Peter Nelson


Office: MC5128

Department of Combinatorics and Optimization
University of Waterloo
Waterloo, ON N2L3G1

I am an Associate Professor in the Department of Combinatorics and Optimization at the University of Waterloo in Ontario, Canada.

My research interests are in structural and extremal matroid theory and graph theory, and their links with coding theory, additive combinatorics and finite geometry, particularly the theory of minor-closed classes, and the binary matroids with the submatroid order and the induced submatroid order. I've also recently become interested in proof formalization in LEAN. I hold an NSERC Discovery Grant and an Early Researcher Award.

Here are the slides for an talk on binary matroids I gave in May 2019 at CanaDAM.

Here is my CV and here are download or arXiv links to my papers in their current state, in roughly reverse chronological order.

38. Typical structure of hereditary properties of binary matroids
Submitted. With Stefan Grosser, Hamed Hatami and Sergey Norin.

37. Excluding a line from complex-representable matroids
Accepted, Memoirs Amer. Math. Soc. With Jim Geelen and Zachary Walsh.

36. The structure of I_4-free and triangle-free binary matroids
SIAM J. Discrete Math. 36 (2022), 1711-1729 , With Kazuhiro Nomoto.

35. The smallest matroids with no large independent flat
Electronic J. Combin. 28 (2021). With Sergey Norin.

34. The critical number of I_{1,t}-free binary matroids.
Submitted. With Kazuhiro Nomoto.

33. Enumeration of extensions of the cycle matroid of a complete graph
Adv. Appl. Math. 138, With Kazuhiro Nomoto.

32. The smallest I_5-free and triangle-free binary matroids.
Submitted. With Kazuhiro Nomoto.

31. The structure of claw-free binary matroids
JCTb 150 (2021), 76-118. With Kazuhiro Nomoto.

30. On the number of biased graphs
SIAM J. Discrete Math. 33, 373-382. With Jorn van der Pol.

29. The structure of binary matroids with no induced claw or Fano plane restriction
Advances in Combinatorics. 1. With Marthe Bonamy, Frantisek Kardos, Tom Kelly and Luke Postle.

28. A Ramsey theorem for biased graphs
SIAM J. Discrete Math. 134 (2020), 2270-2281. With Sophia Park.

27. Bounding χ by a fraction of Δ for graphs without large cliques
Accepted, JCTb. With Marthe Bonamy, Tom Kelly and Luke Postle.

26. Stability and exact Turan numbers for matroids
JCTb 143 (2020), 29-41. With Hong Liu, Sammy Luo and Kazuhiro Nomoto.

25. The extremal function for excluding geometry minors over prime fields
SIAM J. Discrete Math. 36 (2022), 2100-2120. With Zachary Walsh.

24. Matroids with no U_{2,n}-minor and many hyperplanes
Adv. Appl. Math. 100 (2018), 143-147. With Adam Brown.

23. Doubly exponentially many Ingleton matroids
SIAM J. Discrete Math. 32, 1145-1153. With Jorn van der Pol.

22. Almost all matroids are non-representable
Bull. London Math. Soc. 50 (2018), 245-248.

21. The structure of matroids with a spanning clique or projective geometry
JCTb 127 (2017), 65-81. With Jim Geelen.

20. On the probability that a random subgraph contains a circuit
J. Graph Theory 85 (2017), 644-650.

19. The densest matroids in minor-closed classes with exponential growth rate
Trans. Amer. Math. Soc 369 (2017), 6751-6776. With Jim Geelen.

18. The maximum-likelihood decoding threshold for graphic codes
IEEE Trans. Inf. Theory 62 (2016), 5316-5322. With Stefan van Zwam.

17. Linkages in a directed graph with parity restrictions
Not submitted. With Rutger Campbell.

16. The critical number of dense triangle-free binary matroids
JCTb 116 (2016), 238-249. With Jim Geelen.

15. The number of lines in a matroid with no U_{2,n}-minor
European J. Combin. 50 (2016), 115-122. With Jim Geelen.

14. Odd circuits in dense binary matroids
Combinatorica 35 (2015), 730-735. With Jim Geelen.

13. Matroids denser than a clique
JCTb 114 (2015), 51-69. With Jim Geelen

12. Matroids representable over fields with a common subfield
SIAM J. Discrete Math. 29 (2015), 796-810. With Stefan van Zwam.

11. Matroids denser than a projective geometry
SIAM J. Discrete Math 29 (2015), 730-735.

10. On the existence of asymptotically good linear codes in minor-closed classes
IEEE Trans. Inf. Theory 61 (2015), 1153-1158. With Stefan van Zwam.

9. Projective geometries in exponentially dense matroids, II
JCTb 113 (2015), 185-207.

8. Projective geometries in exponentially dense matroids, I
JCTb 113 (2015), 208-219. With Jim Geelen.

7. A density Hales-Jewett theorem for matroids
JCTb 112 (2015), 70-77. With Jim Geelen.

6. An analogue of the Erdős-Stone theorem for finite geometries
Combinatorica 35 (2015), 209-214. With Jim Geelen.

5. The number of rank-k flats in a matroid with no U_{2,n}-minor
JCTb 107 (2014), 140-147.

4. Growth rate functions of dense classes of representable matroids
JCTb 103 (2013), 75-92.

3. On minor-closed classes of matroids with exponential growth rate
Adv. Appl. Math. 50 (2013), 142-154. With Jim Geelen.

2. The number of points in a matroid with no n-point line as a minor
JCTb 100 (2010), 625-630. With Jim Geelen.

1. Sequential automatic algebras
CiE (2008), 84-93. With Michael Brough and Bakhadayr Khoussainov.

0. Exponentially dense matroids
Ph.D. Thesis, University of Waterloo