SM'ART Seminar on Matroids
in Algebra, Representation theory and Topology

Borel Seminar 2016, January 24-29, Les Diablerets, Switzerland.

Description The interaction between matroid theory and other areas of mathematics has recently seen major progress on several fronts. This doctoral school will offer early-career researchers exposure to areas of matroid theory that delve into algebra, topology and geometry, giving an opportunity to gain background for research in these areas and to interact with leading experts.

Lectures - click on the title in order to display the abstract.
Laura Anderson (SUNY Binghamton) -
Combinatorial models for the real and complex Grassmannians
Alex Fink (Queen Mary London) -
Tropical Grassmannians
June Huh (Institute for Advanced Studies, Princeton U, Clay Mathematical Institute) -
Hodge theory for combinatorial geometries
Matteo Varbaro (Genova) -
Commutative algebra of Stanley-Reisner rings
Felipe Rincòn (Oslo) -
Positroids and the positive Grassmannian
Geordie Williamson (MPI Bonn) -
An introduction to the geometry of Kazhdan-Lusztig polynomials
Preparatory lessons will take place Sunday afternoon in a meeting room of the Hôtel Les Sources.
Lesson I - Matroids (Emanuele Delucchi), 15:15 -- 16:30.
Lesson II - Commutative Algebra (Ivan Martino), 17:00 -- 18:15

Sunday Monday Tuesday Wednesday Thursday Friday
9:00 Anderson Anderson Williamson Huh Williamson
10:00 Coffee Coffee Coffee Coffee Coffee
10:20 Varbaro Varbaro Fink Rincón Huh
11:20 Williamson Fink Rincón Williamson Rincón
12:30 Lunch Lunch Lunch Lunch Lunch
15:15 -
- 16:30
Preparation I
(Oriented) Matroids
17:00 -
- 18:15
Preparation II
Commutative Algebra
17:15 Coffee Coffee Coffee Coffee
18:00 Anderson Varbaro Fink Huh
19:15 Dinner Dinner Dinner Dinner Dinner

References and bibliography
  • Combinatorial models for the real and complex grassmannians
  • Tropical grassmannians
    • Matt Baker, Matroids over hyperfields, arXiv:1601.01204
    • David Speyer, Tropical linear spaces, arXiv:0410455
    • Alex Fink and David Speyer, K-classes of matroids and equivariant localization, Duke Math. J. 161 no. 14 (2012), 2699-2723. arxiv:1004.2403
  • Hodge theory for combinatorial geometries
  • Commutative algebra of Stanley-Reisner rings
    • Slides of the lecture can be downloaded here .
    • For the literature on Stanley-Reisner rings: E. Miller, B. Sturmfels Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, 2005.
    • For an account on the Upper Bound Theorem and the g-conjecture:
      R. Stanley The number of faces of simplicial polytopes and spheres, Annals of the New York Academy of Sciences 440, 1985.
    • For a more recent point of view on the proof of the g-theorem for polytopes:
      D.A. Cox, J.B. Little, H.K. Schenck, Toric Varieties, Graduate Studies in Mathematics 124, 2011.
  • Positroids and the positive Grassmannian
    • Federico Ardila, Felipe Rincòn, and Lauren Williams. Positroids and non-crossing partitions. Transactions of the American Mathematical Society 368 (2016): 337-363.
    • Federico Ardila, Felipe Rincòn, and Lauren Williams. Positively oriented matroids are realizable. Journal of the European Mathematical Society, to appear. arXiv: 1310.4159.
    • Allen Knutson, Thomas Lam, and David Speyer. Positroid varieties: juggling and geometry. Compositio Mathematica 149 10 (2013): 1710-1752.
    • Alexander Postnikov, Total positivity, Grassmannians, and networks. Preprint. Available at .
    • Konstanze Rietsch and Lauren Williams. Discrete Morse theory for totally non-negative flag varieties. Advances in Mathematics 223 6 (2010): 1855-1884.
  • Introdution to the geometry of Kazhdan-Lusztig polynomials
    • The original paper defining Kazhdan-Lusztig polynomials:
      Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), no. 2, 165-184.
    • An account which is easier to understand and uses the notation used in the lectures:
      Soergel, Wolfgang Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1 (1997), 83-114 (electronic).
    • Zelevinsky's construction of small resolutions for all Schubert varieties in Grassmannians:
      Zelevinskiĭ, A. V. Small resolutions of singularities of Schubert varieties. (Russian) Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 75-77.
    • Lascoux and Schützenberger's combinatorial formula for Grassmannian Kazhdan-Lusztig polynomials:
      Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Kazhdan & Lusztig pour les grassmanniennes. (French) [Kazhdan-Lusztig polynomials for Grassmannians] Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), pp. 249-266, Astérisque, 87-88, Soc. Math. France, Paris, 1981.
    • Kazhdan and Lusztig's identification of KL polynomials with the graded dimension of IC stalks:
      Kazhdan, David; Lusztig, George Schubert varieties and Poincaré duality. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 185-203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980.
    • Background on constructible and perverse sheaves:
      de Cataldo, Mark Andrea A.; Migliorini, Luca The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535-633.
    • The ``bible'' on perverse sheaves:
      Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers. (French) [Perverse sheaves] Analysis and topology on singular spaces, I (Luminy, 1981), 5-171, Astérisque, 100, Soc. Math. France, Paris, 1982.

Participation in the seminar The seminar's venue has a limited capacity. We invite applications from interested participants to be submitted by e-mail by October 5th, 2015 [Click for details]

Travel directions The venue of the seminar can be comfortably reached by public transportation. [Click for details]