Álvaro Gutiérrez

a.gutierrezcaceres🌀bristol.ac.uk

Hello! I'm a third year PhD student in Mathematics, at the University of Bristol and under the supervision of Mark Wildon. I got my my MSc in the University of Bonn in 2023 and BSc in the University of Seville in 2021.

Fall of 2025 I was visiting ICERM. Until summer, I'm visiting Ilse Fischer at the University of Vienna.

Selected talks, by topic

All talks, chronologically

Projects and involvement

Events (co-)organized

Reviewing

Teaching

Research & output

Interests

My main interests of study are in combinatorics, and combinatorial approaches to algebra. More precisely, symmetric functions theory and representation theory (in its broadest sense).

Papers and preprints

Plethystic lifts of 𝒒-binomial identities (with Á. Martínez, M. Szwej, M. Wildon). Extended abstract, to appear in a special edition of Seminaire Lotharingien Combinatoire for FPSAC (2026).
A geometric and generating function approach to plethysm (with R. Orellana, F. Saliola, A. Schilling, M. Zabrocki). Preprint (2025+) arXiv:2511.02649
Schur log-concavity and the quantum Pascal triangle (with C. Krattenthaler). Preprint (2025+). arXiv:2509.22648
Modular isomorphisms of SL₂(F)-plethysms for Weyl modules labelled by hook partitions (with Á. Martínez, M. Szwej, M. Wildon). Preprint (2025+). arXiv:2509.01490
A new bijective proof of the 𝒒-Pfaff-Saalschütz identity with applications to quantum groups (with Á. Martínez, M. Szwej, M. Wildon). European Journal of Combinatorics 133 (2026). arXiv:2505.08422, doi:10.1016/j.ejc.2025.104321
Towards plethystic 𝖘𝖑₂ crystals. Extended abstract, in a special edition of Seminaire Lotharingien Combinatoire for FPSAC (2025).
Towards plethystic 𝖘𝖑₂ crystals. Combinatorial Theory 6(1) (2026). arXiv:2412.15006
Weak lumping of left-invariant random walks on left cosets of finite groups (with E. Crane, E. Russell, M. Wildon). Preprint (2024+). arXiv:2412.19742
Principal minors of the distance matrix of a tree (with A. Lillo). Preprint (2024+). arXiv:2407.01966
Non-intersecting paths and the determinant of the distance matrix of a tree (with E. Briand, L. Esquivias-Quintero, A. Lillo, M. Rosas). Preprint (2024+). arXiv:2407.01227
Bender-Knuth involutions for types B and C. Electronic Journal of Combinatorics 32(2) (2024). doi:10.37236/12571
Determinant of the distance matrix of a tree (with E. Briand, L. Esquivias-Quintero, A. Lillo, M. Rosas). Extended abstract, in a special edition of Seminaire Lotharingien Combinatoire for FPSAC (2024).
Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients (with M. Rosas). Comptes Rendus Mathématiques 361 (2023). arXiv:2109.11378, doi:10.5802/crmath.468
Partial symmetries of iterated plethysms (with M. Rosas). Annals of Combinatorics 27 (2022). arXiv:2201.00240 doi:10.1007/s00026-023-00652-4
Strategy in Geoguessr: games of competitive Euclidean geometry. Preprint (2021). hal:03688514 (See also Problem 12 of the 30th British Combinatorial Conference.)

Theses

The combinatorics of symmetric polynomials in types A, B, and C. MSc Thesis directed by C. Stroppel and J. Matherne, University of Bonn (2023). Read here. Code available: (.sage, SageMath 9.3 library).
Positivity conditions and hook+column sequences of plethystic coefficients. BSc Thesis directed by M. Rosas, Universidad de Sevilla (2021). Honors grade. idUS:11441/133706. Code available: (.ipynb, SageMath 9.0).

Open source code

A SageMath package for symplectic tableaux. GitHub. Warning: not fully tested.
A SageMath package for Littlewood-Richardson cones and Bernstein-Zelevinsky triangles (with E. Briand). (In preparation)
A SageMath patch for a bijection between Littlewood-Richardson tableaux and Knutson-Tao puzzles. (Under review)

Talks

Upcoming talks

Plethysm, crystals, and generating functions. Combinatorics seminar, Michigan State University (February 2026)
TBA. Discrete Mathematics Working Group (AgDM), joint seminar of U. Vienna and TU Wien (April 2026)
TBA. An Algebraic Combinatorics Virtual Expedition (AlCoVE 2026) (June 2026)

Selected talks, by topic

Plethysmos.
Based on joint work with R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki (arXiv:2511.02649).
- Research seminar, ICERM, University of Brown (October 2025)
Plethysm, categorification, and Stanley's hook-content formula.
Based on joint work with Á. Martínez, M. Szwej, and M. Wildon (arXiv:2505.08422 and arXiv:2509.01490).
- Category Theory, Combinatorics, and Machine Learning, ICERM, Brown University (Poster presentation.) (September 2025)
Towards plethystic 𝖘𝖑₂ crystals.
Based on arXiv:2412.15006
- FPSAC 2025, Hokkaido University (July 2025)
- 6th Graduate Student Meeting in Applied Algebra and Combinatorics, University of Bologna (April 2025)
On the determinant of the distance matrix of a tree.
Based on arXiv:2407.01966 and arXiv:2407.01227
- FPSAC 2024, University of Bochum. (Poster presentation.) (July 2024)
- 30th British Combinatorial Conference, Queen Mary University, London. (July 2024)
- 91st Seminaire Lotharingien de Combinatoire (SLC91). Salobreña, Málaga. (March 2024) Slides

All talks, chronologically

Plethysmos. Algebra and Representation theory seminar, University of Cambridge (February 2026)
Schur polynomials from Ehrhart theory. Geometry seminar, TU Graz (January 2026)
Schur polynomials from Ehrhart theory. Bristol's Junior Algebra and Combinatorics Seminar (October 2025)
Plethysmos. Research seminar, ICERM, University of Brown (October 2025)
Towards plethystic 𝖘𝖑₂ crystals. Early career research seminar, ICERM, University of Brown (October 2025)
Plethysm, categorification, and Stanley's hook-content formula. Category Theory, Combinatorics, and Machine Learning, ICERM, Brown University (Poster presentation.) (September 2025)
Towards plethystic 𝖘𝖑₂ crystals. 37th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2025), Hokkaido University (July 2025)
Towards plethystic 𝖘𝖑₂ crystals. 6th Graduate Student Meeting in Applied Algebra and Combinatorics (GSMAAC25), University of Bologna (April 2025)
Shuffles that frustate card-counters. Mathematics seminar, Royal Holloway, London. (October 2024)
The combinatorics of plethysm. Algebra seminar, University of York (October 2024)
The combinatorics of plethysm. Seminario di Algebra e Geometria, Universitá di Bologna (October 2024)
Determinant of the distance matrix of a tree. Bristol's Junior Algebra and Combinatorics Seminar (September 2024)
What makes a knot better? Welcome week, School of Mathematics of the University of Bristol. (September 2024)
On the determinant of the distance matrix of a tree. FPSAC 2024, University of Bochum. (Poster presentation.) (July 2024)
On the determinant of the distance matrix of a tree. 30th British Combinatorial Conference, Queen Mary University, London. (July 2024)
Symmetric polynomials are symmetric (in types A, B, and C). Symmetric groups and Lie-theoretic friends, University of Birmingham. (May 2024)
Shuffles that frustrate card-counters. Algebra Seminar, University of Bristol. (May 2024)
Symmetric polynomials are symmetric (in types A, B, and C). 5th Graduate Student Meeting in Applied Algebra and Combinatorics, Freie Universität Berlin (Poster presentation.) (April 2024)
On the determinant of the distance matrix of a tree. 91st Seminaire Lotharingien de Combinatoire (SLC91). Salobreña, Málaga. (March 2024) Slides
Partial symmetries of iterated plethysms of Schur functions. Introduction to Modern Advances in Algebra ITMAIA 2024. Based on arXiv:2201.00240. University of Exeter. (February 2024)
Strategy in Geoguessr. PGR seminar, University of Bristol. (December 2023)
Positivity conditions for plethystic coefficients. Seminar on Combinatorics and Lie Theory and Topology, University of Pisa, (December 2023)
The combinatorics of symmetric polynomials in types A, B, and C. University of Bonn, MSc thesis defense. (May 2023)
The coefficients of plethysm. Oberseminar "Kombinatorik". University of Bochum. (April 2023)
Computers in mathematical research. (Young researcher presentation.) Research Data in Discrete Mathematics. Max Planck Institute. Leipzig (March 2023)
Schur-Weyl duality for Lie superalgebras (with Christóforos Miliónis). Graduate Seminar in Representation Theory. University of Bonn. (December 2022)
Trees, derangements, and non-intersecting paths (with E. Briand, L. Esquivias, A. Lillo, M. Rosas). (Poster presentation.) 8th Iberian Mathematics Meeting. Institute of Mathematics of the University of Seville. (October 2022) Poster PDF (to appear soon).
Log-concavity and Huh's medal. Bonn Student Seminar. (September 2022) Slides, recording.
From highest weight to irreducible representations (with Xier Ren). Graduate Seminar in Representation Theory. Based on chapter 12.5 of Hall's book. University of Bonn. (July 2022)
When are plethystic coefficients nonzero? Student Symposium in Combinatorics SCC 2022. Based on arXiv:2109.11378. UNSW Sydney. (June 2022) Recording, slides.
Conway knotation and the Conway polynomial. Graduate Seminar in Number Theory "The Contributions of John H. Conway". Based on an article by John H. Conway. University of Bonn. (May 2022)
The Littlewood-Richardson rule three ways. Graduate Online Combinatorics Colloquium GOCC. (April 2022) Recording, slides.
Partial symmetries of iterated plethysms of Schur functions. Graduate Student in Combinatorics Conference GSCC 2022. Based on arXiv:2201.00240. UC San Diego. (March 2022) Recording.
O-minimality of semialgebraic sets (with Ariadni Karanikola). Graduate Seminar in Advanced Logic. Based on Lou van der Dries' book. University of Bonn. (October 2021) Slides.
El algoritmo de Newton-Puiseux: tres maneras de entender las series de potencias. Instituto de Matemáticas de la Universidad de Sevilla, programa PI3. (September 2021) Vídeo, diapositivas, texto.

(On the Newton-Puiseux algorithm for the resolution of singularities.)

Condiciones de positividad y sucesiones gancho+columna de coeficientes pletísticos. Universidad de Sevilla, defensa del TFG. (June 2021) Diapositivas.

(BSc Thesis defense, on plethystic coefficients.)

Projects and involvement

Events (co-)organized

Heilbronn Focused Research Group on stability of (iterated) plethysms. To happen next September!
7th Graduate Student Meeting in Applied Algebra and Combinatorics (GSMAAC) to happen in Bristol the spring of 2026!
91st Seminaire Lotharingien de Combinatoire (SLC91). (March 2024) Joint with M. Rosas, E. Briand, and C. Krattenthaler.
El Círculo Matemático Virtual. (January - December 2021) Joint with Jaime Benabent, Marta Sánchez, and Álvaro Domínguez. A series of 13 talks from inter- and national researchers directed towards senior undergraduates.

Reviewing

Reviewer for Elec. Jour. Comb., Exper. Math., Proc. of the AMS.
Secondary reviewer for FPSAC36, FPSAC37.

Teaching

Tutor for Programming in C(++) and R. (Prof. Liu and Prof. Fasiolo) University of Bristol, 2023/24.
Tutor for Kombinatorik, Graphen, Matroide. (Prof. Korte and Dr. Brenner) University of Bonn, 2022/23.
Trainer for Mathematic Olympiads (for High School students) . University of Seville, 2020. Here you can find the notes I created for my classes (in Spanish):

Resolución de problemas mediante diagramas. Enunciados, soluciones.
Combinatoria: polinomios cromáticos, triángulos de Pascal y números de Catalan. Enunciados, soluciones.

Please note that I am not the original mind behind all of these problems; most of these were collected from various sources which are not carefully cited. These notes are only meant to be read as informal notes. Let me know if you spot any errors (I know there are some, but can't remember where).

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