## 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.

## 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 | --- | |

May 29th | Memorial day - no class | --- |

May 31st | Talk by Siddharth on "Applications of incidence bounds in point covering problems" | --- |

June 2nd | --- | |

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 | --- |

## READING

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.

- Cutting Algebraic Curves into Pseudo-segments and Applications by Sharir and Zahl. This paper derives the current best point-curve incidence bound in \({\mathbb R}^2\).
- If you like Logic, two recent papers extend incidence bounds to
o-minimal structures.
~~One paper is by Chernikov, Galvin, and Starchenko, and relies on a partitioning technique.~~The other paper is by Basu and Raz and relies on a crossing lemma, as we've seen in the beginning of the course. - Covering lattice points by subspaces and counting point-hyperplane incidences by Balko, Josef Cibulka, and Pavel Valtr. This paper provides new lower bounds for a point-plane incidence problem in \({\mathbb R}^d\).
~~\({\mathbb F}_p\) is locally like \(\mathbb C\) by Grosu. This paper shows how combinatorial results such as the Szemeredi-Trotter theorem apply in finite fields when the sets are sufficiently small with respect to \(p\). As you might expect, this is a very algebraic paper.~~~~Geometric incidence theorems via fourier analysis by Iosevich, Jorati, and Łaba. As you can see from the title, this paper is for students who are interested in Harmonic Analysis.~~-
~~The Szemeredi-Trotter Theorem in the Complex Plane by Toth. This is an older proof of the complex Szemeredi-Trotter theorem, from before the new era of the polynomial methods. It gives a slightly stronger result than the algebraic proof that we see in class.~~ - Rank bounds for design matrices with block entries and geometric applications by Dvir, Garg, Oliveira, and Solymosi. This paper is probably for people who like linear algebra. It studies combinatorial properties of certain matrices and derives applications to incidences and related problems.
~~Polynomials vanishing on grids: The Elekes-Ronyai problem revisited by Raz, Sharir, and Solymosi. A very impressive application of incidences, concerning expanding polynomials.~~~~Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances by Sharir and Solomon. This paper studied several incidence problems in \({\mathbb R}^3\) and several applications. It uses tools from Algebraic Geometry in the same spirit as we have seen in class.~~~~Applications of incidence bounds in point covering problems by Afshani, Berglin, van Duijn, and Nielsen. This is an algorithmic paper, for more CS-oriented people. It relies on incidences to study algorithms for covering problems.~~- New bounds on curve tangencies and orthogonalities by Ellenberg, Solymosi, and Zahl. This paper is not about incidence but rather about a related problem concerning curve tangencies over any field. The techniques used in the paper are also in the same spirit as the ones we see in class.
~~Sharpness of Falconer’s estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains by Iosevich and Senger. The Falconer conjecture can be seen as a continuous analog of the distinct distances problem. Another paper for analysis-oriented people.~~

A few additional papers that come from more surprising fields.

- A surprising work in biology by Zongker. There is a good conference talk about this result available online, which might help you prepare your own talk.
- An interesting work by Armstrong related to Neural Networks. You might also like to look at this related work of Armstrong.