This two-week summer school will present an overview of a selection of topics in the crossroads between geometry, algebra, and combinatorics. It will consist of four one-week mini-courses given by leading experts, complemented with supervised exercise sessions, lectures by invited researchers, and research presentations by junior participants. The school is aimed at young researchers in geometric and algebraic combinatorics and related fields, although any interested researcher is welcome to participate. We expect the school to be an opportunity to bring together researchers from different backgrounds and foster new interactions.
This school is part of the ongoing ANR project CAPPS.
Matroid theory is a combinatorial theory of independence which has its origins in linear algebra and graph theory, and turns out to have deep connections with many other fields. With time, the geometric roots of the field have grown much deeper, bearing many new fruits. The geometric approach to matroid theory has recently led to the development of fascinating mathematics at the intersection of combinatorics, algebra, and geometry, and to the solution of long-standing questions. This course will provide a gentle introduction to matroid theory, with an emphasis on the geometric viewpoint, and an eye towards some recent successes of the theory.
(Part I) This course will concentrate on the study of face numbers of simplicial complexes. Our hope is to convey a flavor of this fascinating field by providing a sample of major results and problems along with a variety of techniques developed over the last fifty years. We will discuss face numbers of simplicial polytopes, spheres, and manifolds, balanced complexes, flag complexes, and centrally symmetric complexes. Be prepared to learn techniques that build on tools from combinatorics and discrete geometry, commutative algebra, and elementary algebraic topology. During the first week we will mainly focus on simplicial polytopes and spheres, while during the second week we will both discuss some generalizations to simplicial manifolds and specializations to particular classes of complexes.
(Part II) This course will concentrate on the study of face numbers of simplicial complexes. Our hope is to convey a flavor of this fascinating field by providing a sample of major results and problems along with a variety of techniques developed over the last fifty years. We will discuss face numbers of simplicial polytopes, spheres, and manifolds, balanced complexes, flag complexes, and centrally symmetric complexes. Be prepared to learn techniques that build on tools from combinatorics and discrete geometry, commutative algebra, and elementary algebraic topology. During the first week we will mainly focus on simplicial polytopes and spheres, while during the second week we will both discuss some generalizations to simplicial manifolds and specializations to particular classes of complexes.
This course, taught by Bernd Sturmfels and Kaie Kubjas, will give an introduction to the geometric combinatorics of log-concave density estimation. This topic belongs to the field of non-parametric statistics, and it opens up some fascnating questions about convex polytopes and their trinagulations. The primary sources are an article by Robeva, Sturmfels and Uhler, and a follow-up paper jointly with Tran. For the relevant background in geometric combinatorics see the book on Triangulations by De Loera, Rambau and Santos. Participants are encouraged to take a look at the exercises now, especially Question 15 on our software for Thursday. The topics four lectures will be as follows: Monday: Regular Subdivisions and Secondary Polytopes Tuesday : Geometry of Log-Concave Density Estimation Wednesday: Getting your Hands Dirty Friday: Being Totally Positive
Tropical varieties are polyhedral complexes in R^n satisfying the so-called balancing condition. It has been known for a long time that a tropical cubic curve in R^2 has genus at most 1. In this talk I will explain how to construct a plane tropical cubic curve of arbitrary genus. In particular, I will resolve the apparent contradiction of the last two sentences. More generally, I will talk about (upper and lower) bounds on Betti numbers of tropical varieties of R^n (and if time permits on tropical Hodge numbers). Generalizing what is written above for cubics, I will show that there is no finite upper bound on the total Betti numbers of projective tropical varieties of degree d and dimension m. This is a joint work with B. Bertrand and L. Lopez de Medrano.
In this talk, we will illustrate on precise problems how the continuous (Riemannian) geometry of surfaces can shed light on the combinatorics of embedded graphs, and vice versa. We will showcase connections between on the first side geodesics, optimal homotopies and sweep-outs, and on the other side non-trivial cycles (edge-width), the complexity of a planar searching problem (homotopy height) and branch decompositions of planar graphs. Based on joint works with Erin Chambers, Gregory Chambers, Éric Colin de Verdière, Alfredo Hubard, Francis Lazarus, Tim Ophelders and Regina Rotman.
Given a cake (identified with the interval [0,1]) and players with different tastes, the envy-free cake-cutting problem asks for a partition of the cake into connected pieces so that it is possible to assign the pieces to the players without making any of them jealous. The Stromquist-Woodall theorem ensures the existence of such an envy-free division under mild conditions. Recently, Segal-Halevi asked whether these conditions could be even further relaxed by allowing that some players dislike the cake (e.g., they know the cake has been poisoned). In the traditional setting, all players are "hungry", and always prefer to get something instead of nothing. We provide a partial answer to that question and propose also other extensions, e.g., when suddenly one player disappears. Based on joint work with Florian Frick, Kelsey Houston-Edwards, Francis E. Su, Shira Zerbib.
Frieze patterns were introduced in the early 70’s by Coxeter. They are elementary algebraic constructions represented as arrangements of numbers in the plane. At first glance they may look like just a mathematical entertainment for a long and boring train ride, but it turns out that friezes are deeply connected to many classical areas of mathematics such as projective geometry, number theory, enumerative combinatorics. Many variants of friezes have been recently studied in connection with more fields: cluster algebras, quiver representations, moduli spaces, integrable systems, algebraic combinatorics,… This talk will give an overview of major results and recent developments in the theory of friezes.
We present a partial order on some decreasing trees which generalizes the weak order on permutations. This is motivated by similar generalizations on the Tamari lattice. We prove that this new order is always a lattice and contains the nu-Tamari lattice as a sublattice. Besides, it has very interesting geometrical properties which leads to beautiful constructions and conjectures.
Every finite collection of points $V$ is the set of solutions to some system of polynomial equations. This is a reasonable way to represent V, in particular when it’s easier to write down the equations than the actual points (think stable sets of a graph). Moreover, every function on V is the restriction of a polynomial. Such a polynomial is not unique and possibly hard to find in practice. Gröbner bases are one way to go as standard monomials give a basis for the polynomial functions on V. What if $V$ carries some interesting combinatorics? Can this be seen on the level of standard monomials? In this talk I will ponder this question. For general 0/1-configurations, it turns out that the standard monomials are encoded by an (interesting?) simplicial complex. For matroid base configurations, it turns out that these simplicial complexes satisfy an interesting deletion-contraction-type relation. And for lattice path matroids, interestingly these complexes can be explicitly described in terms of lattice path combinatorics. This is work in progress with Alexander Engström and Christian Stump.
| Time | Monday 17 | Tuesday 18 | Wednesday 19 | Thursday 20 | Friday 21 |
|---|---|---|---|---|---|
| 9:00 | Registration | ||||
| 9:30 | Ardila | Ardila | Pons | Ardila | Ardila |
| 10:00 | |||||
| 10:30 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |
| 11:00 | Supina | Supina | Supina | Supina | Supina |
| 11:30 | Brugallé | ||||
| 12:00 | |||||
| 12:30 | Lunch break | Lunch break | Lunch break | Lunch break | Lunch break |
| 14:30 | Codenotti | Olarte | Macchia | Bastidas | Scholten |
| 15:00 | Venturello | Venturello | Morier-Genoud | Zheng | Zheng |
| 15:30 | |||||
| 16:00 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |
| 16:30 | Venturello | Venturello | Poster session | Venturello | Venturello |
| 17:00 | |||||
| 17:30 |
| Time | Monday 24 | Tuesday 25 | Wednesday 26 | Thursday 27 | Friday 28 |
|---|---|---|---|---|---|
| 9:00 | Registration | ||||
| 9:30 | Sturmfels | Sturmfels | Meunier | Sturmfels | Sturmfels |
| 10:00 | |||||
| 10:30 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |
| 11:00 | Kubjas | Kubjas | Sorea | Kubjas | Kubjas |
| 11:30 | de Mesmay | ||||
| 12:00 | |||||
| 12:30 | Lunch break | Lunch break | Lunch break | Lunch break | Lunch break |
| 14:00 | Novik | ||||
| 14:30 | Maraj | Görlach | Borger | Bender | |
| 15:00 | Novik | Novik | Sanyal | Novik | Coffee break |
| 15:30 | Zheng | ||||
| 16:00 | Coffee break | Coffee break | Coffee break | Coffee break | |
| 16:30 | Zheng | Zheng | Problem session | Zheng | |
| 17:00 | Awards | ||||
| 17:30 |
