Two intensive mini-courses will be given over one week by experts on polyhedral products and toric topology. These are designed to give graduate students and early career researchers in nearby fields a better understanding of each subject by presenting multiple viewpoints from different lecturers, thus enabling the participants to better engage in the subsequent workshops and seminars.

Mini-course on Polyhedral products

Instructors: Jelena Grbic and Martin Benderky

Oultine:

Lecture 1 : Martin
Preliminary definitions covering basic objects in combinatorics and commutative algebra:
simplicial complexes & polytopes
Stanley-Reisner ring

Lecture 2: Jelena
Main topological definitions:
Davis-Januszkiewicz, moment-angle complex, polyhedral products

Lecture 3: Jelena
Homotopy theoretical properties:
colim \simeq hocolim
homotopy fibration relating DJ and ZK.

Lecture 4: Martin
Tor algebra and resolutions (prep for cohomology of MAC)

Lecture 5: Martin
Cohomology ring of MAC and Hochster theorem

Lecture 6: Jelena
Massey products and Goladness

Lecture 7: Martin
Real moment-angle complexes and Cai’s work on their cohomology

Lecture 8: Jelena
Stable decompositions and J construction

Lecture 9: both
Open problems

Mini-course on the Topology of Toric Spaces and Moment-angle Manifolds

Lecturers: Anthony Bahri, Taras Panov

Topics covered:

1. A. Bahri
Toric manifolds and toric varieties. The Davis-Januszkiewicz construction of Toric (Quasitoric) Manifolds and their
elementary properties.

2. A. Bahri
Moment-angle manifolds and the Buchstaber-Panov formalism for toric manifolds.

3. T. Panov
Smoothing moment-angle manifolds: intersections of quadrics and quotient constructions.

4. T. Panov
Moment maps, symplectic reduction and presymplectic structures.

5. T. Panov
Complex-analytic structures on moment-angle manifolds and their complex geometry.

6. T. Panov
Holomorphic foliations and basic cohomology. 

7. A. Bahri
Weighted  projective spaces as a model for singular toric spaces

8. A. Bahri
More general toric orbifolds

9. Problem session

Literature.

To Part I:
1. Davis, Michael W.; Januszkiewicz, Tadeusz. Convex polytopes, Coxeter orbifolds and torus actions.
Duke Math. J. 62 (1991), no. 2, 417-451.
doi:10.1215/S0012-7094-91-06217-4.
https://projecteuclid.org/euclid.dmj/1077296365
2. A. Bahri, M.Bendersky, F. R. Cohen,  Polyhedral products and features of their homotopy theory,
to appear in  Handbook of Homotopy Theory, Haynes Miller, editor.
https://arxiv.org/pdf/1903.04114.pdf
3.   A. Bahri, M. Bendersky, F. Cohen and S. Gitler, A generalization of
the Davis-Januszkiewicz construction
and applications to toric manifolds and iterated polyhedral products, Perspectives in Lie Theory, F. Callegaro, G. Carnovale, F. Caselli, C. De Concini, A. De Sole (Eds.), Springer INdAM series, (2017), 369-388.
4.   A. Bahri, S. Sarkar and J. Song,  Infinite families of equivariant
formal toric orbifolds,
To appear in Forum Mathematicum   https://arxiv.org/abs/1801.04094
5.   A. Bahri, M. Franz and N. Ray,The Equivariant Cohomology Ring of
Weighted Projective Spaces,
Mathematical Proceedings of the Cambridge Philosophical Society, (2009), 146, 395-404.
6.   A. Bahri, D. Notbohm, S. Sarkar and J. Song,  On the integral
cohomology of certain orbifolds,
To appear in International Mathematics Research Notices.
https://arxiv.org/abs/1711.01748

To Part II:
1. T. Panov. Geometric structures on moment-angle manifolds.
Russian Math. Surveys 68 (2013), no.3, 503-568.
2. V. Buchstaber and T. Panov. Toric Topology. Mathematical Surveys and Monographs, vol.204, AMS, Providence, RI, 2015 (Chapters 5&6).
3. T. Panov, Yu. Ustinovsky and M. Verbitsky Complex geometry of moment-angle manifolds. Math. Zeitschrift 284 (2016), no. 1, 309-333.