# Description

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.

# Mini-Courses

• (San Francisco State University & Universidad de los Andes)
Geometry of Matroids (lecture notes by Connor Simpson)

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.

Exercise sessions by Exercise sheets: 1 2 3 4.
• (University of Osnabrück) / (University of Osnabrück) / (University of Michigan)
Algebraic and combinatorial aspects of face numbers and Stanley-Reisner rings (lecture notes)

(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.

Exercise sessions by Exercise sheets: 1 2 3 4.
• (University of Washington)
Combinatorial and topological aspects of face numbers and Stanley-Reisner rings

(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.

Exercise sessions by Exercise sheets: 1 2 3 4.
• (University of California at Berkeley & Max Planck Institute for Mathematics in the Sciences Leipzig)
Polyhedral Subdivisions in Non-Parametric Statistics (slides of day 2) (slides of day 3) (R tutorial)

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
Friday: Being Totally Positive

Exercise sessions by Exercise sheets: 1.

# Important dates

• Support application deadline: January 30th, 2019.
• Application for participation deadline: February 27th, 2019.
• Contributed talk submission deadline: February 27th, 2019.
• Early registration deadline: April 25th, 2019 (if not complete earlier).
• Registration deadline: May 29th, 2019 (if not complete earlier).
• Summer school: June 17-28, 2019.

# Invited speakers

• (Université de Nantes) Topology of tropical varities (slides)

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.

• (CNRS & Université de Grenoble-Alpes) Bridges between embedded graphs and the geometry of surfaces (slides)

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.

• (École Nationale des Ponts et Chaussées) Envy-free division of a cake: the poisoned case, and other variations (slides)

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.

• (Sorbonne Université) Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics (slides)

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.

• (Université Paris-Sud) The s-Weak order and and s-Permutohedron (slides)

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.

• (Goethe-Universität Frankfurt) Standard monomials, matroids, and lattice paths

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.

# Contributed talks and posters

## Contributed talks

• Jose Bastidas (Cornell University) Characteristic elements for real hyperplane arrangements (slides)
• Matías Bender (Sorbonne Université/INRIA/CNRS - LIP6) Computing Gröbner basis for sparse polynomial systems
• Christopher Borger (Otto-von-Guericke Universität Magdeburg) The mixed degree of a tuple of lattice polytopes (slides)
• Giulia Codenotti (Free University of Berlin) Hollow polytopes of large width (slides)
• Paul Görlach (Max Planck Institute MiS Leipzig) Injective projections of Segre--Veronese varieties
• Antonio Macchia (Freie Universität Berlin) The minimal cellular resolution of the edge ideals of forests (slides)
• Aida Maraj (University of Kentucky) Quantitative Properties of Ideals arising from Hierarchical Models (slides)
• Jorge Alberto Olarte (Freie Universität Berlin) Hypersimplicial subdivisions (slides)
• Georgy Scholten (North Carolina State University) Semi-Inverted Linear Spaces (slides)
• Mariel Supina (University of California, Berkeley) The Hopf monoid of orbit polytopes (slides)
• Miruna-Stefana Sorea (Max Planck Institute for Mathematics in the Sciences) The shapes of level curves of real polynomials near strict local minima (slides)

## Contributed posters

• Markus Baumeister (RWTH Aachen University) Infinite regular extensions of disc triangulations
• Oliver Clarke (University of Bristol) Toric Degenerations of Schubert Varieties
• Aram Dermenjian (Université du Québec à Montréal) The facial weak order on oriented matroids
• Oleksandra Gasanova (Uppsala University) On the Ratliff-Rush closure of monomial ideals
• Oscar Iglesias Valiño (Universidad de Cantabria) The complete clasification of empty 4-simplices
• Olga Kuznetsova (Aalto University) Private Information Retrieval from Databases with Colluding Servers
• Anna Margarethe Limbach (RWTH Aachen University) Polyhedral and Algebraic Approaches to the k-Dimensional Multiplication Table Problem
• Aleksandra Sobieska (Texas A&M University) Toward Free Resolutions Over Scrolls
• Ayush Kumar Tewari (TU Berlin) Point Line Geometry in the tropical plane
• Andrés David Santamaría-Galvis (Univeristy of Primorska) Shellable and collapsible complexes with specified configuration of free edges

# Programme

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

# List of participants

Ali Al Setri (San Fransisco State University)
Doriann Albertin (Université Paris-Est Marne-la-Vallée)
Federico Ardila (San Francisco State University & Universidad de los Andes)
Sergey Avvakumov (IST Austria)
Iman Bahmani Jafarloo (Politecnico di Torino)
İlayda Barış (Bogazici University)
Jose Bastidas (Cornell University)
Markus Baumeister (RWTH Aachen University)
Matías Bender (Sorbonne Université/INRIA/CNRS)
Aenne Benjes (Goethe University Frankfurt)
Christopher Borger (Otto-von-Guericke Universität Magdeburg)
Marin Boyet (CMAP École Polytechnique)
Erwan Brugallé (Université de Nantes)
Jan-Marten Brunink (Osnabrück University)
Jean Cardinal (Université Libre de Bruxelles)
Kyle Celano (University of Miami)
Xiangying Chen (Freie Universität Berlin)
Huanhuan Chen (Paris-Saclay University)
Lorenzo Ciardo (University of Oslo)
Oliver Clarke (University of Bristol)
Giulia Codenotti (Freie Universität Berlin)
Jeshu Dastidar (San Francisco State University)
Esha Datta (UC Davis )
Arnaud de Mesmay (CNRS, GIPSA-lab)
Bishal Deb (Université Paris-Est Marne-la-Vallée)
Aram Dermenjian (Université du Québec a Montréal)
Eliana Duarte (MPI Leipzig )
Clément Dupont (Université de Montpellier)
Sophia Elia (Freie Universität Berlin)
Sabrina Enriquez (University of California Davis)
Oleksandra Gasanova (Uppsala University)
Paul Görlach (Max Planck Institute MiS Leipzig)
Alexandros Grosdos Koutsoumpelias (Osnabrück University)
Derek Hanely (University of Kentucky)
Alfredo Hubard (Université Paris-Est Marne-la-Vallée)
Oscar Iglesias Valiño (Universidad de Cantabria)
R. Amzi Jeffs (University of Washington Seattle)
Martina Juhnke-Kubitzke (University of Osnabrück)
Daniel Kalmbach (RWTH Aachen Universität)
Kunda Kambaso (RWTH Aachen Universität)
Jinha Kim (Seoul National University)
Kaie Kubjas (Sorbonne Université)
Olga Kuznetsova (Aalto University)
Evgeniya Lagoda (FU Berlin, Berlin Mathematical School)
Anna Margarethe Limbach (RWTH Aachen University)
Yefei Ma (ETH Zurich)
Antonio Macchia (Freie Universität Berlin)
Sebastian Manecke (Goethe University Frankfurt)
Aida Maraj (University of Kentucky)
Leonardo Martínez-Sandoval (Sorbonne Université)
Frédéric Meunier (École Nationale des Ponts et Chaussées)
Kim Minki (Technion)
Sophie Morier-Genoud (Sorbonne Université)
Calder Morton-Ferguson (University of Toronto)
Vu Nguyen Dinh (Paris 13 University)
Isabella Novik (University of Washington)
Jorge Alberto Olarte (Freie Universität Berlin)
Zuzana Patáková (IST Austria, Charles University)
Marta Pavelka (University of Miami)
Raul Penaguiao (University of Zurich)
Eva Philippe (ENS Paris)
Vincent Pilaud (École polytechnique)
Viviane Pons (Université Paris-Sud)
Ivan Rasskin (IMAG)
Julian Ritter (École polytechnique)
Rowan Rowlands (University of Washington )
Andrés David Santamaría-Galvis (UP-FAMNIT (University of Primorska))
Raman Sanyal (Goethe-Universität Frankfurt)
Georgy Scholten (North Carolina State University )
Grégoire Sergeant (Uiversité Paris 7)
Connor Simpson (University of Wisconsin -- Madison)
Hannah Sjöberg (Freie Universität Berlin)
Aleksandra Sobieska (Texas A&M University)
Miruna-Stefana Sorea (Max Planck Institute for Mathematics in the Sciences)
Simon Andres Soto Ochoa (Eindhoven University of Technology)
Bernd Sturmfels (University of California at Berkeley & Max Planck Institute for Mathematics in the Sciences Leipzig)
Mariel Supina (University of California, Berkeley)
Soumeya Merwa Tebtoub (University of Science and Technology Houari Boumedienne)
Ayush Kumar Tewari (TU Berlin)
Ling Hei Tsang (The Ohio State University)
Elias Tsigaridas (INRIA Paris)
Lorenzo Venturello (University of Osnabrück)
Andres Vindas Melendez (University of Kentucky)
Pascal Wild (Institute of Science and Technology Austria)
Martin Winter (TU Chemnitz)
Lei Xue (University of Washington)
Hailun Zheng (University of Michigan)

# Venue

The school will take place at the Pierre et Marie Campus of the Faculty of Science & Engineering of Sorbonne Université at Jussieu (check the openstreetmap). Lectures will be at room 15-25-104 at the 1st floor between towers 15 and 25.

# Contact

GACschool@imj-prg.fr

# Related events

## Conferences

• FPSAC 2019, July 1–5, 2019, University of Ljubljana, Slovenia