Research Interests

Publications

Seminars

Conferences

Talks

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Research interests

I like investigating problems of a geometric flavour, which appear in connection to areas in discrete geometry such as convex polytopes, simplicial complexes, Coxeter and reflection groups, subword complexes and tropical geometry. In particular, I appreciate working on problems motivated by their connections to subjects of relevance in other combinatorial and algebraic contexts, for example arising in areas such as Hopf algebras and the theory of diagonal harmonics in representation theory.

Some specific areas in which I am interested in are:

• Discrete Geometry and Polytope Theory
• Algebraic, Enumerative, and Geometric Combinatorics
• Diagonal Harmonics and Representation Theory
• Hopf Algebras and Structure Constants
• Cluster Algebras, Cluster Complexes and Coxeter Groups
• Calabi-Yau Geometry, Toric Varieties and Tropical Geometry

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Publications

My publications on arXiv, Google Scholar, and MathSciNet.

1. A canonical realization of the alt ν-associahedron. arXiv:2401.17204
   

2. The s-weak order and s-permutahedra II: the combinatorial complex of pure intervals. link, arXiv:2309.14261
   with Viviane Pons.
   Electron. J. Combin. 31 (2024), no. 3, Paper No. 3.12.

3. Generalized Heawood Graphs and Triangulations of Tori. arXiv:2307.11859, poster
   with Joseph Doolittle.
   Submitted.

4. On linear intervals in the alt v-Tamari lattices. arXiv:2305.02250
   with Clément Chenevière.
   Accepted in Combinatorial Theory.

5. Lattices of acyclic pipe dreams. arXiv:2303.11025
   with Nantel Bergeron, Noémie Cartier, and Vincent Pilaud.
   Submitted.

6. The s-weak order and s-permutahedra I: combinatorics and lattice structure. arXiv:2212.11556
   with Viviane Pons.
   Accepted in SIAM J. Discrete Math.

7. Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres. arXiv:2206.15461
   with Joseph Doolittle.
   Accepted in Discrete and Computational Geometry, 2023.

8. Revisiting Generalizations of the Dehn-Sommerville Relations. link, arXiv:2108.13145
   with Henri Mühle.
   Séminaire Lotharingien de Combinatoire, 87:Art. B87a, 25, 2022.

9. $F$- and $H$-triangles for $\nu$-associahedra. link, arXiv:2103.04769, poster
   with Henri Mühle.
   Combinatorial Theory, 2(2):Paper No. 3, 26, 2022.

10. Elliptic and q-analogs of the Fibonomial numbers. link, arXiv:1911.12785
    with Nantel Bergeron and Josef Küstner.
    SIGMA 16 (2020), 076, 16 pages.

11. The steep-bounce zeta map in parabolic Cataland . link, arXiv:1903.08515, slides
    with Wenjie Fang and Henri Mühle.
    Journal of Combinatorial Theory, Series A 172 (2020), 105210.

12. Hopf dreams and diagonal harmonics. link, arXiv:1807.03044, slides
    with Nantel Bergeron and Vincent Pilaud.
    Journal of the London Mathematical Society (2022), 105 (3), 1546–1600.

13. Signature Catalan Combinatorics. link, arXiv:1805.03863
    with Rafael S. González D'León.
    Journal of Combinatorics 10 (2019), no. 4, 725-773.

14. The v-Tamari lattice via v-trees, v-bracket vectors, and subword complexes. link, arXiv:1805.03566, extended Abstract
    with Arnau Padrol and Camilo Sarmiento.
    Electron. J. Combin. 27 (2020), no. 1, Paper 1.14, 31 pp.

15. Geometry of v-Tamari lattices in types A and B. link, arXiv:1611.09794
    with Arnau Padrol and Camilo Sarmiento.
    Trans. Amer. Math. Soc. 371 (2019), no. 4, 2575-2622.

16. The configuration space of a robotic arm in a tunnel. link, arXiv:1608.04800
    with Federico Ardila, Hanner Bastidas, and John Guo.
    SIAM J. Discrete Math., 31 (2017), no. 4, 2675-2702.
    Accompanying software and demos.

17. Cluster algebras of type D₄, tropical planes, and the positive tropical Grassmannian. link, arXiv:1511.02699
    with Sarah Brodsky and Jean-Philippe Labbé.
    J. Beitr Algebra Geom 58 (2017), 25-46.

18. A Hopf algebra of subword complexes. link, arXiv:1508.01465, extended abstract
    with Nantel Bergeron.
    Advances in Mathematics 305 (2017), 1163-1201.

19. Combinatorics of the zeta map on rational Dyck paths. link, arXiv:1504.06383, slides
    with Tom Denton and Christopher Hanusa.
    J. Combin. Theory Ser. A 141 (2016), 33-77.

20. Cluster algebras of type D: pseudotriangulations approach. link, arXiv:1504.06377
    with Vincent Pilaud.
    Electron. J. Combin. 22 (2015), no. 4, Paper 4.44, 27 pp.

21. The diameter of type D associahedra and the non-leaving-face property. link, arXiv:1406.0368
    with Vincent Pilaud.
    European Journal of Combinatorics 51 (2016), 109-124.

22. Fan realizations of type A subword complexes and multi-associahedra of rank 3. link, arXiv:1404.7380, extended abstract
    with Nantel Bergeron and Jean-Philippe Labbé.
    Discrete and Computational Geometry 54 (2015), no. 1, 195-231.
    

23. Dyck path triangulations and extendability. link, arXiv:1402.5111, extended abstract
    with Arnau Padrol and Camilo Sarmiento.
    Journal of Combinatorial Theory, Series A 131, (2015) 187-208.

24. Denominator vectors and compatibility degrees in cluster algebras of finite type. link, arXiv:1302.1052, extended abstract
    with Vincent Pilaud.
    Trans. Amer. Math. Soc. 367, (2015), 1421-1439.
    

25. Acyclic systems of permutations and fine mixed subdivisions of simplices. link, arXiv:1111.2966
    with Federico Ardila.
    Discrete and Computational Geometry 49 (3), 2013, 485-510.
    

26. Realizing the associahedron: Mysteries and questions. link, arXiv:1110.4059
    with Günter M. Ziegler.
    Associahedra, Tamari Lattices and Related Structures, Tamari Memorial Festschrift, eds. F. Müller-Hoissen, J. Pallo and J. Stasheff,
    Progress in Mathematics 299 (Birkhäuser, 2012).

27. Many non-equivalent realizations of the associahedron. link, arXiv:1109.5544
    with Francisco Santos and Günter M. Ziegler.
    Combinatorica 35 (2015), no. 5, 513-551.

28. Subword complexes, cluster complexes, and generalized multi-associahedra. link, arXiv:1108.1776, extended abstract
    with Jean-Philippe Labbé and Christian Stump.
    Journal of Algebraic Combinatorics 39 (1), 2014, 17-51.

Conference Proceedings:

1. On linear intervals in the alt ν-tamari lattices. Extended Abstract (12 pages).
   with Clément Chenevière.
   To appear in Séminaire Lotharingien de Combinatoire. FPSAC 2023 (poster).

2. $F$- and $H$-triangles for $\nu$-associahedra. Extended Abstract (12 pages).
   with Henri Mühle.
   Séminaire Lotharingien de Combinatoire 85B (2021). FPSAC 2021 (poster).

3. Elliptic and q-analogs of the Fibonomial numbers. Extended Abstract (12 pages).
   with Nantel Bergeron and Josef Küstner.
   Séminaire Lotharingien de Combinatoire 84B (2020), FPSAC 2020 (poster).

4. Hopf dreams. Extended Abstract (12 pages).
   with Nantel Bergeron and Vincent Pilaud.
   Séminaire Lotharingien de Combinatoire 82B (2019), FPSAC 2019 (talk).

5. The s-weak order and s-permutahedra. Extended Abstract (12 pages).
   with Viviane Pons.
   Séminaire Lotharingien de Combinatoire 82B (2019), FPSAC 2019 (talk).

6. The Steep-Bounce Zeta Map in Parabolic Cataland. Extended Abstract (12 pages).
   with Wenjie Fang and Henri Mühle.
   Séminaire Lotharingien de Combinatoire 82B (2019), FPSAC 2019 (talk).

7. v-Tamari lattices via subword complexes. Extended Abstract (6 pages).
   with Arnau Padrol and Camilo Sarmiento.
   Electronic Notes in Discrete Mathematics, 61 (2017) 215-221.
   Proceedings of the European Conference on Combinatorics, Graph Theory and Applications, Eurocomb 2017, Vienna, August 28 - September 1, 2017.

8. Geometry of v-Tamari lattices in types A and B. Extended Abstract (12 pages).
   with Arnau Padrol and Camilo Sarmiento.
   Séminaire Lotharingien de Combinatoire 78B (2017), FPSAC 2017 (talk).

9. Tropical Catalan Subdivisions. Extended Abstract (6 pages).
   with Arnau Padrol and Camilo Sarmiento.
   Electronic Notes in Discrete Mathematics, 54 (2016) 187-192.
   Proceedings of the Discrete Mathematics Days conference, Barcelona, 6-8 July 2016.

10. The configuration space of a robotic arm in a tunnel of width 2. Extended Abstract (12 pages).
    with Federico Ardila, Hanner Bastidas, and John Guo.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2016 (poster).

11. A Hopf algebra of subword complexes (extended abstract). Extended Abstract (12 pages).
    with Nantel Bergeron.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2016 (poster).

12. Fan realizations of type A subword complexes and multi-associahedra of rank 3. Extended Abstract (12 pages).
    with Nantel Bergeron and Jean-Philippe Labbé.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2015 (poster).

13. Diameters and geodesic properties of generalizations of the associahedron. Extended Abstract (12 pages).
    with Thibault Manneville, Vincent Pilaud, and Lionel Pournin.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2015 (poster).

14. Dyck path triangulations and extendability (extended abstract). Extended Abstract (12 pages).
    with Arnau Padrol and Camilo Sarmiento.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2015 (talk).

15. Denominator vectors and compatibility degrees in cluster algebras of finite type. Extended Abstract (12 pages).
    with Vincent Pilaud.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2013 (talk).

16. Multi-cluster complexes. Extended Abstract (12 pages).
    with Jean-Philippe Labbé and Christian Stump.
    Discrete Math. Theor. Comput. Sci. Proc. FPSAC 2012 (talk).

Theses:

• On associahedra and related topics.
  Doctoral Thesis. pdf, link.
  Freie Universität Berlin, Berlin, Germany, 2012.
  Advisor: Günter M. Ziegler.
  

• A combinatorial model for a torus fibration of a K3 surface in the large complex structure.
  Master Class final Test Problem. pdf.
  Universiteit Utrecht, Utrecht, The Netherlands, 2009.
  Advisor: Jan Stienstra.
  

• The system of permutations of a fine mixed subdivision of a simplex.
  Master Thesis. pdf.
  Universidad de los Andes, Bogota, Colombia, 2008.
  Advisor: Federico Ardila.
  

• Series incondicionalmente convergentes y normalmente sumables en espacios de Banach.
  Bachelor Thesis.
  Universidad del Valle, Cali, Colombia, 2007.
  Advisor: Guillermo Restrepo.
  

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Seminars

Past seminars:

• Combinatorics Working Seminar at the University of Vienna, 2015-2020.
• Rational Catalan Combinatorics Seminar at the University of Vienna, 2015-2016.
• Algebraic Combinatorics Seminar at the Fields Institute, 2013-2015.
• Applied Algebra Seminar at York University, 2013-2015.
• Reading Seminar on Schubert Calculus at the Fields Institute, 2013-2014.
• Seminar on Associahedra and Related Topics at the Freie Universität Berlin, 2011-2012.

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Conferences

• 6th Encuentro Colombiano de Combinatoria ECCO 2018
• 5th Encuentro Colombiano de Combinatoria ECCO 2016
• 4th Encuentro Colombiano de Combinatoria ECCO 2014

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Talks

(Last update: January 2022)

• Hopf Algebras and Diagonal Harmonics.
  Online Seminar, Algebraic and Combinatorial Perspectives in the Mathematical Sciences, January 2022.

• Reconstruction of polytopes and Kalai's conjecture on reconstruction of spheres.
  Geometry Seminar, TU Graz, Austria, January 2022.

• F- and H-Triangles for ν-Associahedra. (poster)
  FPSAC 2021, Israel, Online, January 2022.

• What is a combinatorial reciprocity.
  Graduate Online Combinatorics Colloquium (GOCC), November 2021.

• F- and H-Triangles for ν-Associahedra. (poster)
  AlCoVE: an Algebraic Combinatorics Virtual Expedition, June 2021.

• The Dehn-Sommerville relations: a survey.
  Geometry Seminar, TU Graz, Austria, June 2021.

• Hopf algebras and diagonal harmonics.
  Ghent Methusalem Junior Analysis and PDE Seminar, Ghent University, Belgium, May 2021.

• Hyperplane arrangements.
  Habilitation didactic talk, TU Graz, Austria, Mrach 2021.

• Experiencias. Youtube
  Conexión U, Universidad del Valle, Colombia, February 2021.

• The geometry of a robotic arm in a tunnel.
  Gonit Sora Webinar, India, November 2020.

• Combinatorics, geometry, algebra, and computation.
  Interview Talk, Graz, Austria, November 2020.

• Hopf dreams and diagonal harmonics.
  Advanced topics in discrete mathematics, Graz, Austria, October 2020.

• Hopf dreams and diagonal harmonics.
  Combinatorial Coworkspace, Haus Bergkranz, Kleinwalsertal, Austria, March 2020.

• Permutahedral matchings, zonotopal tilings, and d-partitions.
  Plateau Saclay Combinatorics Seminar, École Polytechnique, Paris, France, November 2019.

• The s-weak order and s-permutahedra.
  FPSAC 2019, Ljubljana, Slovenia, July 2019.

• Hopf dreams.
  FPSAC 2019, Ljubljana, Slovenia, July 2019.

• The steep-bounce zeta map in parabolic Cataland.
  FPSAC 2019, Ljubljana, Slovenia, July 2019.

• The s-weak order and s-permutahedra
  Discrete Geometry Seminar, Freie Universität Berlin, Germany, June 2019.

• Hopf dreams and diagonal harmonics.
  Math Colloquium, Universidad de Los Andes, Bogota, Colombia, May 2019.

• Hopf dreams and diagonal harmonics.
  Foro del Departamento, Universidad del Valle, Cali, Colombia, April 2019.

• Hopf dreams and diagonal harmonics.
  Algebra and Incidence Geometry Seminar, Vrije Universiteit Brussel, Belgium, March 2019.

• Hopf dreams and diagonal harmonics.
  Algebra and Number Theory Seminar, Graz, Austria, October 2018.

• The steep-bounce conjecture.
  Joint Meeting of the Czech, Slovenian, Austrian, Slovak and Catalan Mathematical Societies CSASC 2018, Bratislava, Slovakia, September 2018.

• On s-Catalan combinatorics.
  Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, April 2018.

• Hopf dreams.
  Séminaire Philippe Flajolet, l'Institut Henri Poincaré, Paris, France, April 2018.

• Combinatorics and geometry of v-Tamari lattices.
  Oberseminar Geometry, University of Fribourg, Switzerland, February 2018.

• Hopf dreams.
  SFB Status Seminar, Strobl, Austria, December 2017.

• Combinatorics and geometry of v-Tamari lattices.
  19th ÖMG Congress and Annual DMV Meeting, Salzburg, Austria, September 2017.

• The configuration space of a robotic arm in a tunnel.
  19th ÖMG Congress and Annual DMV Meeting, Salzburg, Austria, September 2017.

• v-Tamari lattices via subword complexes.
  Eurocomb 2017, European Conference on Combinatorics, Graph Theory and Applications, Vienna, Austria, August 2017.

• Geometry of v-Tamari lattices in types A and B.
  FPSAC 2017, London, England, July 2017.

• Combinatorics and geometry of v-Tamari lattices.
  Algebraic and Geometric Combinatorics of Reflection Groups, Centre de Recherches Math\'ematiques, Montreal, Canada, June 2017.

• The configuration space of a robotic arm in a tunnel.
  Geometry and Analysis on Groups Research Seminar, University of Vienna, Austria, April 2017.

• v-Tamari lattices via subword complexes.
  The 78th Séminaire Lotharingien de Combinatoire, Domaine Saint-Jacques, France, March 2017.

• The configuration space of a robotic arm in a tunnel.
  Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, March 2017.

• v-Tamari lattices via subword complexes.
  Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, December 2016.

• Subword complexes in algebra, combinatorics, and geometry.
  Algebra and Number Theory Seminar, Universität Graz, Austria, November 2016.

• Tropical Catalan Subdivisions.
  Seminar at École Polytechnique, Palaiseau, France, April 2016.

• Tropical Catalan Subdivisions.
  The 76th Séminaire Lotharingien de Combinatoire, Domaine Saint-Jacques, France, April 2016.

• Subword complexes in algebra, combinatorics, and geometry.
  SFB-AEC Statusseminar 2015, Strobl am Wolfgangsee, Austria, November 2015.

• Combinatorics of the zeta map on rational Dyck paths.
  Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, November 2015.

• Dyck path triangulations and extendability.
  FPSAC 2015, Daejeon, South Korea, July 2015.

• Subword complexes in algebra, combinatorics, and geometry.
  UBC Math Colloquium, University of British Columbia, Canada, January 2015.

• Combinatorics of the zeta map on rational Dyck paths.
  Coloquio Latino Americano de Algebra, Lima, Peru, December 2014.

• A Hopf algebra of subword complexes.
  Coloquio Latino Americano de Algebra, Lima, Peru, December 2014.

• Subword complexes in algebra, combinatorics, and geometry.
  Discrete Geometry and Combinatorics Seminar, Cornell University, USA, November 2014.

• Denominators vectors, Hopf algebras, and multi-associahedra.
  Hall and Cluster Algebras, Centre de Recherches Mathématiques (CRM), Canada, May 2014.

• Dyck path triangulations and extendability.
  Combinatorics Seminar, University of Washington, March 2014.

• Dyck path triangulations and extendability.
  Applied Algebra Seminar, York University, Canada, February 2014.

• Mini-course: Topics in discrete, tropical and Calabi-Yau geometry.
  60th anniversary of the Mathematics Department, Universidad del Valle, Cali, Colombia, January 2014.

• Subword complexes in algebra, combinatorics, and geometry.
  Foro del Departamento, Universidad del Valle, Cali, Colombia, August 2013.

• Subword complexes in algebra, combinatorics, and geometry.
  XIX Colombian Mathematics Conference, Barranquilla, Colombia, July 2013.

• Denominator vectors and compatibility degrees in cluster algebras of finite types.
  FPSAC 2013, Paris, France, June 2013.

• Subword complexes in algebra, combinatorics, and geometry.
  Combinatorics Seminar, University of Minnesota, USA, May 2013.

• Multi-cluster complexes.
  FPSAC 2012, Nagoya, Japan, July 2012.

• Subword complexes, cluster complexes, and generalized multi-associahedra.
  MDS Monday Colloquium, Berlin, Germany, January 2012.

• Subword complexes, cluster complexes, and generalized multi-associahedra.
  Algebra-Geometry-Combinatorics Seminar, San Francisco State University, USA, November 2011.

• Subword complexes, cluster complexes, and generalized multi-associahedra.
  Algebra and Discrete Mathematics Seminar, UC Davis, USA, November 2011.

• Subword complexes, cluster complexes, and generalized multi-associahedra.
  Applied Algebra Seminar, York University, Canada, November 2011.

• Subword complexes, cluster complexes, and generalized multi-associahedra.
  Student-Postdoc Seminar, CIRGET-LaCIM, UQAM, Canada, November 2011.

• Many non-equivalent realizations of the associahedron.
  LaCIM Seminar, LaCIM, UQAM, Canada, November 2011.

• What is tropical geometry?.
  What is Seminar, Berlin Mathematical School, Germany, October 2011.

• Many non-equivalent realizations of the associahedron.
  BMS Days, Berlin, Germany, February 2011.

• What is tropical geometry?.
  University of Zurich Graduate Colloquium, Zurich, Switzerland, November 2010.

• Una breve introducción a la geometría de Calabi-Yau.
  Universidad Autónoma de Occidente, Cali, Colombia. September 2009.

• The system of permutations of a fine mixed subdivision of a simplex.
  MDS Monday Colloquium, Berlin, Germany, May 2009.

• Coloraciones de subdivisiones mixtas de simplices.
  ALTENCOA, Bucaramanga, Colombia, July 2008.
  

• Colorings of fine mixed subdivisions of a simplex.
  Graduate Student Combinatorics Seminar, San Francisco State University, USA, April 2008.

• Colorings of fine mixed subdivisions of a simplex.
  Graduate Student Combinatorics Conference, UC Davis, USA, April 2008.

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