federico ardila
matemático
professor . san francisco state university
profesor adjunto . universidad de los andes . colombia


. research . teaching . society . outreach . cv . interviews . other .

      . writing . talks .

            . research . expository .



Research monograph.

Hopf monoids and generalized permutahedra.
        (Memoirs of the American Mathematical Society, 289 Number 1437, September 2023.)
        with Marcelo Aguiar


Research preprints.

Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly .
        (submitted)
        with Naya Banerjee and Coleson Weir.

The tropical critical points of an affine matroid.
        (SIAM Journal of Discrete Mathematics, to appear)
        with Chris Eur and Raul Penaguiao.

The degree of a tropical root surface of type A.
        (Orbita Mathematica, to appear)
        with Mont Cordero Aguilar.


Research papers.

Intersection theory of matroids: variations on a theme.
        (Clay Lecture, British Combinatorial Conference, 2024)

The geometry of geometries: matroid theory, old and new.
        (Proceedings of the International Congress of Mathematicians 6 (2022) 4510-4541.)

Lagrangian geometry of matroids.
        (Journal of the American Mathematical Society, 36 (2023) 727–794.)
        with Graham Denham and June Huh

Lagrangian combinatorics of matroids.
        (Algebraic Combinatorics, 6 (2023) 387–411.)
        with Graham Denham and June Huh

The bipermutahedron.
        (Combinatorial Theory 2 (3) 2022, #1.)

Valuations and the Hopf monoid of generalized permutahedra.
        (International Mathematics Research Notices (2022) rnab355.)
        with Mario Sanchez

The harmonic polytope.
        (Selecta Mathematica 27 (2021) 1-31.)
        with Laura Escobar

The arithmetic of Coxeter permutahedra.
        (Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales, 44 (173) (2020) 1152-1166.)
        with Matthias Beck and Jodi McWhirter

CAT(0) geometry, robots, and society.
        (Notices of the American Mathematical Society (67) (2020), 977--987.)

The equivariant Ehrhart theory of the permutahedron.
        (Proceedings of the American Mathematical Society, 148 (2020) 5091-5107.)
        with Mariel Supina and Andrés Vindas-Meléndez

Deformations of Coxeter permutahedra and Coxeter submodular functions.
        (Advances in Mathematics, 365 (2020) 107039.)
        with Federico Castillo, Chris Eur, and Alex Postnikov

The geometry of matroids.
        (Notices of the American Math. Society, 65 (2018) 902-908.)

The equivariant volumes of the permutahedron.
        (Discrete and Computational Geometry, 65 (2021) 618-635.)
        with Anna Schindler and Andrés Vindas-Meléndez

Algebraic structures on polytopes.
        Notices of the American Math. Society 65 (2018), 11-12.
        (The Notices of the AMS asked me to write this snapshot as a preview to my plenary talk at JMM 2018.)

The configuration space of a robotic arm in a tunnel.
        with Hanner Bastidas, César Ceballos, John Guo
        (SIAM Journal of Discrete Mathematics, 31 (2017) 2675-2702.)
        Accompanying software and demos

Positively oriented matroids are representable.
        with Felipe Rincon and Lauren Williams
        (Journal of the European Math. Society, 19 (2017) 815-833.)

The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial.
        with Erwan Brugallé
        (International Math. Research Notices, 2017 (2017) 614-641.)

The closure of a linear space in a product of lines.
        with Adam Boocher
        (Journal of Algebraic Combinatorics, 43 (2016) 199-235.)

The topology of the external activity complex of a matroid.
        with Federico Castillo and Jose Samper
        (Electronic Journal of Combinatorics, 23 (2016) #P3.8.)

The configuration space of a robotic arm in a tunnel of width 2.
        Federico Ardila, Hanner Bastidas, Cesar Ceballos, John Guo
        (Proceedings of FPSAC 2016, Discrete Mathematics and Theoretical Computer Science, BC (2016) 61-72.)
        Accompanying software and demos

Positroids and non-crossing partitions.
        with Felipe Rincon and Lauren Williams
        (Transactions of the American Math. Society, (2016) 368 337-363.)

The arithmetic Tutte polynomials of the classical root systems.
        with Federico Castillo and Mike Henley
        (International Math. Research Notices, (2015) 2015 3830-3877.)

Moving robots efficiently using the combinatorics of CAT(0) cubical complexes.
        with Tia Baker and Rika Yatchak
        (SIAM Journal of Discrete and Applied Mathematics 28 (2014) 986-1007.)

Two counterexamples for power ideals of hyperplane arrangements.
        with Alex Postnikov
        (Transactions of the American Mathematical Society, 367 (2015) 3759-3762.)

Universal polynomials for Severi degrees of toric surfaces.
        with Florian Block
        (Advances in Mathematics. 237 (2013) 165-193.)

Lifted generalized permutahedra and composition polynomials.
        with Jeff Doker
        (Advances in Applied Mathematics, 50 (2013) 607-633.)

Acyclic systems of permutations and fine mixed subdivisions of simplices.
        with César Ceballos
        (Discrete and Computational Geometry, 49 (2013) 485-510.)

Staircase Schur functions are P-Schur positive.
        with Luis Serrano
        (Journal of Algebraic Combinatorics, 36 (2012) 409-423.)

Geodesics in CAT(0) cubical complexes.
        with Megan Owen and Seth Sullivant
        (Advances in Applied Mathematics, 48 (2012) 142-163.)

Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes.
        with Thomas Bliem and Dido Salazar
        (Journal of Combinatorial Theory Series A, 118 (2011) 2454-2462.)

Combinatorics and geometry of power ideals.
        with Alex Postnikov
        (Transactions of the American Mathematical Society 362 (2010) 4357-4384.)

Root polytopes and growth series of root lattices.
        with Matt Beck, Serkan Hosten, Julian Pfeifle, and Kim Seashore
        (SIAM Journal of Discrete Mathematics 25 (2011) 360-378.)

Matroid polytopes and their volumes.
        with Carolina Benedetti and Jeff Doker
        (Discrete and Computational Geometry 43 (2010) 841-854.)

When do two planted graphs have the same cotransversal matroid?
        with Amanda Ruiz
        (Boletin de la Sociedad Matematica Mexicana, 16 (2010) 63-73.)

Valuations for matroid polytope subdivisions.
        with Alex Fink and Felipe Rincon
        (Canadian Journal of Mathematics, 62 (2010) 1228-1245.)

Tropical hyperplane arrangements and oriented matroids.
        with Mike Develin
        (Mathematische Zeitschrift, 262 (2009) 795-816.)

Pruning processes and a new characterization of convex geometries.
        with Elitza Maneva
        (Discrete Mathematics, 309 (2009) 3083-3091.)

Flag arrangements and triangulations of products of simplices.
        with Sara Billey
        (Advances in Mathematics, 214 (2007) 495-524.)

Transversal and cotransversal matroids via their representations.
        (Electronic Journal of Combinatorics, 14 (2007), #N6.)

Bergman complexes, Coxeter arrangements, and graph associahedra.
        with Vic Reiner and Lauren Williams
        (Seminaire Lotharingien de Combinatoire, 54A (2006), Article B54Aj)

The positive Bergman complex of an oriented matroid.
        with Carly Klivans and Lauren Williams
        (European Journal of Combinatorics, 27 (2006), 577-591.)

The Bergman complex of a matroid and phylogenetic trees.
        with Carly Klivans
        (Journal of Combinatorial Theory, Series B, 96 (2006), 38-49.)

Subdominant matroid ultrametrics.
        (Annals of Combinatorics, 8 (2004), 379-389.)

Semimatroids and their Tutte polynomial.
        (Revista Colombiana de Matemáticas, 41 (2007) 39-66.)

Computing the Tutte polynomial of a hyperplane arrangement.
        (Pacific Journal of Mathematics, 230 (2007), 1-26.)

The Catalan matroid.
        (Journal of Combinatorial Theory, Series A., 104 (2003), 49-62.)

Enumerative and algebraic aspects of matroids and hyperplane arrangements.
        (my Ph.D. thesis, MIT, 2003).

The number of halving circles.
        (American Mathematical Monthly, 111 (2004), 586-592)

The coefficients of a Fibonacci power series.
        (Fibonacci Quarterly, 42 (2004), 202-204)

A generalization of Euler's R>=2r.
        (Crux Mathematicorum, 21 (1995), 1-2.) (Note.)



Expository work for mathematicians.

NOTE: Expository work for the general public is here.

The geometry of geometries: matroid theory, old and new.
        Proceedings of the International Congress of Mathematicians 2022, to appear.

Algebraic and geometric methods in enumerative combinatorics
is one of the two long chapters of the Handbook of Enumerative Combinatorics
        edited by Miklos Bona.
        (Discrete Mathematics and Its Applications 87, CRC Press -- Chapman Hall, 2015.)

Tutte polynomials of hyperplane arrangements and the finite field method.
        (Handbook on the Tutte Polynomial, Chapter 29.)
        edited by Joanna Ellis-Monaghan, Iain Moffatt.

Catalan numbers.
        on the Catalunyan separatist movement and combinatorics
        (Mathematical Intelligencer 38 (2016) 4-5.)

Números de Catalán.
        sobre el movimiento separatista de Cataluña y la combinatoria
        (universo.math 2 (2016) No. 3, Artículo 2.)

Tres lecciones en combinatoria algebraica, I. Matrices totalmente negativas y funciones simétricas.
        with Emerson León, Mercedes Rosas, and Marcos Skandera
        (universo.math 1 (2014) No. 1, Artículo 5.)

Tres lecciones en combinatoria algebraica, II. Funciones simétricas y teoría de representaciones.
        with Emerson León, Mercedes Rosas, and Marcos Skandera
        (universo.math 1 (2014) No. 2, Artículo 5.)

Tres lecciones en combinatoria algebraica, III. Arreglos de hiperplanos.
        with Emerson León, Mercedes Rosas, and Marcos Skandera
        (universo.math 1 (2014) No. 3, Artículo 3.)

Pflasterungen.
        translation of "Tilings" to German by Gunter Ziegler
        (Mathematische Semesterberichte, 53 (2006), 17-43.)

Teselaciones.
        translation of "Tilings" to Spanish by Carolina Benedetti
        (Gaceta de la Real Sociedad Matematica Espanola 14 (2011) 463-490.)

54th International Mathematical Olympiad, Shortlisted Problems with Solutions
        with Ilya Bogdanov, Geza Kós, Carlos Gustavo Tamm de Araújo Moreira and Christian Reiher
        (Santa Marta, Colombia, 2013.)

Some problems I have proposed.
        (an incomplete list)