MCS 591 Discrete Geometry, Spring 2015
Instructor: Andrew Suk
Time and Place: LH 215, 11-1150 MWF.
Office hours: Fri: 2-3, and by appointment, SEO 521.
References: (Recommended) Lectures in Discrete Geometry by Jiri Matousek, and Combinatorial Geometry by Janos Pach and Pankaj Agarwal.
Grades: Will be based on homework only.
Course Description: The aim of this course is to give an introduction to discrete geometry and present several recent results in the field. Topics include: Incidence geometry, geometric Ramsey theory, and intersection patterns of geometric objects with applications to graph drawing.
Announcements Classes on 26, 28, 30 of January are cancelled. Makeup classes will be scheduled later (sometime in Feb, March, and/or April). Homework 1 may be turned in on Wednesday Feb 4.
Homework 1. Due Monday Feb. 2 in class. (Note: a typo was corrected on Problem 4c: "of" to "or"). Solutions.
Homework 2. Due Monday March 2 in class.
Homework 3. Due Monday April 20 in class.
Lecture 1 (12 January), Kovari-Sos-Turan's Theorem and Szemeredi-Trotter (statement).
Lecture 2 (14 January), Crossing Lemma and Szekely's proof of Szemeredi-Trotter.
Lecture 3 (16 January), Crossing lemma for mutli-graphs and Sums versus Product.
Lecture 4 (21 January), Cell decomposition, weak cutting lemma.
Lecutre 5 (23 January), Simplicial partitions.
Lecture 6 (2 February), Lower bound consruction for point-line incidences. Notes (by David Conlon) on the triangle removal lemma and Roth's Theorem are here.
Lecture 7 (4 February), Solymosi's theorem on dense arrangements of points and lines.
Lecture 8 (6 February), Solymosi's theorem on sums versus product.
Lecture 9 (9 February), Caratheodory's theorem and Radon's Theorem.
Lecture 10 (11 February), Helly's theorem and the center point theorem. Larry Guth's notes on the ham sandwich theorem and the polynomial ham sandwich theorem.
Lecture 11 (13 February), Polynomial Ham Sandwich Theorem, Erdos-Szekeres Happy Ending Theorem.
Lecture 12 (16 February), the Erdos-Szekeres convex polygon theorem.
Lecture 13 (18 February), Ramsey-type results for monotone paths in ordered graphs and 3-uniform hypergraphs.
Lecture 14 (20 Februrary), The fractional Helly theorem.
Lecture 15 (23 Februrary), Tverberg's Theorem (Sarkaria's proof).
Lecture 16 (25 February), Colorful Caratheordory's theorem, and the first selection lemma.
Lecture 17 (27 February), The colored Tverberg's Theorem and the second selection lemma.
Lecture 18 (2 March), the Erdos-Simonovitz theorem and Ramsey's theorem. See Jacob Fox's notes on Ramsey's Theorem here.
Lecture 19 (4 March), The Erdos-Hajnal Conjecture, and their theorem on forbidden induced subgraphs.
Lecture 20 (6 March), The Erdos-Hajnal Theorem and the strong Erdos-Hajnal Property.
Lecture 21 (9 March), The Erdos Hajnal property and interesection graphs.
Lecture 22 (11 March), Intersection graphs have the strong Erdos-Hajnal property
Lecture 23 (13 March), Density theorems for intersection graphs, and the weak regularity lemma.
Lecture 24 (16 March), Triangle free intersection graphs of segments with large chromatic number.
Lecture 25 (18 March), Set systems with bounded VC-dimension.
Lecture 26 (20 March), Proof of the epsilon-net theorem.
Lecture 27 (30 March), Packing Lemma for set systems with bounded VC-dimension.
Lecture 29 (1 April), Graphs defined geometrically and graphs with bounded VC-dimension.
Lecture 30 (3 April), Lovasz-Szegedy regularity lemma for graphs with bounded VC-dimension.
Lecture 31 (6 April), Nonlinear lower bound for epsilon nets, and weak epsilon nets for convex sets.
Lecture 32 (8 April), The weak epsilon net theorem for convex sets.
Lecture 33 (10 April), The (p,q)-problem of Hadwiger and Debrunner, the fractional transversal number.
Lecture 34 (13 April), The (p,q)-theorem of Alon and Kleitman.
Lecture 35 (15 April), Bounded VC-dimension implies a fractional Helly theorem, which implies a (p,q)-theorem.
Lecture 36 (17 April), The distinct distances and unit distance problem.
Lecture 37 (20 April), Unit distances among n points in convex position, distinct distances among n points in general position.
Lecture 38 (22 April), Distinct distances between two lines.
Lecture 39 (24 April), The joints theorem.
Lecture 40 (27 April), Distinct distances: rigid motions and lines in 3-space.
Lecture 41 (29 April), Polynomial partitioning, estimating r-rich points among lines in 3-space.
Lecture 42 (1 May), n points in the plane determine n^{1 - epsilon} distinct distances.