This research school is part of the
MDM FOCUSED RESEARCH PROGRAMME ON COMBINATORIAL GEOMETRIES & GEOMETRIC COMBINATORICS
Research school on Combinatorial Geometries and Geometric Combinatorics - Polytope week
Sign into October 10, 2025
Venue: Centre de Recerca Matemàtica
Room: Auditorium
description
Perspectives on flags of matroids and positroids
Carolina Benedetti
Universidad de los Andes
Abstract: The uniform matroid U_{k,n} of rank k over [n] can be thought of as the hypersimplex D_{k,n}. Subdivisions of D_{k,n} into positroids have gained much attention recently, in particular for their connection to nonnegativity in tropical geometry. Now, the Minkowski sum D_n= D_{1,n}+\cdots+D_{n,n} is the polytope of the flag matroid of uniform matroids. What can be said about subdivisions of D_n into flags of positroids?
We aim to use this minicourse as a way to start understanding this question, by refining it and asking about subdivisions of D_n into flags of lattice path matroids. To this end we will introduce positroids, flags of matroids, (flag) matroid polytopes and, of course, lattice path matroids. At the end of the minicourse several open problems will be shared with the participants in relation with such subdivisions. These lectures will be based on current work by Benedetti, and previous work by Benedetti-Knauer.
Moduli polytopes: parametrizing combinatorial structures with polytopes
Raman Sanyal
Goethe-Universität Frankfurt
Abstract: Combinatorial structures such as permutations, rooted trees, and matroids can be realized as vertices of polytopes, with the face structure reflecting combinatorial similarities. This perspective provides a powerful geometric way for studying discrete objects. In this minicourse we will explore such ‘moduli polytopes’, focusing on those arising as fiber polytopes and pivot rule polytopes. Along the way, we will encounter (poset) permutahedra, associahedra, and flag matroid polytopes in various guises, and highlight open problems and research opportunities.
Organising Committee
Kolja Knauer | Universitat de Barcelona – Centre de Recerca Matemàtica
Anna de Mier | Universitat Politècnica de Catalunya
Arnau Padrol | Universitat de Barcelona – Centre de Recerca Matemàtica
Julian Pfeifle | Universitat Politècnica de Catalunya
Vincent Pilaud | Universitat de Barcelona – Centre de Recerca Matemàtica
Scientific Committee
Martina Juhnke | Universität Osnabrück
Peter Nelson | University of Waterloo
Felipe Rincón | Queen Mary University of London
Martha Yip | University of Kentucky
schedule
Monday October 6th | Tuesday October 7th | Wednesday October 8th | Thursday October 9th | Friday October 10th | |
9:30 – 9:50 | Registration | ||||
9:50 – 10:00 | Welcome | ||||
10:00 – 11:00 | Course 1 Perspectives on flags of matroids and positroids Carolina Benedetti Universidad de los Andes | Course 1 Perspectives on flags of matroids and positroids Carolina Benedetti Universidad de los Andes | Course 1 Perspectives on flags of matroids and positroids Carolina Benedetti Universidad de los Andes | Course 2 Moduli polytopes: parametrizing combinatorial structures with polytopes Raman Sanyal Goethe-Universität Frankfurt | Course 2 Moduli polytopes: parametrizing combinatorial structures with polytopes Raman Sanyal Goethe-Universität Frankfurt |
11:00 – 11:30 | Coffee break | Group picture + coffee break | Coffee break | ||
11:30 – 13:00 | Exercise session | Exercise session | Exercise session | Exercise session | Exercise session |
13:00 – 14:30 | Lunch | ||||
14:30 – 15:30 | Course 2 Moduli polytopes: parametrizing combinatorial structures with polytopes Raman Sanyal Goethe-Universität Frankfurt | Course 2 Moduli polytopes: parametrizing combinatorial structures with polytopes Raman Sanyal Goethe-Universität Frankfurt | Course 1 Perspectives on flags of matroids and positroids Carolina Benedetti Universidad de los Andes | Presentation of research projects | |
15:30 – 15:45 | Break | Break | |||
15:45 – 17:15 | Exercise session | Exercise session | Exercise session | ||
20:00 | Social dinner | ||||
LIST OF PARTICIPANTS
| Name | Institution |
|---|---|
| Vincent Pilaud | Universitat de Barcelona |
| Arnau Padrol Sureda | Universitat de Barcelona |
| Anouk Brose | University of California Davis |
| Leo Jiang | University of Toronto |
| William Dugan | Universitat de Barcelona |
| Yirong Yang | University of Washington - Seattle |
| Marguerite Bin | University of Lorraine |
| Chiara Giardino | Ruhr University Bochum |
| Chenying Lin | University of Regensburg |
| Félix Gélinas | York University |
| Egor Bakaev | University of Copenhagen |
| Natasha Crepeau | University of Washington - Seattle |
| Leonardo Saud Maia Leite | Royal Institute of Technology |
| Jhon Bladimir Caicedo Portilla | University of Osnabrück |
| Teemu Lundström | Aalto University |
| Niraj Velankar | University of Greifswald |
| Alexandros Leivaditis | Ruhr University Bochum |
| Torben Donzelmann | University of Osnabrück |
| Juliana Curtis | University of Washington - Seattle |
| David Alcántara | Universidad de Cantabria |
| Jan Stricker | Goethe University Frankfurt |
| Hung Hoang | Vienna University of Technology |
| Hoi Ping Luk | Západoceská univerzita v Plzni |
| Dmitry Mineev | Bar-Ilan University |
| Carolina Benedetti Velásquez | Universidad de los Andes, Bogotá |
| Bilal Khan | Henan Academy of Sciences |
| Katarina Krivokuca | Free University of Berlin |
| Spencer Backman | University of Vermont |
| Sania Asif | Henan Academy of Sciences |
| Jette Gutzeit | University of Zurich |
| Lauren Nowak | University of Washington - Seattle |
| Francesco Verciani | University of Kassel |
| Jon Pål Hamre | Royal Institute of Technology |
| Danai Deligeorgaki | Universitat de Barcelona |
| Ileana Gonzalez Escalante | Universidad Nacional Autonoma de Mexico |
| Florian Rieg | Goethe University Frankfurt |
| Chi Ho Yuen | National Yang Ming Chiao Tung University |
| Raman Sanyal | Goethe University Frankfurt |
| Juan Luis Valerdi Cabrera | Independent |
| Sophia Keip | FernUniversität in Hagen |
| Federico Lazzeri | Royal Institute of Technology |
| Kolja Knauer | Universitat de Barcelona – Centre de Recerca Matemàtica |
| Julian Pfeifle | Universitat Politècnica de Catalunya |
| Mónica Blanco | Universidad de Cantabria |
| Francisco Santos Leal | Universidad de Cantabria |
| Tähvend Uustalu | Independent |
| Yannic Vargas | CUNEF Universidad |
| Marcel Celaya | Cardiff University |
| Pallavi Panda | University of Paris 13 |
| Emil Verkama | Royal Institute of Technology |
| Sienna Unter | University of Vermont |
| Anna de Mier | Universitat Politècnica de Catalunya |
| Geoff Whittle | Victoria University of Wellington |
| Elna Guardiola | Universitat de Barcelona |
| Júlia Sabiote Cartes | Universitat de Barcelona |
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Acknowledgements
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