## Thomas Lam's Publications and Preprints

1. The totally nonnegative Grassmannian is a ball (with Pavel Galashin and Steven Karp)
We prove three spaces are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.
arxiv: 1707.02010

2. From dimers to webs (with Chris Fraser and Ian Le)
We connect the dimer model with webs by producing a SL_r tensor invariant from the r-fold dimer model of a bipartite graph. 
arxiv: 1705.09424

3. A conjectural Peterson isomorphism in K-theory (with Changzheng Li, Leonardo Mihalcea, and Mark Shimozono)
We give a conjectural isomorphism between localizations of the equivariant quantum K-theory of a flag variety and the equivariant K-homology of the affine Grassmannian.
arxiv: 1705.03435

4. The mirror conjecture for minuscule flag varieties (with Nicolas Templier)
We prove Rietsch's mirror conjecture for minuscule flag varieties. The main idea is to recognize the quantum D-module (A-model) as Galois and the Landau-Ginzburg model (B-model) as Automorphic, then apply ideas from the Langlands program.
arxiv: 1705.00758

5. Positive geometries and canonical forms (with Nima Arkani-Hamed and Yuntao Bai)
We define and study a class of positive geometries, characterized by the existence of a canonical form with simple poles. Key examples include polytopes, positive parts of toric varieties, and totally positive parts of flag varieties.
JHEP, (2017) 2017: 39.
arxiv: 1703.04541

6. On the cluster nature and quantization of geometric R-matrices (with Rei Inoue and Pavlo Pylyavskyy)
We construct a quantization of the affine geometric R-matrix, and explain the relation to cluster mutation.
Publ. RIMS, to appear.
arxiv: 1607.00722

7. Cohomology of cluster varieties. I. Locally acyclic case (with David Speyer)
We initiate a systematic study of the cohomology of cluster varieties.
arxiv: 1604.06843

8. The Amplituhedron and the One-loop Grassmannian Measure (with Yuntao Bai and Song He)
We study the one-loop Grassmannian and the one-loop amplituhedron. We construct canonical measures on the one-loop Grassmannian, and give integral formulae for one-loop scattering amplitudes in super Yang Mills.
JHEP, (2016) 2016: 112.
arxiv: 1510.03553

9. Totally nonnegative Grassmannian and Grassmann polytopes
These are lecture notes for my second CDM lecture.
Current Developments in Mathematics, 2014.
arxiv: 1506.00603

10. Toric networks, geometric R-matrices and generalized discrete Toda lattices (with Rei Inoue and Pavlo Pylyavskyy)
We study the dynamics of the double affine geometric R-matrix acting on a toric network. We construct two lattices of commuting time evolutions and linearize these actions on the Jacboian of the spectral curve.
Comm. Math. Phys., 347 (2016), 799-855.
arxiv: 1504.03448

11. Rigged Configurations and Cylindric Loop Schur Functions (with Pavlo Pylyavskyy and Reiho Sakamoto )
We conjecture a formula for the shapes of a rigged configuration associated to a tensor product of one-row crystals, as the tropicalization of a cylindric loop Schur function. We prove this formula for the first shape, thus giving a piecewise-linear formula for the lengths of the solitions of a box-ball system.
Annales de l'Institut Henri Poincare D, to appear.
arxiv: 1410.4455

12. Grassmannians for scattering amplitudes in 4d N=4 SYM and 3d ABJM (with Henriette Elvang, Yu-tin Huang, Cynthia Keeler, Timothy Olson, Samuel Roland, and David Speyer)
We study four integral formulae for scattering amplitudes. For super Yang Mills theory these are the (known) integrals over Grassmannians in momentum space and momentum twistor space. For ABJM theory, these are integrals over orthogonal Grassmannians. The ABJM integral in momentum twistor space is new.
JHEP, (2014) 2014: 181.
arxiv: 1410.0621

13. Amplituhedron cells and Stanley symmetric functions
We compute the cohomology class of a tree amplituhedron subvariety of the Grassmannian to be the truncation of an affine Stanley symmetric function.
Comm. Math. Phys., (2016) 343, 1025-1037.
arxiv: 1408.5531

14. The uncrossing partial order on matchings is Eulerian
We prove that the partial order on the set of matchings of 2n points on a circle, given by resolving crossings, is an Eulerian poset.
J. Combin. Theory A, 135 (2015), 105-111.
arxiv: 1406.5671

15. Dimers, webs, and positroids
We study the dimer, double-dimer, and triple-dimer models of a planar bipartite graph embedded into a disk. In particular, we relate triple dimer configurations to Kuperberg's webs, and find a connection between square moves (or urban renewal) of planar bipartite graphs and reduction moves of webs.
J. London. Math. Soc., 92 (2015), 633-656.
arxiv: 1404.3317

16. Electroid varieties and a compactification of the space of electrical networks
We construct a compactification of the space of circular planar electrical networks as a linear slice of the totally nonnegative Grassmannian. We study an electroid stratification of our space, which is an analogue of the positroid stratification of the TNN Grassmannian.
arxiv: 1402.6261

17. Whittaker functions, geometric crystals, and quantum Schubert calculus
This is a mostly expository article explaining the relations between the three objects in the title. It contains a proof of the equivariant version of Rietsch's mirror conjecture for flag varieties, which appears to be new.
Advanced Studies in Pure Mathematics, Volume 71 (Schubert Calculus - Osaka 2012).
arxiv: 1308.5451

18. k-Schur functions and affine Schubert calculus (with Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, and Mike Zabrocki)
This book is an exposition of the subject spanning k-Schur functions and affine Schubert calculus, based on lectures at the Fields Institute in 2010.
Fields Institute Monographs, Volume 33.
arxiv: 1301.3569

19. Infinite reduced words and the Tits boundary of a Coxeter group (with Anne Thomas)
We connect the limit weak order on infinite reduced words of a Coxeter group with the Tits metric on the visual boundary of the Davis complex.
IMRN, (2015), 7690-7733.
arxiv: 1301.0873

20. Linear Laurent phenomenon algebras (with Pavlo Pylyavskyy)
We give an explicit description of Laurent phenomenon algebras with a linear initial seed arising from a graph. In particular, any graph associahedron is shown to be the dual cluster complex for some Laurent phenomenon algebra.
IMRN, (2016), 3163-3203.
arxiv: 1206.2612

21. Laurent phenomenon algebras (with Pavlo Pylyavskyy)
We generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials.
Camb. J. Math., 4 (2016), 121-162.
arxiv: 1206.2611

22. Total positivity, Schubert positivity, and Geometric Satake (with Konstanze Rietsch)
We compare three notions (total, Schubert, Mirkovic-Vilonen) of positivity on the centralizer subgroup of a principal nilpotent element, in a complex simple algebraic group. We explicitly parametrize this positive part, establishing a conjecture of Rietsch.
J. Algebra, 460 (2016), 284-319.
arxiv: 1203.1682

23. Alcoved Polytopes II (with Alex Postnikov)
We continue our study of polytopes arising from affine Coxeter arrangements. In particular, we give compatible definitions of descent numbers, major indices, and a q-analogue of Weyl's formula for the order of a Weyl group.
Kostant Memorial Volume, Birkhauser, to appear.
arxiv: 1202.4015

24. Positroid Varieties: Juggling and Geometry (with Allen Knutson and David Speyer)
We study positroid varieties, the intersections of cyclically rotated Schubert varieties in the Grassmannian. This paper supercedes "Positroid Varieties I: Juggling and Geometry" which has been broken up into this paper, and "Projections of Richardson Varieties".
Compositio Mathematica, 149 (2013), 1710-1752.
arxiv: 1111.3660

25. From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice (with Mark Shimozono)
We show that k-double Schur functions and quantum double Schubert polynomials can be obtained from each other by an explicit rational substitution. The main new ingredient is an explicit computation of Kostant's solution to the Toda lattice in terms of equivariant Schubert classes.
arxiv: 1109.2193

26. Quantum double Schubert polynomials represent Schubert classes (with Mark Shimozono)
We show that quantum double Schubert polynomials represent equivariant quantum Schubert classes, and define parabolic analogues.
Proc. AMS, 142 (2014), 835-850.
arxiv: 1108.4958

27. Projected Richardson varieties and affine Schubert varieties (with Xuhua He)
We compare projections of Richardson varieties from G/B to G/P with Schubert varieties in the affine flag variety. Two comparisons are made: a combinatorial one on the level of the closure order poset, and a cohomological comparison of the corresponding classes.
Annales de l'Institut Fourier, 65 (2015), 2385-2412.
arxiv: 1106.2586

28. Equivariant Pieri Rule for the homology of the affine Grassmannian (with Mark Shimozono)
We give a Chevalley formula for equivariant homology in the affine Grassmannian of an arbitrary simple algebraic group. We give an explicit Pieri formula for the affine Grassmannian of type A.
Journal of Algebraic Combinatorics, 36 (2012), 623-648.
arxiv: 1105.5154

29. k-Double Schur functions and equivariant (co)homology of the affine Grassmannian (with Mark Shimozono)
The Schubert bases of the torus-equivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called k-double Schur functions and affine double Schur functions.
Math. Annalen, 356 (2013), 1379-1404.
arxiv: 1105.2170

30. Inverse problem in electrical cylindrical networks (with Pavlo Pylyavskyy)
We study the inverse Dirichlet-to-Neumann problem for cylindrical electrical networks, and define an "electrical-R-matrix" which conjecturally solves the inverse problem.
SIAM J. of Applied Math., 72 (2012), 767-788.
arxiv: 1104.4998

31. Electrical networks and Lie theory (with Pavlo Pylyavskyy)
We introduce a new class of "electrical" Lie algebras and Lie groups which act on electrical networks. An interview by Scott Baldridge!
Algebra and Number Theory, 9 (2015), 1401-1418.
arxiv: 1103.3475

32. A Markov chain on the symmetric group which is Schubert positive? (with Lauren Williams)
We study a multivariate Markov chain on the symmetric group and conjecture that the invariant distribution is Schubert positive.
Experimental Mathematics, 21 (2012), 189-192.
arxiv: 1102.4406

33. The shape of a random affine Weyl group element, and random core partitions
We show that large random affine Weyl group elements almost surely have one of finitely many "shapes". In the type A case, one obtains that the shape of a random n-core is a piecewise-linear graph, in the sense of Vershik and Kerov.
Ann. of Probability, 43 (2015), 1643-1662.
arxiv: 1102.4405

34. Loop symmetric functions and factorizing matrix polynomials
A survey of results obtained with Pavlo Pylyavskyy, concerning the ring of loop symmetric functions, and relations to networks on surfaces, total positivity, crystal graphs, and discrete integrable systems.
Fifth International Congress of Chinese Mathematicians, AMS/IP Studies in Advanced Mathematics, 51 (2012), 609-628.
arxiv: 1012.1262

35. Box-basket-ball systems (with Pavlo Pylyavskyy and Reiho Sakamoto)
We define a new discrete solitonic system, which consists of boxes, baskets, and balls. We classify the solitons and study their scattering.
Reviews in Math. Phys , 24 (2012), 23 pages.
arxiv: 1011.5930

36. From quantum Schubert polynomials to k-Schur functions via the Toda lattice (with Mark Shimozono)
We show that quantum Schubert polynomials and k-Schur functions can be obtained from each other via a rational substitution.
Math Research Letters, 19 (2012), 81-93.
arxiv: 1010.4047

37. Projections of Richardson Varieties (with Allen Knutson and David Speyer)
We prove that projected Richardson varieties are normal, Cohen-Macaulay, have rational singularities, and are exactly the compatibly Frobenius-split subvarieties of G/P with respect to the standard splitting. We also show that parabolic projections of the order complex of a Bruhat interval is shellable.
Crelle's journal, 687 (2014), 133-157.
arxiv: 1008.3939

38. Crystals and total positivity on orientable surfaces (with Pavlo Pylyavskyy)
We study a model of combinatorial networks on orientable surfaces, and apply this to the theory of geometric crystals, and to total positivity.
Selecta Math., 19 (2013), 173-235.
arxiv: 1008.1949

39. k-shape poset and branching of k-Schur functions (with Luc Lapointe, Jennifer Morse, and Mark Shimozono)
We introduce a new class of partitions, called k-shapes, and use them to study branching of k-Schur functions.
Memoirs of the AMS, 223 (2013), no. 1050.
arxiv: 1007.5334

40. Stanley symmetric functions and Peterson algebras
Lecture notes on affine Stanley symmetric functions for the summer school on affine Schubert calculus at the Fields Institute 2010.
arxiv: 1007.2871

41. Affine geometric crystals in unipotent loop groups (with Pavlo Pylyavskyy)
The quotient by the R-matrix of products of the affine geometric crystal of type A corresponding to symmetric powers is constructed in the unipotent loop group.
Representation Theory, 15 (2011), 719-728.
arxiv: 1004.2233

42. Intrinsic energy is a loop Schur function (with Pavlo Pylyavskyy)
We identify the intrinsic energy function of tensor products of Kirillov-Reshetikhin crystals for symmetric powers of type A with the tropicalization of loop Schur functions.
Journal of Combinatorics, 4 (2013), 387-401.
arxiv: 1003.3948

43. Skew Littlewood-Richardson rules for Hopf algebras (with Aaron Lauve and Frank Sottile)
We use Hopf algebra theory to prove "skew" versions of Littlewood-Richardson rules, establishing a conjecture of Assaf and McNamara.
IMRN, (2011), 1205-1219.
arxiv: 0908.3714

44. (Appendix in) A Pieri rule for skew shapes
An algebraic proof of Assaf and McNamara's (who are the main authors) skew Pieri rule.
J. Combin. Theory Ser. A, 118 (2011), 277-290.
arxiv: 0908.0345

45. Total positivity for loop groups II: Chevalley generators (with Pavlo Pylyavskyy)
This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper we study infinite products of Chevalley generators.
Transformation Groups, 18 (2013), 179-231.
arxiv: 0906.0610

46. Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras
We suggest that Schubert classes for the affine Grassmannian of a simple algebraic group can be viewed as Schur positive symmetric functions.
Bulletin of the LMS, 2011; doi:10.1112/blms/bdq110.
arxiv: 0906.0385

47. Positroid varieties I: juggling and geometry (with Allen Knutson and David Speyer)
We study positroid varieties, the intersections of cyclically rotated Schubert varieties in the Grassmannian.
arxiv: 0903.3694

48. Combinatorial Hopf algebras and Towers of Algebras: Dimension, Quantization, and Functorality (with Nantel Bergeron and Huilan Li)
We study the relations between combinatorial Hopf algebras, towers of algebras, and dual graded graphs. We prove a dimension formula, describe a quantization and a categorial version of our constructions. This paper is a full version of the similarly titled summary below.
Algebras and Representation Theory, to appear.
arxiv: 0903.1381

49. K-theory Schubert calculus of the affine Grassmannian (with Anne Schilling and Mark Shimozono)
We study the K-theory Schubert calculus of the affine Grassmannian of a simple algebraic group G. For the case G = SL_n, we identify the Schubert classes in K-(co)homology as K-k-Schur functions and affine stable Grothendieck polynomials.
Compositio Mathematica, 146 (2010), 811-852.
arxiv: 0901.1506

50. Total positivity for loop groups I: whirls and curls (with Pavlo Pylyavskyy)
We initiate the study of a theory of total positivity for loop groups.
Adv. in Math., 230 (2012), 1222-1271.
arxiv: 0812.0840

We study quantizations of dual graded graphs, and construct examples of them using Fibonacci posets, permutations, standard Young tableaux, and plane binary trees.
Elec. J. Combin., 17 (2010), no.1, Research Paper 88, 11pp.
arxiv: 0808.0345

52. Tiling with commutative rings
An expository article explaining an approach to tiling problems using commutative algebra, written for the Harvard undergraduate math journal.
Harvard College Mathematics Review, 2 (2008), 55-60.
[ps] [pdf]

53. Combinatorial Hopf algebras and Towers of Algebras (with Nantel Bergeron and Huilan Li)
We prove that a tower of algebras A = {A_n} whose Grothendieck groups give rise to graded dual Hopf algebras via induction and restriction must have dimension dim(A_n) = r^n n!.
Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, France (Proc. FPSAC 2008), 2008, 52-60.
arxiv: 0710.3744

54. Total positivity for cominuscule Grassmannians (with Lauren Williams)
We study the totally non-negative cells of cominuscule Grassmannians. In particular, we define and study Le-diagrams.
New York J. of Math., 14 (2008), 53-99.
arxiv: 0710.2932

55. Schubert polynomials for the affine Grassmannian of the symplectic group (with Anne Schilling and Mark Shimozono)
Using Schur P and Q-functions, we define and study Schubert polynomials for the affine Grassmannian of the symplectic group.
Math. Z., 264 (2010), 765-811.
arxiv: 0710.2720

56. Combinatorial Hopf algebras and K-homology of Grassmannians (with Pavlo Pylyavskyy)
Motivated by work of Buch on set-valued tableaux, we define and study six "K-theoretic" combinatorial Hopf algebras.
International Mathematics Research Notices, 2007 (2007), rnm 125, 48 pages.
arxiv: 0705.2189

57. Quantum cohomology of G/P and homology of affine Grassmannian (with Mark Shimozono)
We prove an unpublished result of Dale Peterson showing that quantum and affine homology Schubert calculi are equivalent.
Acta. Math., 204 (2010), 49-90.
arxiv: 0705.1386

58. Dual graded graphs for Kac-Moody algebras (with Mark Shimozono)
We define and study a family of dual graded graphs with vertex set equal to the Weyl group of a Kac-Moody algebra.
Algebra and Number Theory, 1 (2007), 451-488.
arxiv: math.CO/0702090

59. Temperley-Lieb Pfaffinants and Schur Q-positivity conjectures (with Pavlo Pylyavskyy)
We define and study a pfaffian analogue of immanants, focusing on Temperley-Lieb pfaffinants.
Adv. in Math., 218 (2008), 1654-1684.
arxiv: math.CO/0612842

60. Signed differential posets and sign-imbalance
We study a signed analogue of differential posets, and relate them to sign-imbalance.
J. Comb. Theory Series A, 115 (2008), 466-484.
arxiv: math.CO/0611296

61. On domino insertion and Kazhdan-Lusztig cells in type B_n (with Cedric Bonnafe, Meinolf Geck, and Lacri Iancu)
We give a conjectural characterization of all Kazhdan-Lusztig cells for B_n with unequal parameters, via domino insertion.
Representation theory of algebraic groups and quantum groups (Nagoya 2006; eds. A. Gyoja et al.) , 33-54, Progress in Math. 284, Birkhauser, 2010.
arxiv: math.RT/0609279

62. P-partition products and fundamental quasi-symmetric function positivity (with Pavlo Pylyavskyy)
We show that certain differences of products of P-partition generating functions are positive combinations of fundamental quasi-symmetric functions.
Adv. Appl. Math., 40 (2008), 271-294.
arxiv: math.CO/0609249

63. Affine insertion and Pieri rules for the affine Grassmannian (with Luc Lapointe, Jennifer Morse, and Mark Shimozono)
We prove an affine insertion algorithm, obtaining Pieri rules for the (co)homology of the affine Grassmannian as a consequence.
Memoirs of the AMS, 208 (2010), no. 977.
arxiv: math.CO/0609110

64. On Sjostrand's skew sign-imbalance identity
We give a quick proof of a sign-imbalance identity due to Sjostrand.
[ps] [pdf]
arxiv: math.CO/0607516

65. Schubert polynomials for the affine Grassmannian
We identify the (co)homology Schubert basis of the affine Grassmannian as the (dual)k-Schur functions. The files here are an extended abstract which appeared in Proc. FPSAC San Diego, 2006; the arXiv version is the real one.
J. Amer. Math. Soc., 21 (2008), 259-281.
[ps] [pdf]
arxiv: math.CO/0603125

66. A Little bijection for affine Stanley symmetric functions (with Mark Shimozono)
David Little developed a combinatorial algorithm to study the Schur-positivity of Stanley symmetric functions and the Lascoux-Sch\"{u}tzenberger tree. We generalize this algorithm to affine Stanley symmetric functions.
Seminaire Lotharingien de Combinatoire, 54A (2006), B54Ai.
arxiv: math.CO/0601483

67. A combinatorial generalization of the Boson-Fermion correspondence
We explain the ubiquity of tableaux and Pieri and Cauchy identities for many families of symmetric functions, using representations of Heisenberg algebras.
Math. Res. Letters, 13 (2006), 377-392.
[ps] [pdf]
arxiv: math.CO/0507341

68. Cell Transfer and Monomial Positivity (with Pavlo Pylyavaskyy)
We show that certain differences of products of Schur functions are monomial positive and give a generalisation of this to arbitrary labelled posets.
J. Alg. Combin., 26 (2007), 209-224.
arxiv: math.CO/0505273

69. Combinatorics of Ribbon Tableaux
My Ph.D. Thesis written under the guidance of Richard Stanley. This contains my papers on ribbon Schur operators and ribbon tableaux and the Heisenberg algebra, together with a combinatorial generalisation of the Boson-Fermion correspondence.
[ps] [pdf]

70. Schur positivity and Schur log-concavity (with Alex Postnikov and Pavlo Pylyavskyy)
We prove Schur positivity conjectures of Okounkov, of Lascoux, Leclerc and Thibon and of Fomin, Fulton, Li and Poon.
Amer. J. Math., 129 (2007), 1611-1622.
arxiv: math.CO/0502446

71. Alcoved Polytopes I (with Alex Postnikov)
We study certain polytopes arising from the affine Coxeter arrangement (in type A).
Disc. and Comp. Geom., 38 (2007), 453-478.
arxiv: math.CO/0501246

72. Affine Stanley Symmetric Functions
We define and study a new family of symmetric functions which are affine analogues of Stanley symmetric functions.
American J. of Math., 128 (2006), 1553-1586 .
arxiv: math.CO/0501335

73. Ribbon Schur Operators
A new combinatorial approach to the ribbon tableaux generating functions and $q$-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested, following methods of Fomin and Greene.
European J. of Combinatorics, 29 (2008), 343-359.
arxiv: math.CO/0409463

74. A note on graphs without short even cycles (with Jacques Verstraete)
We show that any n-vertex graph without even cycles of length at most 2k has at most (1/2)n^{1 + 1/2} + O(n) edges.
Electronic Journal of Combinatorics, 12/1 (2005), N5.
arxiv: math.CO/0503623

75. On symmetry and positivity for domino and ribbon tableaux
We show the symmetry of LLT-ribbon functions, and describe the product of a Schur function and a domino function.
Annals of Combinatorics, 9 (2005), 293-300.
[ps] [pdf]
arxiv: math.CO/0407184

76. Affine Stanley Symmetric Functions (extended abstract)
We define and study a new family of symmetric functions which are affine analogues of Stanley symmetric functions.
Proceedings of FPSAC, 2005, Taormina.
[ps] [pdf]

77. Ribbon Tableaux and the Heisenberg Algebra
We prove Pieri, Cauchy and Murnaghan-Nakayama formulae for the ribbon tableaux generating functions of Lascoux, Leclerc and Thibon. (The version on the arXiv is a longer, older version.)
Mathematische Zeitschrift, 250 (2005), 685-710.
[ps] [pdf]
arxiv: math.QA/0310250

78. Growth diagrams, domino insertion, and sign-imbalance
We study some properties of domino insertion and settle Stanley's `2^{n/2}' conjecture on sign-imbalance.
Journal of Combinatorial Theory Ser. A., 107 (2004), 87-115.
[ps] [pdf]
arxiv: math.CO/0308265

79. Pieri and Cauchy formulae for Ribbon Tableaux
This is an extended abstract of "Ribbon Tableaux and the Heisenberg Algebra", where I focus on more combinatorial aspects.
Proceeedings of FPSAC, 2004, Vancouver.
[ps] [pdf]

80. A result on 2k-cycle free bipartite graphs
A bipartite graph with parts of sizes N >= M and no cycles of length 2l, for all l \in [2,2k], has number of edges less than M^{1/2}N^{(k+1)/2k} + O(N).
Australasian Journal of Combinatorics, 32 (2005), 163.
[ps] [pdf]

81. Graphs without cycles of even length
We prove that a bipartite graph with parts of sizes M and N, having no cycles of even length less than or equal to 2(2k+1), has at most (NM)^{ \frac{k+1}{2k+1}} + O(N+M) edges.
Bulletin of the Australian Mathematical Society, 63, (2001) 435-440.

82. Graphs without cycles of even length
Honours thesis for my B.Sc. at University of New South Wales (supervised by Professor Terence Tao), 2001.