Description
The interaction between matroid theory and other areas of mathematics
has recently seen major progress on several fronts. This doctoral school will offer earlycareer 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
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Laura Anderson (SUNY Binghamton) 

Combinatorial models for the real and complex
Grassmannians
TBA

Alex Fink (Queen Mary London) 

Tropical Grassmannians
TBA

June Huh (Institute for Advanced Studies, Princeton U, Clay Mathematical Institute) 

Hodge theory for combinatorial geometries
A conjecture of Read predicts that the coefficients of the chromatic polynomial of any graph form a logconcave sequence. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a logconcave sequence for any configuration of vectors in a vector space. All known proofs use Hodge theory for projective algebraic varieties, and the more general conjecture of Heron, Rota, and Welsh for possibly ‘nonrealizable’ matroids is still open. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the HodgeRiemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This cohomology theory gives strong restrictions on numerical invariants of tropical varieties, and in particular those conjectured by Heron, Rota, and Welsh. Joint work with Karim Adiprasito and Eric Katz.

Matteo Varbaro (Genova) 

Commutative algebra of StanleyReisner rings
TBA

Felipe Rincòn (Oslo) 

Positroids and the positive Grassmannian
TBA

Geordie Williamson (MPI Bonn) 

An introduction to the geometry of KazhdanLusztig
polynomials
This course will be an introduction to the geometric interpretation of KazhdanLusztig polynomials as the graded dimension of local intersection cohomology of Schubert varieties. I will try to focus on concrete examples (e.g. Grassmannians), where the geometry and topology can be understood explicitly. If time permits I will also discuss some Hodge theoretic aspects, again with the examples of Grassmannians in mind.
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
Schedule

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, Kclasses of matroids and equivariant
localization, Duke Math. J. 161 no. 14 (2012), 26992723.
arxiv:1004.2403

Hodge theory for combinatorial geometries

Commutative algebra of StanleyReisner rings

Slides of the lecture can be downloaded here .

For the literature on StanleyReisner rings:
E. Miller, B. Sturmfels Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, 2005.

For an account on the Upper Bound Theorem and the gconjecture:
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 gtheorem 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 noncrossing partitions. Transactions of the American Mathematical Society 368 (2016): 337363.

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): 17101752.

Alexander Postnikov, Total positivity, Grassmannians, and networks. Preprint. Available at http://wwwmath.mit.edu/~apost/papers/tpgrass.pdf .

Konstanze Rietsch and Lauren Williams. Discrete Morse theory for totally nonnegative flag varieties. Advances in Mathematics 223 6 (2010): 18551884.

Introdution to the geometry of KazhdanLusztig polynomials

The original paper defining KazhdanLusztig polynomials:
Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), no. 2, 165184.

An account which is easier to understand and uses the notation used in the lectures:
Soergel, Wolfgang KazhdanLusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1 (1997), 83114 (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, 7577.

Lascoux and Schützenberger's combinatorial formula for Grassmannian KazhdanLusztig polynomials:
Lascoux, Alain; Schützenberger, MarcelPaul
Polynômes de Kazhdan & Lusztig pour les grassmanniennes. (French) [KazhdanLusztig polynomials for Grassmannians] Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), pp. 249266,
Astérisque, 8788, 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. 185203,
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, 535633.

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), 5171,
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
email by
October 5th, 2015
[Click for details]
Who can apply: everybody.
How to apply: send an email to one of the organizers
explaining your research interests and the reasons of your interest in / motivation to attend
the seminar. Graduate students and postdocs should include a
CV in pdf format. For graduate students we also ask for a letter
of support from the supervisor.
Notification of acceptance: by October 16.
Financial aspects. The seminar participants are housed at
''Hôtel les sources''; the cost, comprehensive of lodging
and full board, is 143 CHF per person and per night (double room)
or 167 CHF per person and night (single room). Graduate students
from CUSO institutions will be fully reimbursed and must also
subscribe to this activity at the CUSO's website . Other applicants
should indicate whether they need financial support towards
lodging or travel. As the seminar's organization and budgeting
progress, we will be happy to use any available funds in order to
offer (partial) support, with preference going to younger participants.
Travel directions
The venue of the seminar can be comfortably reached by
public transportation.
[Click for details]
The best way to reach Les
Diablerets is by train (train station "Les Diablerets"). The website of the Swiss
Federal Railways has a very handy and reliable electronic timetable which can be used to research train
schedules, even with departure/arrival in other countries.
The nearest airports are Geneva, Zurich and Basel. It must be noted
that the stops "Geneva airport" and "Zurich airport" are located
directly under the airports themselves, so the connection is very
practical. From either Geneva or Zurich you'll have to change trains
at least once in order to get to Les Diablerets: do not worry if the
electronic timetable gives you quite tight connections, these usually
work out.
Notice: The hotel rate is comprehensive of the dinner on
Sunday, until the lunch on Friday.
Funding