Ma 191c-sec2:  Geometric Incidences (Spring 2016-17)

## COURSE DESCRIPTION

Geometric incidences are a family of combinatorial problems. While these problems existed for several decades, in the past few years they have been experiencing a renaissance: Many new incidence results are being derived by using algebraic methods, while at the same time interesting connections between incidences and other parts of mathematics are being exposed (such as Harmonic Analysis and Theoretical Computer Science). This is currently an active research field which seems to attract the interest of various prominent mathematicians. In this class we will study this subfield and some of its connections to other parts of mathematics.

Each course participant is expected to read one related paper and to present it in class. This webpage will contain a list of possible papers. Students are also expected to attend most of the classes.

## PREREQUISITES

The class requires a basic mathematical understanding, such as basic familiarity with combinatorics, probability, and linear algebra. We will go over most of the mathematical concepts that we rely on.

## SCHEDULE

MWF, 15:00 - 15:55, 103 Downs

## INSTRUCTORS

HARRY BATEMAN INSTRUCTOR IN MATHEMATICS
Sloan 276
626-395-4347

By appointment.

## TEXTBOOKS

We will rely on this draft of Adam's book "Incidence Theory". This draft does not yet contain the entire material, and additional chapters will be added as the term continues. See below for the relevant sections for each class. Adam will be very happy to hear about any mistakes, typos, or even unclear formulations that you find in this draft.

Other relevant resources are Larry Guth's book on polynomial methods in combinatorics and Ze'ev Dvir's survey.

## LECTURE NOTES

The "Sections" column contains the relevant section numbers of the book.

Date Description Sections
April 3rd Introduction to incidences 1.1,1.2,
April 5th The Szemeredi-Trotter theorem and unit distances 1.3,1.4,1.5
April 7th First applications of incidences 1.6,1.8
April 10th Basic Algebraic Geometry in $${\mathbb R}^2$$ 2.1,2.2
April 12th Basics of polynomial partitioning and incidences with planar curves 2.2, 3.1,3.2
April 14th More polynomial partitioning and incidences with planar curves 3.2
April 17th Proving the polynomial partitioning theorem, lattice points on curves 3.3,3.4
April 19th Basic Algebraic Geometry in $${\mathbb R}^d$$ 4.1,4.2,4.3
April 21st More Algebraic Geometry in $${\mathbb R}^d$$ 4.3,4.4,4.5
April 24th The joints problem 5.1,5.2
April 26th Introduction to incidences in $${\mathbb R}^d$$ 6.1,6.2,6.3
April 28th The Szemeredi-Trotter theorem in $${\mathbb C}^2$$ 6.3,6.4
May 1st Incidences with arbitrary curves in $${\mathbb C}^2$$ 6.5
May 3rd Introduction to incidences over finite fields 7.1,7.2
May 5th Finite field Kakeya and introduction to projective spaces 7.2,7.3,7.4
May 8th Vinh's bound and planes in $${\mathbb F}_q^3$$ 7.4,7.5
May 10th Rudnev's point-plane incidence bound in $${\mathbb F}_q^3$$ 7.5
May 12th Talk by Alex on "Sharpness of Falconer’s estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice" ---
May 15th Talk by Noah on "Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes" ---
May 17th Talk by Luke on "$${\mathbb F}_p$$ is locally like $$\mathbb C$$" ---
May 19th Talk by Cosmin on "Polynomials vanishing on grids: The Elekes-Ronyai problem revisited" ---
May 22nd Talk by Zach on "Geometric incidence theorems via fourier analysis" ---
May 24th Talk by Aaron on "Cutting lemma and Zarankiewicz's problem in distal structures" ---
May 26th Talk by Sam on a reduction to the distinct distances problem in $${\mathbb R}^d$$ Ditch day! ---
May 29th Memorial day - no class ---
May 31st Talk by Siddharth on "Applications of incidence bounds in point covering problems" ---
June 2nd Talk by Lazar on "The Szemeredi-Trotter Theorem in the Complex Plane" ---
June 5th Talk by Sam on a reduction to the distinct distances problem in $${\mathbb R}^d$$ ---
June 7th Talk by Lazar on "The Szemeredi-Trotter Theorem in the Complex Plane" ---
June 9th After the end of the year for grad students and seniors ---

The following is a list of papers for students who give a talk in class. I might add a few more papers. You are welcome to suggest papers that are not on the list, or ask Adam whether there are papers related to some specific subject.

A few additional papers that come from more surprising fields.