Teaching

VRG on Geometric Combinatorics and Fourier Analysis

Fall 2008 & Spring 2009

Thursdays from 2:00-3:30 in Boyd 323

Notes:

From harmonic analysis to arithmetic combinatorics (Izabella Laba)

Erdos distance problem (Alex Iosevich and Julia Garibaldi)

Crossing number inequality (Terry Tao)

Fourier analysis and geometric combinatorics (Alex Iosevich)

Curvature, cominatorics, and the Fourier transform (Alex Iosevich)

Sum-product estimates

An upper bound on the multiplicative energy (Solymosi's most recently improvement)

The beautiful argument of Elekes is discussed in Tao's notes on the crossing number inequality

The Kakeya conjecture

In addition to the notes you already have (Laba's AMS Bulletin paper) other sources (there are many) include:

Ben Green's Part III notes on Restriction & Kakeya (lectures 1-3 for now)
Alex Iosevich also has 5 Lectures on Kakeya
Terry Tao and Izabella Laba have expositions of Dvir's recent solution to the finite field analogue of the Kakeya conjecture here and here

Schedule:

Erdos Distance Problem

	Aug 28	Two proofs of the N^{1/2} bound	Nathan Walters
		Further investigation: How many integers from 1 to N can be written as the sum of two squares?
	Sept 4	Overview and general discussion	Neil Lyall
		Investigate different metrics and higher dimensions
	Sept 11	Incidence geometry, the crossing number inequality, and a first proof of the N^{2/3} bound	Stacy Musgrave & David Krumm
	Sept 18	Moser's proof of the N^{2/3} bound & the Erdos integer distance principle	Nham Ngo & Katherine Thompson
		Obtain the bound N^{3/4} (N^{d/2-1/d^2} if d>2) for "well-distributed/homogeneous" sets
		Integral sets, generalizations of the EIDP, and their relation to the distance problem
	Sept 28	Crossing number inequality for multi-graphs and a sketch of how we can reach the N^{4/5} plateau	Neil Lyall
	Oct 2	Using bisectors and Szemeredi-Trotter to get Szekely's N^{4/5} bound	Nathan Walters
		Can one obtain the bound N^{4/5} for different metrics?
		Obtain the bound N^{6/7} for the Euclidean metric (arithmetic enters the picture)

Sum-Product Estimates

	Oct 16	The beautiful argument of Elekes that gives \|A\|^{5/4}	David Krumm
		Extremal cases: Conjecture holds if either \|A+A\| or \|AA\| is small
		Solymosi's \|A\|^{4/3} bound
	Oct 23	General discussion	Nham Ngo and Neil Lyall
		The sum-product problem in the complex numbers
		The sum-product problem in finite fields

The Kakeya Conjecture

	Oct 30	Construction of a Kakeya set in the plane	Stacy Musgrave & Nham Ngo
	Nov 6	Minkowski dimension, the Kakeya conjecture, and establishing the conjecture in the plane	Katherine Thompson & Nathan Walters
	Nov 13	The (n+1)/2 bound in higher dimensions	Neil Lyall
		The finite field Kakeya conjecture and Wolff's (n+2)/2 bound in the finite field setting	David Krumm
		Obtain the bound (4n+3)/7 in the finite field setting (arithmetic enters the picture)

	Nov 20	Dvir's solution to the finite field Kakeya conjecture	Pete L. Clark
		Further investigation: Dvir's method
	Dec 4