Seminar on classic results, Spring 2017

1914 Walter Crane


Wednesdays 10:30–noon and 5:00–approx. 6:00


Wean Hall 6202


Math is pretty, and much of its beauty lies sleeping on the pages of classic papers. The aim of this course is to have fun reading mathematics, and to explain it to each other. Each week, one of the students presents a topic of their choice to an inquisitive group of peers. Fun ensues.

The seminar takes places in two session of total length of 2.5 hours. This permits detailed examination of arguments, exploration of side arguments, and lots of discussion.

Below is a list of potential topics, combining my suggestions with those of the students. Additions are welcome.

Ramsey theory
Euclidean Ramsey theory
All triangles are RamseyFrankl and RödlMathSciNet
A partition property of simplices in Euclidean spaceFrankl and Rödlpdf
Strong Ramsey properties of simplicesFrankl and RödlMathSciNet
Permutation groups in Euclidean Ramsey theoryKřížMathSciNet
Hales--Jewett-type approach to Euclidean Ramsey theory:
Transitive Sets in Euclidean Ramsey TheoryLeader, Russell, WaltersarXiv
Transitive Sets and Cyclic QuadrilateralsLeader, Russell, WaltersarXiv
Arithmetic combinatorics
Solutions to linear equations
Linear problems in combinatorial number theory Komlós, Sulyok, SzemerédiMathSciNet
Solving a linear equation in a set of integers. I and IIRuzsaMathSciNet
Three-term arithmetic progressions
On subsets of finite abelian groups with no 3-term arithmetic progressions.MeshulamMathSciNet
On large subsets of Fqn with no three-term arithmetic progressionEllenberg and GijswijtarXiv
A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset boundTaoblog
Extremal (hyper)graph theory
Undecidability of linear inequalities in graph homomorphism densitiesHatami and NorinearXiv
Probabilistic combinatorics
Algorithmic local lemma
A constructive proof of the general Lovasz Local LemmaMoser and TardósarXiv
Entropy compression argumentTaoblog
(Application) A new approach to nonrepetitive sequencesGrytczuk, Kozik, MicekarXiv
Hypergraph containers
Counting independent sets in graphsSamotijpdf
Simple containers for simple hypergraphsSaxton and ThomasonarXiv
Independent sets in hypergraphsBalogh, Morris, and Samotijpdf
Removal and regularity lemmas
Graph removal lemmasConlon and FoxarXiv
A short proof of Gowers' lower bound for the regularity lemmaMoshkovitz and ShapiraarXiv
Algebraic methods
Linear algebra method
Intersection theorems with geometric consequencesFrankl and WilsonMathSciNet
Counterexample to Borsuk's conjectureKahn and KalaiMathSciNet
On the number of zero-patterns of a sequence of polynomialsRónyai, Babai, GanapathyMathSciNet
Random polynomial constructions
Random algebraic construction of extremal graphsBukharXiv
Rational exponents in extremal graph theoryBukh and ConlonarXiv
On the size of Kakeya sets in finite fieldsDvirarXiv
The joints problem in RnQuilodránarXiv
Slice rank method
On large subsets of Fqn with no three-term arithmetic progressionEllenberg and GijswijtarXiv
A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset boundTaoblog
Notes on “slice rank” of tensorsTao and Sawinblog
Extremal combinatorics
Entropy method
An entropy proof of the Kahn-Lovász theoremCutler and RadcliffeMathSciNet
An entropy approach to the hard-core model on bipartite graphsKahnMathSciNet
The number of independent sets in a regular graphZhaoarXiv
Discrepancy theory
Combinatorial discrepancy
Six standard deviations sufficeSpencerMathSciNet
Constructive discrepancy minimization by walking on the edgesLovett and MekaarXiv
Factorization norms and hereditary discrepancyMatoušek, Nikolov and TalwararXiv
Semidefinite programming
Approximating cut-norm via Grothendieck's inequalityAlon and NaorMathSciNet