Raul Penaguiao

Post Doc at the Max Planck Institute for the Sciences

Algebraic Combinatorics

20 anos 20 valores conference - Lisbon, 2013

About me

I will be teaching a course in algebraic methods in combinatorics in the Spring semester 2023.

Research interests: Hopf algebras and Hopf monoids; Combinatorics of graphs, permutation patterns, posets and polytopes; root systems and Coxeter groups.

I'm currently a PostDoc at the MPI in Leipzig, Germany.

My research projects as of 30th of September, 2022.

I finished my PhD at Universität Zürich, under the supervision of Valentin Feray. My PhD thesis can be found here.

You can also see my master thesis, and a work on the chromatic symmetric function developed during my masters.

Check out the seminar Algebra, Geometry and Combinatorics, that I am organizing this Fall semester.

You can also see more photos of me: building an oven, in my masters cerimony, hiking in Kyrgyzstan and in Switzerland.

A modular relation on nestohedra

Contact Me

Raul Penaguiao
Location Contact
MPI Leipzig raul.penaguiao(at)mis.mpg.de
Inselstraße 22 +49 (0) 341 - 9959 - 811
Office number: G3 04
Leipzig, Germany

Publications and Research

Papers accepted for publication

Papers submited for publication

Conference proceedings

Talks and Seminares

  1. The discrete signature Veronese variety, at Probabilistic methods, Signatures, Cubature and Geometry, York UK - January 2023 - slides
  2. Computing degrees of Bergman fans in a funny way, at MaTroCom, London UK - January 2023 - slides
  3. MPI Nonlinear Algebra Seminar, Leipzig Germany - November 2022 - slides
  4. Introduction to Tropical Gemetry with Matroids, at Seminar Algebra and Combinatorics - slides
  5. A new cancellation-free formula for permutation patterns Hopf algebra, at ACPMS, Online - August 2022 - slides
  6. Cancellation-free formulas on pattern Hopf algebras, at Summer school in algebraic combinatorics, Krakow Poland - July 2022 - poster
  7. Pattern Hopf algebras in combinatorial presheaves, at Rough Algebra Workshop, Berlin Germany - March, 2022 - slides.
  8. Using intersection of fans to construct more fans?, at AGC Seminar SFSU San Francisco - February 2022 - slides.
  9. The Birkoff von Neumann polytope, at Alternating sign matrices FU Berlin Seminar FU Berlin - June 2021 - slides.
  10. Feasible regions on permutation patters - with dimension results, at Permutation Patterns 2021, Virtual Workshop Online - June 2021 - slides.
  11. Matroids: the five ways to think about them, at a Villa Student Seminar in FU Berlin - April 2021 - slides.
  12. Feasible regions meet pattern avoidance. The awaited 3rd part on feasible regions, at Algebra, Geometry, and Combinatorics Seminar SFSU San Francisco US - February 2021 - slides.
  13. The UC Berkeley combinatorics seminar, Berkeley US - March 2021 - slides
  14. The Virtual 10th Combinatorics Days, Coimbra Portugal - November 2020 - slides.
  15. University of Florida Combinatorics Seminar, Florida USA - November 2020 - slides.
  16. Another set composition Hopf algebra and a commutative diagram with polytopes, at Hopf District seminar - January 2021 - slides.
  17. Yet another cancellation-free antipode formula - new and old methods, at your group seminar? - transparencies.
  18. Hopf algebras and combinatorics, at my thesis defense - Zürich Switzerland, May 2020 - slides and video of the defense.
  19. Quantum probability and density matrices - at reading group on quantum computing - May, 2020 Beamer
  20. The feasible region for consecutive occurrences of large permutations, at Formal Power Series and Algebraic Combinatorics, video-poster session - July, 2020 poster.
  21. Séminaire Combinatoire et Théorie des Nombres ICJ, Lyon France - December 2019.
  22. CMUC seminar, Coimbra Portugal - January, 2020.
  23. Pattern Hopf algebras in combinatorial presheaves, at Rencontre du GDR Renormalisation 2019, Calais France - October, 2019 - slides.
  24. Permutation Patterns 2019, Zürich - June, 2019 - slides.
  25. 82nd Séminaire Lotharingien de Combinatoire, Curia Portugal - April, 2019 - slides.
  26. Zurich Graduate Seminar, Zurich Switzerland - November 2019 - slides.
  27. Chromatic symmetric functions on graphs and nestohedra, at Formal Power Series and Algebraic Combinatorics, Darmouth USA - July, 2018 - slides.
  28. CMUC seminar, Coimbra Portugal - June, 2018 - slides.
  29. 8th Combinatorics day, Porto Portugal - June, 2018 - slides.
  30. 80th Séminaire Lotharingien de Combinatoire, Lyon France - March, 2018 - slides.
  31. Discrete mathematics seminar, University of Zürich - December, 2017 - slides.
  32. Coefficients of a random invariant in random graphs, at Discrete mathematics seminar, University of Zürich - September, 2016 - slides.
  33. Flips in triangulations, at Class seminar in "Geometry: Combinatorics and Algorithms", ETH Zürich, (May, 2015) - slides.
  34. Non-repetitive sequences in paths and graphs, at Mittagseminar, ETH Zürich, (May, 2015) - slides.
  35. Quandle Structures On Knots, at Diagonal School, Universidade Nova de Lisboa, (February, 2014) - slides.

You can also find me on the following events:

  1. MaTroComb - London, January 2023 link.
  2. Probabilistic methods, Signatures, Cubature and Geometry - York, January 2023.
  3. Macaulay2 Bootcamp - Leipzig, Dezember 2022 link.
  4. Two combinatorial viewpoints on geometry - Online, December 2022.
  5. Pfaffians, Tensors, and Applications - Leipzig, Dezember 2022 link.
  6. MOM workshop on MaRDI, OSCAR and MATHREPO - Berlin, November 2022 link.
  7. Geometry and Renormalisation - Potsdam, September 2022 link.
  8. Vector Bundles on Algebraic Curves - Warwick, July 2022 link.
  9. Summer School in Algebraic Combinatorics - Krakow, July 2022 link.
  10. AlCoVE: an Algebraic Combinatorics Virtual Expedition - Online, June 2021 link.
  11. Permutation Patterns 2021, Virtual Workshop - Online, June 2021 link.
  12. AMS Special Session, Algebraic and Combinatorial Aspects of Polytopes - Online, May 2021 link.
  13. (Polytop)ics: Recent advances on polytopes - Max Planck Institute (Online), April 2021 link.
  14. Hopf Extravaganza - Online, January 2021.
  15. FPSAC 2020 Ramat-Gan - Ramat-Gan Israel, July 2019 link.
  16. Algebraic Combinatorics Krakow 2020 - Krakow Poland, July 2019 link.
  17. ECCO 2020 - Bogota Colombia, June 2019 link.
  18. 84th SLC - Joyeuse France, March 2019 link.
  19. Rencontre du GDR Renormalisation - Calais France, September 2019 link.
  20. FPSAC'19, University of Ljubljana - Ljubljana Slovenia, July 2019 link.
  21. Summer School on Geometric and Algebraic Combinatorics, Paris France, June 2019 - link.
  22. Permutation Patterns 2019, University of Zurich - Zurich Switzerland, June 2019 - link.
  23. Permutation Patterns Pre-Conference Workshop, University of Zurich - Zurich Switzerland, June 2019 - link.
  24. 82nd SLC, Coimbra - Curia Portugal, April 2019 - link.
  25. Einstein workshop on Algebraic Combinatorics, FU Berlin - Berlin Germany, November 2018 - link.
  26. 81st SLC - KrattentalerFest, Strobl Austria, September 2018 - link.
  27. 4th AEC summer school, Hagenberg Austria, August 2018 - link.
  28. FPSAC'18, Dartmouth univeristy - Hanover NH USA, July 2018 - link.
  29. 8th Combinatorics Day, Faculdade de Ciencias da Unversidade do Porto - Porto Portgual, June 2018 - link.
  30. 80th SLC, Lyon 1 - Lyon France, March 2018 - link.
  31. 79th SLC, Bertinoro Italy, September 2017 - link.
  32. FPSAC'17, Queen Mary university - London UK, July 2017 - link.
  33. 78th SLC - Thibon Fest, Muhlhouse France, March 2017 - link.
  34. 3rd AEC summer school, Hagenberg Austria, August 2016 - link.

What are student projects

If you have asked me to do a student project with me (be it a master thesis or a semester paper) I probably directed you to this section. This is not an exaustive list of all the projects that I propose, so after giving them a short look you can ask me your question or suggest another topic.

However, this is not supposed to tell the whole story of your project: Be patient, and if you have picked a topic that thrills you, you will notice a natural development that transcends the first impressions.

Do not endeavout in these projects without talking with me. It is important to set up guidelines and expectations, adapt to certain timelines, etc. If you find a book or a project here that spikes your interest, feel free to send me an e-mail.

What are the student projects

  • The cycle polytope is a graph invariant that still has many hidden properties. Understand some chromatic graph invariants, for instance exploring the book Graph theory. Investigate which graph invariants can you obtain from the cycle polytope, etc. See this paper for an introduction to the cycle polytope.
  • Understand spectal theory of graphs, for instance exploring the book Spectra of graphs. You can apply these methods to the cycle polytope introduced in this paper.
  • Analyse new methods in machine learning: A new paper from Michael Joswig details an intermediate polytope that can be useful for machine learning methods in classification problems.
  • A new and exciting problem, by Christoph Hertrich has been proposed. This takes advantage of a new connection between a specific neural network architecture (ReLU), tropical geometry and the Minkowski sum of polytopes, see the paper. This paper translates the question of representability of certain function in ReLU into a geometrical problem. Its deceiving simplicity may be the begining of several new methods and applications of geometry in ReLU neural networks.
  • In this paper, some polytopes related to permutation patterns is studied, so called the restricted feasible region for consecutive patterns. There, the restricted feasible region avoiding the patterns 123 and 132 is calculated, and some methods that can be applied to some other patterns of permutations are laid out.
  • In this paper, a symmetric function is associated to each graph. As it turns out, the same can be done in the context of any graded Hopf algebra, see this paper. Your task is to understand the shuffle algebra and obtain symmetric function invariants from this algebra.
  • Take notes on The Probabilistic Method.
  • Take notes on Combinatorial Theory.
  • Take notes on Symmetry groups and Coxeter groups.


Assistant at courses in San Francisco State University

Principal instructor in 2021 fall semester MATH 226 (Calculus) - Course webpage

Assistant in the online student seminar on polytopes in 2020 fall semester - Federico Ardila. A report by Christina Nguyen on tree polytopes.

Assistant at courses in Universitat of Zürich

Assistant in the student seminar on automatic proofs of binomial identities in 2019 fall semester - Valentin Féray - Seminar webpage

Assistant in probability II in 2019 fall semester - Valentin Féray - Course webpage

Assistant in stochastics in 2018 fall semester - Christoph Luchsinger - Course webpage

Assistant in linear algebra I in 2018 fall semester - Alberto Cattaneo - Course webpage

Assistant in Hopf algebras in 2018 spring semester - Benedict Stufler - Course webpage

Assistant in probability II in 2017 fall semester - Valentin Féray - Course webpage

Assistant in linear algebra II in 2017 spring semester - Andrew Kresch - Course webpage

Assistant in foundations of mathematics in 2016 fall semester - Mathilde Bouvel - Course webpage

Assistant at courses in ETH Zürich

Assistant in Analysis II in 2016 spring semester Course webpage

Assistant in MMP I in 2015 fall semester - Course webpage

My Vita

You can download my CV here, as of the 11th of August, 2020.

  • MPI, Leipzig, Germany
  • Post Doctoral researcher supported by Max Planck Institute.

    September 2022 - August 2023

  • SFSU, San Francisco, US
  • Post Doctoral researcher supported by Early PostDoc.Mobility grant from SNF.

    August 2021 - August 2022

  • FU Berlin, Berlin, Germany
  • Post Doctoral researcher supported by Early PostDoc.Mobility grant from SNF.

    April 2021 - July 2021

  • Zurich University, Zurich, Switzerland
  • Doctoral researcher supported by SNF.

    October 2016 - August 2020


Doctorate Studies

UZH, University of Zürich, Switzerland

Doctoral studies in mathematics - Degree (PhD) in Pure Mathematics

September 2016 - May 2020


ETH Zürich, Federal University of Zürich, Switzerland

Masters of Science in Mathematics - Degree (MSc) in Pure Mathematics

September 2014 - June 2016


Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal.

Degree (BSc) in Applied Mathematics and Computation

September 2011 - August 2014

Software Habilities
Technical languages General purpose languages Markup languages
Mathematica C++ HTML
MatLab Python
SAGE math

Languages spoken
Portuguese English German Spanish French
Mother tongue Proficient - C1 Intermediate - B1 Beginner - A2 Beginner - A1



Reports and projects
An abstract for a Workshop in permutation patterns in 2021, see their website. PDF(EN)
A report on Quandules, an algebraic structure related to knots. PDF(EN)
A report on one of the proofs from the book, Gale Shapeley algorithm and Dinitz conjecture. PDF(EN), PDF(PT)
My master thesis in the chromatic symmetric function of random graphs. PDF(EN)
A report done on the chromatic symmetric function and Stanley's conjecture. PDF(EN)
A presentation on quantum probability and density matrices for a reading group on quantum computing. PDF(EN)
Exercises on Hopf algebras
Exercise 1 - link Solutions 1 - link
Exercise 2 - link Solutions 2 - link
Exercise 3 - link Solutions 3 - link
Exercise 4 - link Solutions 4 - link
Exercise 5 - link Solutions 5 - link
Exercise 6 - link Solutions 6 - link
Exercise 7 - link Solutions 7 - link
Exercise 8 - link Solutions 8 - link
Exercises on Probability 2
Exercise 0 - link Solutions 0 - link
Exercise 1 - link Solutions 1 - link
Exercise 2 - link Solutions 2 - link
Exercise 3 - link Solutions 3 - link
Exercise 4 - link Solutions 4 - link
Exercise 5 - link Solutions 5 - link
Exercise 6 - link Solutions 6 - link
Exercise 7 - link Solutions 7 - link
Exercise 8 - link Solutions 8 - link
Exercise 9 - link Solutions 9 - link
Exercise 10 - link Solutions 10 - link
Exercise 11 - link Solutions 11 - link
Exercise 12 - link Solutions 12 - link
Exercise 13 - link Solutions 13 - link
Projects on Calculus
Projects derivation- link Projects integration - link