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Professor of Mathematics
Ph.D., Dartmouth College, 1989
Member, GW Academy of Distinguished Teachers
2009 Oscar and Shoshana Trachtenberg Prize for Teaching Excellence
2023 WID Best Assignment Design Award
2024 Morton A. Bender Teaching Award
MathSciNet, SCOPUS, and ORCID pages.
Address:

Joseph E. Bonin
Department of Mathematics
Phillips Hall, Room 720A
The George Washington University
801 22nd Street, NW
Washington, DC, 20052		 Phone: (202) 994-6273
Fax: (202) 994-6760
e-mail: jbonin x gwu.edu (where x is the at symbol)
Photo (2005)

 

Research Interests  Expository Papers   Research Publications  Preprints Slides of Recent Talks   Research Blog Posts   PhD Students

GW Combinatorics and Algebra Seminar Schedule

Research Interests

My primary research interest is matroid theory, which is a branch of combinatorics. Matroid theory was founded in the 1930’s by Hassler Whitney, who noticed a common thread in certain ideas of independence in linear algebra and graph theory. The two-way interplay between matroid theory and its many fields of application yields a rich and powerful theory. At a basic level, linear algebra contributes the ideas of flats (generalizing subspaces) and rank (generalizing dimension); graph theory contributes the ideas of circuits and cocircuits (the latter generalizing minimal edge cut-sets). Flats and rank are then new tools for exploring graphs; circuits and cocircuits shed light on linear independence. As one pursues the field further, and as one exploits the simple but powerful idea of matroid duality, this interplay gets rich and deep.

The 1960's and 70's witnessed an explosive growth in the field, spurred partly by newly discovered connections with optimization: for instance, matroids are the simplicial complexes on which the greedy algorithm yields optimal solutions. More recent advances (within the last decade) include extending the Robertson/Seymour graph minors theorem to prove Rota's conjecture (there are only finitely many excluded minors for matroids that are representable over a given finite field) and to prove well-quasi-ordering for matroids that are representable over a given finite field, along with deep connections with algebraic geometry that have resolved the unimodality conjectures (for which June Huh won a Fields medal). Matroid theory also has important connections with coding theory, arrangements of hyperplanes, the rigidity of bar and joint frameworks, knot theory, and many other areas.

I work on whatever topics catch my interest and on which I think I can make contributions, so my work in matroid theory has spanned a fair number of facets of the field. Among the recurring themes are: Dowling lattices, the critical problem, extremal matroid theory, the Tutte polynomial, the G-invariant, cyclic flats, transversal matroids, lattice path matroids, basis-exchange properties, matroid constructions, polymatroids, . . .

Expository Papers

  1. J. Bonin and A. de Mier, Tutte uniqueness and Tutte equivalence, in Handbook of the Tutte Polynomial and Related Topics, pages 100-138.
  2. You can download a brief introduction to matroid theory (35 pages; 2001). This is intended as a gentle introduction to the parts of matroid theory that connect most closely with some of my work; this is not a representative survey of the entire field.
  3. You can also download an introduction to extremal matroid theory with an emphasis on the geometric perspective . (75 pages.) This document contains the notes for a short course in extremal matroid (Universitat Politecnica de Catalunya, Barcelona, Spring 2003).
  4. You can also download an introduction to transversal matroids. (27 pages, in pdf format; 2010.)

Research Publications

  1. J. Bonin, A characterization of positroids, with applications to amalgams and excluded minors, European Journal of Combinatorics, 122, December 2024. DOI DOI
  2. J. Bonin and K. Long, Connectivity gaps among matroids with the same enumerative invariants. Advances in Applied Mathematics, 154, March 2024. DOI
  3. J. Bonin and K. Long, The excluded minors for three classes of 2-polymatroids having special types of natural matroids, SIAM Journal on Discrete Mathematics, 37 (2023) 1715–1737. DOI
  4. J. Bonin, C. Chun, and T. Fife, The natural matroid of an integer polymatroid, SIAM Journal on Discrete Mathematics, 37 (2023) 1751–1770. DOI
  5. J. Bonin and K. Long, The free m-cone of a matroid and its G-invariant, Annals of Combinatorics, 26 (2022) 1021–1039. DOI
  6. J. Bonin, C. Chun, and T. Fife, The excluded minors for lattice path polymatroids, The Electronic Journal of Combinatorics, 29(2) (2022) # P2.38 (19 pages).   DOI
  7. J. Bonin, Matroids with different configurations and the same G-invariant Journal of Combinatorial Theory, Series A, Volume 190, August 2022.   DOI
  8. J. Bonin, C. Chun, and S. Noble, Delta-matroids as subsystems of sequences of Higgs lifts, Advances in Applied Mathematics, Volume 126, May 2021.   DOI
  9. J. Bonin, C. Chun, and S. Noble, The excluded 3-minors for vf-safe delta-matroids, Advances in Applied Mathematics, Volume 126, May 2021.   DOI
  10. J. Bonin and C. Chun, Decomposable polymatroids and connections with graph coloring European Journal of Combinatorics, Volume 89, October 2020.   DOI
  11. J. Bonin and J.P.S.Kung, The G-invariant and catenary data of a matroid, Advances in Applied Mathematics, 94 (2018) 39-70.   DOI
  12. J. Bonin, Lattices related to extensions of presentations of transversal matroids, The Electronic Journal of Combinatorics, 24(1) (2017) #P1.49 (22 pages).   DOI
  13. J. Bonin and T. Savitsky, An infinite family of excluded minors for strong base-orderability, Linear Algebra and its Applications, 488 (2016) 396-429.   DOI
  14. J. Bonin and A. de Mier, Extensions and presentations of transversal matroids, European Journal of Combinatorics, Special Issue in Memory of Michel Las Vergnas, 50 (2015) 18-29.   DOI
  15. J. Bonin and J. P. S. Kung, Semidirect sums of matroids, Annals of Combinatorics, 19 (2015) 7-27.   DOI
  16. J. Bonin, Basis-exchange properties of sparse paving matroids, Advances in Applied Mathematics, 50 (2013) 6-15.   DOI
  17. J. Bonin, A note on the sticky matroid conjecture, Annals of Combinatorics, 15 (2011) 619-624.   DOI
  18. J. Bonin and W. Schmitt, Splicing matroids, European Journal of Combinatorics, Special Issue in Memory of Thomas Brylawski, 32 (2011) 722-744.   DOI
  19. J. Bonin, J. P. S. Kung, and A. de Mier, Characterizations of transversal and fundamental transversal matroids, The Electronic Journal of Combinatorics, 18 (2011) #P106 (16 pages).   DOI
  20. J. Bonin, A construction of infinite sets of intertwines for pairs of matroids, SIAM Journal on Discrete Mathematics, 24 (2010) 1742-1752.   DOI
  21. J. Bonin, Lattice path matroids: the excluded minors, Journal of Combinatorial Theory Series B, 100 (2010) 585-599.   DOI
  22. J. Bonin, R. Chen, and K. Xiang, Amalgams of extremal matroids with no U2,l+2-minor, Discrete Mathematics, 310 (2010) 2317-2322.   DOI
  23. J. Bonin and A. de Mier, The lattice of cyclic flats of a matroid, Annals of Combinatorics, 12 (2008) 155-170.   DOI
  24. J. Bonin, Transversal lattices, The Electronic Journal of Combinatorics, 15 (2008) #R15 (11 pages).   DOI
  25. J. Bonin and O. Gimenez, Multi-path matroids, Combinatorics, Probability, and Computing, 16 (2007) 193-217.   DOI
  26. J. Bonin, Extending a matroid by a cocircuit, Discrete Mathematics, 306 (2006) 812-819.   DOI
  27. J. Bonin and A. de Mier, Lattice path matroids: Structural properties, European Journal of Combinatorics, 27 (2006) 701-738.   DOI
  28. J. Bonin and A. de Mier, Tutte polynomials of generalized parallel connections, Advances in Applied Mathematics, 32 (2004) 31-43.   DOI
  29. J. Bonin and A. de Mier, T-uniqueness of some families of k-chordal matroids, Advances in Applied Mathematics, 32 (2004) 10-30.   DOI
  30. J. Bonin, A. de Mier, and M. Noy, Lattice path matroids: enumerative aspects and Tutte polynomials, Journal of Combinatorial Theory, Series A, 104 (2003) 63-94.   DOI
  31. J. Bonin, Strongly inequivalent representations and Tutte polynomials of matroids, Algebra Universalis, Special Issue in Memory of Gian-Carlo Rota, 49 (2003) 289-303.   DOI
  32. R. M. Ankney and J. Bonin, Characterizations of PG(n-1,q)\PG(k-1,q) by numerical and polynomial invariants, Advances in Applied Mathematics, Special Issue in Memory of Rodica Simion, 28 (2002) 287-301.   DOI
  33. J. Bonin and H. Qin, Tutte polynomials of q-cones, Discrete Mathematics, 232 (2001) 95-103.   DOI
  34. J. Bonin and T. J. Reid, Simple matroids with bounded cocircuit size, Combinatorics, Probability, and Computing, 9 (2000) 407-419.   DOI
  35. J. Bonin, Involutions of connected binary matroids, Combinatorics, Probability, and Computing, 9 (2000) 305-308.   DOI
  36. J. Bonin and H. Qin, Size functions of subgeometry-closed classes of representable combinatorial geometries, Discrete Mathematics, 224 (2000) 37-60.   DOI
  37. J. Bonin and W. P. Miller, Characterizing combinatorial geometries by numerical invariants, European Journal of Combinatorics, 20 (1999) 713-724.   DOI
  38. J. Bonin, J. McNulty, and T. J. Reid, The matroid Ramsey number n(6,6), Combinatorics, Probability, and Computing, 8 (1999) 229-235.   DOI
  39. J. Bonin, On basis-exchange properties for matroids, Discrete Mathematics, 187 (1998) 265-268.   DOI
  40. J. Bonin and J. P. S. Kung, The number of points in a combinatorial geometry with no 8-point-line minor, in: Mathematical Essays in Honor of Gian-Carlo Rota, B. Sagan and R. Stanley, eds., Birkhauser, 1998, 271-284.   DOI
  41. J. Bonin, Matroids with no (q+2)-point-line minors, Advances in Applied Mathematics, 17 (1996) 460-476.   DOI
  42. K. P. Bogart, J. Bonin, and J. Mittas, Interval orders based on weak orders, Discrete Applied Mathematics, 60 (1995) 93-98.   DOI
  43. J. Bonin, Automorphisms of Dowling lattices and related geometries, Combinatorics, Probability, and Computing, 4 (1995) 1-9.   DOI
  44. J. Bonin and J. P. S. Kung, Every group is the automorphism group of a rank-3 matroid, Geometriae Dedicata, 50 (1994) 243-246.   DOI
  45. R. D. Baker, J. Bonin, F. Lazebnik, and E. Shustin, On the number of nowhere zero points in linear mappings, Combinatorica, 14 (1994) 149-157.   DOI
  46. M. K. Bennett, K. P. Bogart, and J. Bonin, The geometry of Dowling lattices, Advances in Mathematics, 103 (1994) 131-161.   DOI
  47. J. Bonin, Modular elements of higher-weight Dowling lattices, Discrete Mathematics, 119 (1993) 3-11.   DOI
  48. J. Bonin, Automorphism groups of higher-weight Dowling geometries, Journal of Combinatorial Theory, Series B, 58 (1993) 161-173.   DOI
  49. J. Bonin, L. Shapiro, and R. Simion, Some q-analogs of the Schroeder numbers arising from combinatorial statistics on lattice paths, Journal of Statistical Planning and Inference, 34 (1993) 35-55.   DOI
  50. J. Bonin and K. P. Bogart, A geometric characterization of Dowling lattices, Journal of Combinatorial Theory, Series A, 56 (1991) 195-202.   DOI

Slides of Recent Talks

  1. An overview of recent developments on non-isomorphic matroids that have the same G-invariant. (The Fields Institute for Research in Mathematical Sciences, Matroids - Combinatorics, Algebra and Geometry Seminar, March 21, 2023.)
  2. New Connections between Polymatroids and Graph Coloring. (CanaDAM 2019, Simon Fraser University, Vancouver.) (Version for Louisiana State University.)
  3. Delta-matroids as subsystems of sequences of Higgs lifts, or a way to think about delta-matroids from the perspective of matroids. (PRIMA 2017, Oaxaca, Mexico, August 2017.) Combinatorial Geometries 2018, Marseille, France, CIRM, September 2018, Video of the CIRM talk.
  4. Minor-Closed Classes of Polymatroids. (CanaDAM, Toronto, June 2017.) (GW Seminar Version.) (AMS Sectional Meeting, Vanderbilt University.)
  5. A New Perspective on the G-Invariant of a Matroid. (2016 International Workshop on Structure in Graphs and Matroids, Eindhoven, The Netherlands; George Mason University, Combinatorics, Algebra, and Geometry Seminar, October 2016.)
    University of Delaware Combinatorics Seminar Version, Combinatorics Seminar, Louisiana State University, November 2018
  6. Lattice Path Matroids, Tutte Polynomials, and the G-Invariant. (Colloquium, U.S. Naval Academy, 2016.)
  7. Cyclic flats of matroids and their connections to Tutte polynomials and other matroid invariants. (Workshop on New Directions for the Tutte Polynomial: Extensions, Interrelations, and Applications, Royal Holloway University of London, July 2015.)
  8. Excluded Minors for (Strongly) Base Orderable Matroids. (CanaDAM, Saskatoon, June 2015.)
  9. Presentations and Extensions of Transversal Matroids. (2014 International Workshop on Structure in Graphs and Matroids, Princeton University, July 2014.) GW Version
  10. Semidirect Sums of Matroids. (Third Workshop on Graphs and Matroids, Maastricht, The Netherlands, August 2012.)
  11. Characterizations of Fundamental Transversal Matroids. (AMS Special Session, New Orleans, January 2011.)
  12. An Introduction to Transversal Matroids. (MAA Short Course, New Orleans, January 2011.)
  13. The Excluded Minors of Lattice Path Matroids. (Second Workshop on Graphs and Matroids, Maastricht, The Netherlands, August 2010; Special Session on Algebraic and Geometric Aspects of Matroids, AMS Meeting, Wake Forest University, NC, September 2011.)
  14. Cyclic Flats, Sticky Matroids, and Intertwines. (Workshop on Invariant Theory and Combinatorics, George Mason University, March 2010; Combinatorics Seminar, Universitat Politecnica de Catalunya, Barcelona, Spain, April 2010.)
  15. A Construction of Infinite Sets of Intertwines for Pairs of Matroids. (AMS Meeting, Lexington KY, March 2010; SIAM Meeting, Austin TX, June 2010.)
  16. An Introduction to Matroid Theory Through Lattice Paths. (Colloquium, Wichita State University, November 2009. Colloquium, Howard University, February 2011, The College of William and Mary, April 2011. Discrete Math Seminar, Virginia Commonwealth University, March 2016.)
  17. Recent Progress on the Sticky Matroid Conjecture (with a brief introduction to matroid theory). (Combinatorics Seminar, GWU, October 2010.)
  18. Old and new connections between matroids and codes: a short introduction to two fields. (Graduate Colloquium, Louisiana State University, November 2018.)
  19. What do lattice paths have to do with matrices, and what is beyond both?. (Undergraduate Colloquium, Gettysburg College, November 2010. Undergraduate Colloquium, U.S. Naval Academy, April 2016.) Undergraduate Colloquium, Louisiana State University, November 2018.

Research Blog Posts

Ph.D. Students

  • William P. Miller, Approaches to Matroid Reconstruction Problems, 1995.
  • Hongxun Qin, Tutte Polynomials and Matroid Constructions, 2000.
  • Rachelle Ankney, The Geometries PG(n-1,q)\PG(k-1,q), 2001.
  • Ken Shoda, Large Families of Matroids with the Same Tutte Polynomial, 2012.
  • Thomas Savitsky, Some Problems on Matroids and Integer Polymatroids, 2015.
  • Kevin Long, Some Problems on Matroid Invariants and Integer Polymatroids, 2022.

Updated 6 August 2024

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