Joseph E. Bonin

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Professor of Mathematics
Ph.D., Dartmouth College, 1989
2009 Winner, Oscar and Shoshana Trachtenberg Prize for Teaching Excellence
Member, GW Academy of Distinguished Teachers

   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

                    Photo (2005)

  Research Interests     Expository Papers     Research Publications
  Slides of Recent Talks     Ph.D. Students

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. 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.

My work in matroid theory has spanned a fair number of facets of the field, including the following.

Expository Papers

  1. You can download a brief introduction to matroid theory (35 pages, in postscript; 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.
  2. You can also download an introduction to extremal matroid theory with an emphasis on the geometric perspective . (75 pages, in postscript.) This document contains the notes for a short course in extremal matroid (Universitat Politecnica de Catalunya, Barcelona, Spring 2003).
  3. You can also download an introduction to transversal matroids. (27 pages, in pdf format; 2010.)

Research Publications

  1. J. Bonin and T. Savitsky, An infinite family of excluded minors for strong base-orderability, Linear Algebra and its Applications, 488 (2016) 396-429.
  2. 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.
  3. J. Bonin and J. P. S. Kung, Semidirect sums of matroids, Annals of Combinatorics, 19 (2015) 7-27.
  4. J. Bonin, Basis-exchange properties of sparse paving matroids, Advances in Applied Mathematics, 50 (2013) 6-15.
  5. J. Bonin, A note on the sticky matroid conjecture, Annals of Combinatorics, 15 (2011) 619-624.
  6. J. Bonin and W. Schmitt, Splicing matroids, European Journal of Combinatorics, Special Issue in Memory of Thomas Brylawski, 32 (2011) 722-744.
  7. J. Bonin, J. P. S. Kung, and A. de Mier, Characterizations of transversal and fundamental transversal matroids, The Electronic Journal of Combinatorics, May 8, 2011.
  8. J. Bonin, A construction of infinite sets of intertwines for pairs of matroids, SIAM Journal on Discrete Mathematics, 24 (2010) 1742-1752.
  9. J. Bonin, Lattice path matroids: the excluded minors, Journal of Combinatorial Theory Series B, 100 (2010) 585-599.
  10. J. Bonin, R. Chen, and K. Xiang, Amalgams of extremal matroids with no U2,l+2-minor, Discrete Mathematics, 310 (2010) 2317-2322.
  11. J. Bonin and A. de Mier, The lattice of cyclic flats of a matroid, Annals of Combinatorics, 12 (2008) 155-170.
  12. J. Bonin, Transversal lattices, The Electronic Journal of Combinatorics, January 14, 2008.
  13. J. Bonin and O. Gimenez, Multi-path matroids, Combinatorics, Probability, and Computing, 16 (2007) 193-217.
  14. J. Bonin, Extending a matroid by a cocircuit, Discrete Mathematics, 306 (2006) 812-819.
  15. J. Bonin and A. de Mier, Lattice path matroids: Structural properties, European Journal of Combinatorics, 27 (2006) 701-738.
  16. J. Bonin and A. de Mier, Tutte polynomials of generalized parallel connections, Advances in Applied Mathematics, 32 (2004) 31-43.
  17. J. Bonin and A. de Mier, T-uniqueness of some families of k-chordal matroids, Advances in Applied Mathematics, 32 (2004) 10-30.
  18. 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.
  19. J. Bonin, Strongly inequivalent representations and Tutte polynomials of matroids, Algebra Universalis, Special Issue in Memory of Gian-Carlo Rota, 49 (2003) 289-303.
  20. 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.
  21. J. Bonin and H. Qin, Tutte polynomials of q-cones, Discrete Mathematics, 232 (2001) 95-103.
  22. J. Bonin and T. J. Reid, Simple matroids with bounded cocircuit size, Combinatorics, Probability, and Computing, 9 (2000) 407-419.
  23. J. Bonin, Involutions of connected binary matroids, Combinatorics, Probability, and Computing, 9 (2000) 305-308.
  24. J. Bonin and H. Qin, Size functions of subgeometry-closed classes of representable combinatorial geometries, Discrete Mathematics, 224 (2000) 37-60.
  25. J. Bonin and W. P. Miller, Characterizing combinatorial geometries by numerical invariants, European Journal of Combinatorics, 20 (1999) 713-724.
  26. J. Bonin, J. McNulty, and T. J. Reid, The matroid Ramsey number n(6,6), Combinatorics, Probability, and Computing, 8 (1999) 229-235.
  27. J. Bonin, On basis-exchange properties for matroids, Discrete Mathematics, 187 (1998) 265-268.
  28. 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.
  29. J. Bonin, Matroids with no (q+2)-point-line minors, Advances in Applied Mathematics, 17 (1996) 460-476.
  30. K. P. Bogart, J. Bonin, and J. Mittas, Interval orders based on weak orders, Discrete Applied Mathematics, 60 (1995) 93-98.
  31. J. Bonin, Automorphisms of Dowling lattices and related geometries, Combinatorics, Probability, and Computing, 4 (1995) 1-9.
  32. J. Bonin and J. P. S. Kung, Every group is the automorphism group of a rank-3 matroid, Geometriae Dedicata, 50 (1994) 243-246.
  33. 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.
  34. M. K. Bennett, K. P. Bogart, and J. Bonin, The geometry of Dowling lattices, Advances in Mathematics, 103 (1994) 131-161.
  35. J. Bonin, Modular elements of higher-weight Dowling lattices, Discrete Mathematics, 119 (1993) 3-11.
  36. J. Bonin, Automorphism groups of higher-weight Dowling geometries, Journal of Combinatorial Theory, Series B, 58 (1993) 161-173.
  37. 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.
  38. J. Bonin and K. P. Bogart, A geometric characterization of Dowling lattices, Journal of Combinatorial Theory, Series A, 56 (1991) 195-202.


  1. J. Bonin, Lattices related to extensions of presentations of transversal matroids.
  2. J. Bonin and J.P.S.Kung, The G-invariant and catenary data of a matroid.

Slides of Recent Talks

  1. A New Perspective on the G-Invariant of a Matroid. (2016 International Workshop on Structure in Graphs and Matroids, Eindhoven.)
  2. Lattice Path Matroids, Tutte Polynomials, and the G-Invariant. (Colloquium, U.S. Naval Academy, 2016.)
  3. 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.)
  4. Excluded Minors for (Strongly) Base Orderable Matroids. (CanaDAM, Saskatoon, June 2015.)
  5. Presentations and Extensions of Transversal Matroids. (2014 International Workshop on Structure in Graphs and Matroids, Princeton University, July 2014.)
  6. Semidirect Sums of Matroids. (Third Workshop on Graphs and Matroids, Maastricht, The Netherlands, August 2012.)
  7. Characterizations of Fundamental Transversal Matroids. (AMS Special Session, New Orleans, January 2011.)
  8. An Introduction to Transversal Matroids. (MAA Short Course, New Orleans, January 2011; see also item 3 under Expository Papers.)
  9. 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.)
  10. 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.)
  11. A Construction of Infinite Sets of Intertwines for Pairs of Matroids. (AMS Meeting, Lexington KY, March 2010; SIAM Meeting, Austin TX, June 2010.)
  12. 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.)
  13. Recent Progress on the Sticky Matroid Conjecture (with a brief introduction to matroid theory). (Combinatorics Seminar, GWU, October 2010.)
  14. 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.)

Ph.D. Students

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

Updated 14 July 2016.