Álvaro

Gutiérrez Cáceres

a.gutierrezcaceres🌀bristol.ac.uk

Welcome

Hello! I'm Álvaro Gutiérrez, welcome to my personal page. I'm doing a PhD in Mathematics at the University of Bristol under the supervision of Mark Wildon. I got my BSc in the University of Seville in 2021 and my MSc in the University of Bonn in 2023.

Research

Research 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

Weak lumping of Markov chains on finite groups (with E. Crane, E. Russell, M. Wildon). In preparation.
Bender-Knuth involutions for types B and C. Accepted to Electronic Journal of Combinatorics, arXiv:2311.10659 (2023)
Determinant of the distance matrix of a tree (with E. Briand, L. Esquivias, A. Lillo, M. Rosas). Accepted to FPSAC24 (extended abstract). Full text in preparation.
Partial symmetries of iterated plethysms (with M. Rosas). Annals of Combinatorics, arXiv:2201.00240 (doi) (2022).
Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients (with M. Rosas). Comptes Rendus Mathématiques, arXiv:2109.11378 (doi) (2023).
Strategy in Geoguessr: games of competitive Euclidean geometry. Preprint, ResearchGate (2021).

Theses:

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

Talks

Upcoming talks:

Shuffles that frustrate card-counters. Algebra Seminar, University of Bristol. (May 2024)
Symmetric polynomials are symmetric (in types A, B, and C). Symmetric groups and Lie-theoretic friends, University of Birmingham. (May 2024)
On the determinant of the distance matrix of a tree. 30th British Combinatorial Conference, Queen Mary University, London. (July 2024)
On the determinant of the distance matrix of a tree. FPSAC 2024, University of Bochum. (Poster presentation.) (July 2024)

Past talk archive:

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

Conferences co-organized

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

Secondary reviewer for FPSAC 2024.

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).