Source: http://coxeter-conv.lacim.uqam.ca/
Timestamp: 2019-04-25 02:03:52+00:00

Document:
imaginary cones and infinite root systems.
we also obtain new families of interesting polytopes, which is very useful in convex geometry since such families are fairly rare.
August 20-22: discussions and collaborations.
The registration is now closed. For any information, please contact coxeter-conv@lacim.ca.
Reception: Monday 13th, from 8h30 to 9h, in the LaCIM central room PK-4214 (4th floor of President-Kennedy building, see section Location).
Minicourses: The lectures will take place in room PK-5115 (5th floor).
(*) During the second slot of the afternoon (15h30-17h... or more), the lecturers will be available to answer questions, or talk about the exercises. This is also a time devoted for blackboard discussions between the participants (we provide coffee and you'll make the theorems!).
You can download the pdf version with a more detailed schedule.
The description of the lectures can be found in the section Lectures below.
Workshop: The problems stated during the lectures of the first week, as well as problems that may be proposed by the participants, will be the basis for the discussions of the workshop days.
You can now consult the tentative schedule of the workshop.
Finite reflection groups and permutahedra. In this first lecture we will put the emphasis on a presentation of finite Coxeter groups by the mean of a family of convex polytopes named permutahedra. Then we will give a construction of a second class of convex polytopes, generalized associahedra, that is intimately linked to permutahedra.
On the geometry of infinite root systems. The aim of this second lecture is to present some open problems that arise when trying to generalize some combinatorial and geometrical properties seen in the first lecture to infinite Coxeter groups.
 M. Dyer, On the Weak Order of Coxeter groups, arXiv:1108.5557.
 C. Hohlweg, Permutahedra and Associahedra: Generalized associahedra from the geometry of finite reflection groups in « Associahedra, Tamari Lattices and Related Structures », Tamari Memorial Festschrift, editors F. Mueller-Hoissen, J. Pallo and J. Stasheff, Progress in Mathematics, vol. 299, (Birkhauser, 2012), arXiv:1112.3255.
 C. Hohlweg, J. P. Labbé and V. Ripoll, Asymptotic behaviour of roots of infinite Coxeter groups I, arXiv:1112.5415.
 J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press.
Introduction to the theory of polytopes. The first lecture is a basic introduction to polyhedral geometry and polytopes. Our goal is to clarify different notions of equality for polytopes and explain possible relations between distinct families of equal polytopes.
Cambrian theory and associahedra. We use associahedra in the second lecture as a fundamental example to show how certain phenomena that occur in the combinatorics of Coxeter groups can be described geometrically.
Generalized permutahedra. The third lecture puts these phenomena into the unifying geometric picture of generalized permutahedra and discusses some aspects of generalized permutahedra.
 C. Ceballos, F. Santos, G. M. Ziegler, Many non-equivalent realizations of the associahedron, arXiv:1109.5544.
 C. Hohlweg, C. Lange, H. Thomas, Permutahedra and generalized associahedra, Adv. Math. 226 (2011), 608-640.
 J. Morton, L. Pachter, A. Shiu, B. Sturmfels, O. Wienand, Convex rank tests and semigraphoids, SIAM J. Discrete Math. 23 (2009), 1117--1134.
 N. Reading, D. Speyer, Cambrian fans, J. Europ. Math. Soc. (JEMS) 11 (2009), 407-447.
 A. Postnikov, Permutahedra, associahedra and beyond, Int. Math. Res. Not. (2009), 1026-1106.
 A. Postnikov, V. Reiner, L. Williams, Faces of generalized permutahedra, Documenta Math. 13 (2008) 207-273.
We will present the construction of the brick polytope of a subword complex, which was developed in type A in [PS2] and in general finite types in [PS1]. The first lecture will be devoted to the combinatorics of subword complexes and to the definition of the brick polytope, while the second one will cover the geometric properties of the brick polytope and its application to generalized associahedra. We will essentially follow the presentation of [PS1].
[PS1] V. Pilaud and C. Stump. Brick polytopes of spherical subword complexes: a new approach to generalized associahedra (arXiv:1111.3349).
[PS2] V. Pilaud and F. Santos. The brick polytope of a sorting network. European J. Combin. 2012.
[KM] A. Knutson and E. Miller. Subword complexes in Coxeter groups. Adv. Math. 2004.
[PP] V. Pilaud and M. Pocchiola. Multitriangulations, pseudotriangulations and primitive sorting networks. Discrete and Comput. Geom. 2012.
[CLS] C. Ceballos, J.-P. Labbé and C. Stump. Subword complexes, cluster complexes, and generalized multi-associahedra (arXiv:1108.1776).
[P] V. Pilaud. The greedy flip tree of a subword complex (arXiv:1203.2323).
Vector labelled face lattice of a polyhedral cone. Motivated by 1), we attach to a polyhedral cone in a Euclidean space the Hasse diagram of its face lattice with edges of the Hasse diagram labelled by suitable vectors (). The face lattice is again the face poset of a regular CW complex of a sphere.
Deodhar's conjecture and the nil Hecke ring. We discuss Deodhar's conjecture, which may be viewed as the analogue for Bruhat order of the trivial fact that a d-dimensional convex polytope has at least (d+1) vertices. The proof (see ) involves rational functions from Kostant and Kumar's nil Hecke ring, which arises in the study of T-equivariant cohomology of flag varieties of Kac-Moody groups.
Volume formula for polyhedral cones. We indicate the analogous arguments to those in 3) in the context 2) instead of 1), which turn out to involve rational functions describing volume formulae for frustums of cones.
Conjecture on Bruhat intervals and polyhedral cones. Partly motivated by 3)-4) (and more fully by ) we present a part of some conjectures which suggest there is a natural common generalization of the structures considered in 1) and 2).
Coxeter groups, Bruhat order: Ch 5 of , Ch 2 of .
Polyhedral cones, polytopes, homogenization: Lectures 0-2 of .
Posets, regular CW complexes, shellability: Appendix 4.7 of .
 Humphreys, James E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990.
 Björner, Anders and Brenti, Francesco, Combinatorics of Coxeter groups, Graduate Texts in Mathematics 231, Springer, New York, 2005.
 Ziegler, Günter M., Lectures on polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York,1995.
 Björner, Anders and Las Vergnas, Michel and Sturmfels, Bernd and White, Neil and Ziegler, Günter M., Oriented matroids, Encyclopedia of Mathematics and its Applications 46, Cambridge University Press, Cambridge, 1999.
 Dyer, M. J., Bruhat intervals, polyhedral cones and Kazhdan-Lusztig-Stanley polynomials, Math. Z. 215 (1994), 223-236.
 Dyer, M. J., The nil Hecke ring and Deodhar's conjecture on Bruhat intervals, Invent. Math. 111, (1993), 571--574.
 Dyer, M. J., Representation theories from Coxeter groups, in Representations of groups (Banff, AB, 1994), CMS Conf. Proc. 16, 105--139, Amer. Math. Soc., Providence, RI, 1995.
The minicourses and the workshop will take place in the President-Kennedy building on the UQÀM campus. The entrance is located at 201, President-Kennedy Street (it is also one of the entrances of the metro station Place des Arts).
The following hotels are within walking distance to the conference venue, and have an agreement with UQÀM to offer special rates. When you make your booking, ask for the corporate rate for UQÀM.
N.B.: prices are indicated per night, in Canadian dollars, without taxes.
Armor Manoir Sherbrooke: 99$ for a superior room: queen size bed or two single beds (twin), private bathroom, includes breakfast.
L'Appartement Hôtel: Single occupancy: studio apartment (queen size bed + kitchenette): 127$ (115$ if you stay 7 nights or more). Double occupancy: suite (one bedroom with queen size bed + living-room with sofa bed + kitchenette): $133 (150$ if you stay 7 nights or more). Includes breakfast.
Delta Montréal: $138 for single occupancy.
Important: for the participants who have financial support from the LaCIM, we remind you to keep all your receipts (invoice of plane tickets, boarding passes, hotel bill, gas bill, restaurant receipts); this will make much easier the process of reimbursement.
Montreal is served by the Pierre Elliott Trudeau Airport (YUL). From the airport, a taxi to the downtown core costs about $35-$45. More economically, you can take the Airport Express bus #747, which operates 24 hours per day and costs $8 (exact fare necessary coins only!). The ticket provides also with a transit pass valid during 24h for all the STM network.
The closest stop to UQAM is stop number 8 (René Lévesque and Jeanne-Mance).
Buses arrive and depart from the Station Centrale d'autobus, which is a short walk to/from UQAM. Intercity bus service is offered by Megabus, Coach Canada, Adirondack Trailways, Greyhound.
Trains arrive and depart from the Gare Centrale, which is a short walk to/from UQAM. Train service is provided by Via Rail Canada and Amtrak.
More information can be found on the WikiTravel website for Montreal.
Thursday dinner: We reserved in the restaurant Le Nil Bleu (ethiopian food) on Thursday 16th at 6:30pm (3706 rue St-Denis, corner Avenue des Pins). Please let us know by Tuesday noon about your presence and the number of accompanying people.
Information document: Here is a pdf document with general information on the conference and on Montreal, in particular a map of the restaurants near LaCIM (will be distributed on Monday).
Photos: the group photos can be found on that page.
Funding: We are grateful for the financial support from the LaCIM.

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