For the time in-between there will be additional project work.

The course is also open for students of all three Berlin universities, who can get credits equivalent to a regular course "Discrete Geometry III" of TU Berlin. There was a first meeting on Tuesday, Nov 21, 10:00 in MA 621, where mathematical requirements, the organization of the projects etc were discussed.

It will be assumed that the participants have some basic knowledge in the following subjects: cellular homology and cohomology, polytopes and linear programming, smooth manifolds, group actions.

Personal application (via email) is mandatory since there is a limit on the number of participants.

Tentative Schedule for the week of Mar 19 - Mar 23, 2018 [Bedlewo]

Monday, Mar 19 14:30-16:00 Joswig h-vectors of polytopes and spheres
16:30-18:00 Exercises
Tuesday, Mar 20 9:00-10:30 Januszkiewicz Symplectic geometry, hamiltonian group actions, convexity theorems
11:00-12:30 Exercises
14:30-16:00 Joswig The g-theorem
16:30-18:00 Exercises
Wednesday, Mar 21 9:00-10:30 Januszkiewicz Goresky-MacPherson-Kottwitz and other cohomology computations
11:00-12:30 Exercises
14:30-16:00 Joswig Computing face lattices and f-vectors
16:30-18:00 Exercises
Thursday, Mar 22 9:00-10:30 Januszkiewicz Singular symplection structures and group actions: symplectic origami and related singularities
11:00-12:30 Exercises

Schedule for the week of Nov 27 - Dec 1, 2017 [Berlin]

Monday, Nov 27 9:00-10:30 Januszkiewicz General strategy and simple examples
11:00-12:30 Exercises
14:30-16:00 Joswig Affine toric varieties
16:30-18:00 Exercises
Tuesday, Nov 28 9:00-10:30 Joswig Projective toric varieties
11:00-12:30 Exercises
14:30-16:00 Januszkiewicz Right angled Coxeter groups
16:30-18:00 Exercises
Wednesday, Nov 29 9:00-10:30 Kastner polymake Demo: Polytopes and Toric Varieties
11:00-12:30 Exercises
Thursday, Nov 30 9:00-10:30 Januszkiewicz Topological toric manifolds
11:00-12:30 Exercises
14:30-16:00 Joswig Even simple polytopes
16:30-18:00 Exercises
Friday, Dec 1 9:00-10:30 Joswig A colorful Lebesgue theorem
11:00-12:30 Exercises
14:30-16:00 Januszkiewicz Right angled buildings
16:30-18:00 Exercises

Text book references

  1. Abramenko & Brown: Buildings - theory and applications, Springer 2008
  2. Buchstaber & Panov: Torus actions and their applications in topology and combinatorics, AMS 2002
  3. Buchstaber & Panov: Toric topology, AMS 2015
  4. Cox, Little & Schenck: Toric varieties, AMS 2011
  5. Davis: The geometry and topology of Coxeter groups, Princeton 2008
  6. Dold: Lectures on algebraic topology, reprint of the 1972 edition, Springer 1995
  7. Humphreys: Reflection groups and Coxeter groups, Cambridge 1990
  8. Joswig & Theobald: Polyhedral and algebraic methods in computational geometry, Springer 2013
  9. Stanley: Combinatorics and commutative algebra, 2nd ed., Birkhäuser 1996
  10. Warner: Foundations of differentiable manifolds and Lie groups, corrected reprint of the 1971 edition, Springer 1983
  11. Ziegler: Lectures on polytopes, Springer 1995

Special topics

  1. Bahri, Bendersky, Cohen & Gitler: The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces. Adv. Math. 225 (2010), no. 3, 1634-1668.
  2. Baralić & Živaljević: Colorful versions of the Lebesgue, KKM, and Hex theorem, J. Combin. Theory Ser. A (2017)
  3. Davis & Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. (1991)
  4. Joswig: Projectivities in simplicial complexes and colorings of simple polytopes, Math. Z. (2002)
  5. Park, Park & Park: Graph cubeahedra and graph associahedra in toric topology, arXiv:1801.00296

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Last modified: Fri Mar 08:47:40 UTC 2018 by joswig