Thomas Lam's Publications and Preprints

Back stable Schubert calculus (with Seung Jin Lee and Mark Shimozono)
We study the back stable Schubert calculus of the infinite flag variety.
arxiv: 1806.11233

Parity duality for the amplituhedron. (with Pavel Galashin)
We prove a bijection between triangulations of parity dual amplituhedra.
arxiv: 1805.00600

The totally nonnegative part of G/P is a ball (with Pavel Galashin and Steven Karp)
We prove that the totally nonnegative part of G/P, in the sense of Lusztig, is homeomorphic to a closed ball.
arxiv: 1801.08953

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

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 rfold dimer model of a bipartite graph.
arxiv: 1705.09424

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

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 Dmodule (Amodel) as Galois and the LandauGinzburg model (Bmodel) as Automorphic, then apply ideas from the Langlands program.
arxiv: 1705.00758

Positive geometries and canonical forms (with Nima ArkaniHamed 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

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

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

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

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

Toric networks, geometric Rmatrices and generalized discrete Toda lattices (with Rei Inoue and Pavlo Pylyavskyy)
We study the dynamics of the double affine geometric Rmatrix 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), 799855.
arxiv: 1504.03448

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 onerow crystals, as the tropicalization of a cylindric loop Schur function. We prove this formula for the first shape, thus giving a piecewiselinear formula for the lengths of the solitions of a boxball system.
Annales de l'Institut Henri Poincare D, to appear.
arxiv: 1410.4455

Grassmannians for scattering amplitudes in 4d N=4 SYM and 3d ABJM (with Henriette Elvang, Yutin 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

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, 10251037.
arxiv: 1408.5531

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), 105111.
arxiv: 1406.5671

Dimers, webs, and positroids
We study the dimer, doubledimer, and tripledimer 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), 633656.
arxiv: 1404.3317

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

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

kSchur 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 kSchur functions and affine Schubert calculus, based on lectures at the Fields Institute in 2010.
Fields Institute Monographs, Volume 33.
arxiv: 1301.3569

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), 76907733.
arxiv: 1301.0873

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), 31633203.
arxiv: 1206.2612

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), 121162.
arxiv: 1206.2611

Total positivity, Schubert positivity, and Geometric Satake (with Konstanze Rietsch)
We compare three notions (total, Schubert, MirkovicVilonen) 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), 284319.
arxiv: 1203.1682

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 qanalogue of Weyl's formula for the order of a Weyl group.
Kostant Memorial Volume, Birkhauser, to appear.
arxiv: 1202.4015

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), 17101752.
arxiv: 1111.3660

From double quantum Schubert polynomials to kdouble Schur functions via the Toda lattice (with Mark Shimozono)
We show that kdouble 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

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), 835850.
arxiv: 1108.4958

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), 23852412.
arxiv: 1106.2586

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), 623648.
arxiv: 1105.5154

kDouble Schur functions and equivariant (co)homology of the affine Grassmannian (with Mark Shimozono)
The Schubert bases of the torusequivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called kdouble Schur functions and affine double Schur functions.
Math. Annalen, 356 (2013), 13791404.
arxiv: 1105.2170

Inverse problem in electrical cylindrical networks (with Pavlo Pylyavskyy)
We study the inverse DirichlettoNeumann problem for cylindrical electrical networks, and define an "electricalRmatrix" which conjecturally solves the inverse problem.
SIAM J. of Applied Math., 72 (2012), 767788.
arxiv: 1104.4998

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), 14011418.
arxiv: 1103.3475

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), 189192.
arxiv: 1102.4406

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 ncore is a piecewiselinear graph, in the sense of Vershik and Kerov.
Ann. of Probability, 43 (2015), 16431662.
arxiv: 1102.4405

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), 609628.
arxiv: 1012.1262

Boxbasketball 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

From quantum Schubert polynomials to kSchur functions via the Toda lattice (with Mark Shimozono)
We show that quantum Schubert polynomials and kSchur functions can be obtained from each other via a rational substitution.
Math Research Letters, 19 (2012), 8193.
arxiv: 1010.4047

Projections of Richardson Varieties (with Allen Knutson and David Speyer)
We prove that projected Richardson varieties are normal, CohenMacaulay, have rational singularities, and are exactly the compatibly Frobeniussplit 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), 133157.
arxiv: 1008.3939

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), 173235.
arxiv: 1008.1949

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

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

Affine geometric crystals in unipotent loop groups (with Pavlo Pylyavskyy)
The quotient by the Rmatrix 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), 719728.
arxiv: 1004.2233

Intrinsic energy is a loop Schur function (with Pavlo Pylyavskyy)
We identify the intrinsic energy function of tensor products of KirillovReshetikhin crystals for symmetric powers of type A with the tropicalization of loop Schur functions.
Journal of Combinatorics, 4 (2013), 387401.
arxiv: 1003.3948

Skew LittlewoodRichardson rules for Hopf algebras (with Aaron Lauve and Frank Sottile)
We use Hopf algebra theory to prove "skew" versions of LittlewoodRichardson rules, establishing a conjecture of Assaf and McNamara.
IMRN, (2011), 12051219.
arxiv: 0908.3714

(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), 277290.
arxiv: 0908.0345

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), 179231.
arxiv: 0906.0610

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

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

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

Ktheory Schubert calculus of the affine Grassmannian (with Anne Schilling and Mark Shimozono)
We study the Ktheory 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 KkSchur functions and affine stable Grothendieck polynomials.
Compositio Mathematica, 146 (2010), 811852.
arxiv: 0901.1506

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), 12221271.
arxiv: 0812.0840

Quantized dual graded graphs
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

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), 5560.
[ps] [pdf]

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, 5260.
arxiv: 0710.3744

Total positivity for cominuscule Grassmannians (with Lauren Williams)
We study the totally nonnegative cells of cominuscule Grassmannians. In particular, we define and study Lediagrams.
New York J. of Math., 14 (2008), 5399.
arxiv: 0710.2932

Schubert polynomials for the affine Grassmannian of the symplectic group (with Anne Schilling and Mark Shimozono)
Using Schur P and Qfunctions, we define and study Schubert polynomials for the affine Grassmannian of the symplectic group.
Math. Z., 264 (2010), 765811.
arxiv: 0710.2720

Combinatorial Hopf algebras and Khomology of Grassmannians (with Pavlo Pylyavskyy)
Motivated by work of Buch on setvalued tableaux, we define and study six "Ktheoretic" combinatorial Hopf algebras.
International Mathematics Research Notices, 2007 (2007), rnm 125, 48 pages.
arxiv: 0705.2189

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), 4990.
arxiv: 0705.1386

Dual graded graphs for KacMoody algebras (with Mark Shimozono)
We define and study a family of dual graded graphs with vertex set equal to the Weyl group of a KacMoody algebra.
Algebra and Number Theory, 1 (2007), 451488.
arxiv: math.CO/0702090

TemperleyLieb Pfaffinants and Schur Qpositivity conjectures (with Pavlo Pylyavskyy)
We define and study a pfaffian analogue of immanants, focusing on TemperleyLieb pfaffinants.
Adv. in Math., 218 (2008), 16541684.
arxiv: math.CO/0612842

Signed differential posets and signimbalance
We study a signed analogue of differential posets, and relate them to signimbalance.
J. Comb. Theory Series A, 115 (2008), 466484.
arxiv: math.CO/0611296

On domino insertion and KazhdanLusztig cells in type B_n (with Cedric Bonnafe, Meinolf Geck, and Lacri Iancu)
We give a conjectural characterization of all KazhdanLusztig 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.) , 3354, Progress in Math. 284, Birkhauser, 2010.
arxiv: math.RT/0609279

Ppartition products and fundamental quasisymmetric function positivity (with Pavlo Pylyavskyy)
We show that certain differences of products of Ppartition generating functions are positive combinations of fundamental quasisymmetric functions.
Adv. Appl. Math., 40 (2008), 271294.
arxiv: math.CO/0609249

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

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

Schubert polynomials for the affine Grassmannian
We identify the (co)homology Schubert basis of the affine Grassmannian as the (dual)kSchur 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), 259281.
[ps] [pdf]
arxiv: math.CO/0603125

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

A combinatorial generalization of the BosonFermion 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), 377392.
[ps] [pdf]
arxiv: math.CO/0507341

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), 209224.
arxiv: math.CO/0505273

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 BosonFermion correspondence.
[ps] [pdf]

Schur positivity and Schur logconcavity (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), 16111622.
arxiv: math.CO/0502446

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), 453478.
arxiv: math.CO/0501246

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), 15531586 .
arxiv: math.CO/0501335

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), 343359.
arxiv: math.CO/0409463

A note on graphs without short even cycles (with Jacques Verstraete)
We show that any nvertex 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

On symmetry and positivity for domino and ribbon tableaux
We show the symmetry of LLTribbon functions, and describe the product of a Schur function and a domino function.
Annals of Combinatorics, 9 (2005), 293300.
[ps] [pdf]
arxiv: math.CO/0407184

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]

Ribbon Tableaux and the Heisenberg Algebra
We prove Pieri, Cauchy and MurnaghanNakayama 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), 685710.
[ps] [pdf]
arxiv: math.QA/0310250

Growth diagrams, domino insertion, and signimbalance
We study some properties of domino insertion and settle Stanley's `2^{n/2}' conjecture on signimbalance.
Journal of Combinatorial Theory Ser. A., 107 (2004), 87115.
[ps] [pdf]
arxiv: math.CO/0308265

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]

A result on 2kcycle 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]

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) 435440.

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