## ANNOUNCEMENTS

- If you are giving a talk, please read this document.
- I am very happy to hear any comments about the lecture notes - even small issues like typos or sentences that are not completely clear. If you notice something, please tell me.

## COURSE DESCRIPTION

The "polynomial method" is a new sub-field of mathematical research, which emerged about 5-6 years ago and is currently very active. It revolves around the use of tools from algebraic geometry to solve problems in discrete geometry (the study of combinatorial problems with some geometric flavor). Moreover, it already has some connections with harmonic analysis, computer science, number theory, and more.

Since this is a new sub-field, there are still no textbooks available for it. However, detailed lecture notes will be placed on the website. The material partially overlaps with a course that was recently given by Larry Guth in MIT, and with a course that was recently given by Zeev Dvir in Princeton. Lecture notes of these courses can be found here and here.

Each course participant is expected to read one related paper and to present it. This webpage will contain a list of possible papers. If more than a few students will enroll (which currently seems likely), some of the undergrad participants would have 3-4 homework assignments instead. Students are also expected to attend more than half of the classes.

## PREREQUISITES

The course requires no background in either discrete geometry or algebraic geometry. We will go over all of the relevant background. However, basic mathematical background is assumed (Ma1, very basic probability, etc.).

## INSTRUCTOR

Adam Sheffer

HARRY BATEMAN INSTRUCTOR IN MATHEMATICS

276 Sloan, 626-395-4347,

adamsh@caltech.edu

## TOPICS COVERED

The following list contains some of the topics that we will cover. It is tentative and might change.- Guth and Katz's distinct distances result.
- Joints problems.
- Incidences in higher dimensions.
- Dvir's finite field Kakeya result.
- Etc.

## LECTURE NOTES

sDate | Description | Notes |
---|---|---|

March 30th | A brief non-rigorous introduction to the polynomial method | None |

April 1st | Incidences via crossings | chapter 1 |

April 3rd | Distinct distances and unit distances | chapter 1 |

April 6th | Ideals and varieties | Chapter 2 |

April 8th | Basic properties of real varieties: dimension, degree, and singularities. | Chapter 2 |

April 10th | Real curves and introduction to polynomial partitioning | Chapter 2 and Chapter 3 |

April 13th | Incidences with curves | Chapter 3 |

April 15th | Proving the polynomial partitioning theorem | Chapter 3 |

April 17th | Curves containing many lattice points. Constant-degree partitioning | Chapter 3 and Chapter 4 |

April 20th | Szemerédi–Trotter in the complex plane | Chapter 4 |

April 22nd | The Joints problem | Chapter 5 |

April 24th | Distinct distances between points on two lines | Chapter 6 |

April 27th | The Elekes-Sharir-Guth-Katz framework | Chapter 6 |

April 29th | From intersections to incidences | Chapter 7 |

May 1st | Points with at least three lines through them. | Chapter 7 |

May 4th | Flat points and incidences in the partition | Chapter 7 |

May 6th | Degree reduction and ruled surfaces | Chapter 8 |

May 8th | Completing the distinct distances proof | Chapter 8 |

May 11th | Two distinct distances variants | Chapter 9 |

May 13th | Lecture by Kevin Yin about A note on rich lines in truly high dimensional sets / Zahl. | |

May 15th. Special room - Downs 119 | Lecture by Emory Brown about Polynomial partitioning for a set of varieties / Guth. | |

May 18th | Lecture by Jenish Mehta about On the size of Kakeya sets in finite fields / Dvir. | |

May 20th | Lecture by Karlming Chen on Szemerédi--Trotter-type theorems in dimension 3 / Kollár. | |

May 27th | Lecture by Alex Mun about The number of unit-area triangles in the plane: Theme and variations / Raz and Sharir. | |

May 29th | Lecture by Daodi Lu about Algebraic curves, rich points, and doubly-ruled surfaces / Guth and Zahl. | |

June 1st | Lecture by Bella Guo about Cutting circles into pseudo-segments and improved bounds for incidences / Aronov and Sharir . | |

June 3rd | Lecture by Milica Kolundžija about Distinct distances on algebraic curves in the plane / Pach and de Zeeuw | |

June 3rd Sloan 153 at 3pm | Lecture by Nikola Kovachki about Generalizations of the Szemerédi-Trotter Theorem / Kalia and Yang | |

June 5th | Lecture by Gaurav Sinha about about An Exposition of Bourgain’s 2-Source Extractor / Anup Rao. |

The following is a list of papers for students who give a talk in class. You may also suggest papers that are not on this list.

~~Polynomial partitioning for a set of varieties / Guth. Constructs partitioning polynomials for a set of varieties (instead of for a set of points).~~~~Algebraic curves, rich points, and doubly-ruled surfaces / Guth and Zahl. A recent paper that I am very interested to hear more about. Two people may do this one together (using two classes to talk about this).~~- Distinct distance estimates and low degree polynomial partitioning / Guth. A simplified proof of the distinct distances result, although yielding a slightly weaker bound.
~~A note on rich lines in truly high dimensional sets / Zahl. Incidences with lines in complex spaces.~~~~Szemerédi--Trotter-type theorems in dimension 3 / Kollár. Preferably for someone with more advanced knowledge in algebraic geometry and/or an interest in finite fields.~~- Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions / Basu and Sombra. Incidences in 4-dimensional spaces. Based on Hilbert functions.
- On the use of Klein quadric for geometric incidence problems in two dimensions / Rudnev and Selig. Presents a generalized framework for the one that is used to prove Guth and Katz's distinct distances theorem.
- Incidences between points and lines in R^4 / Sharir and Solomon. Can be done by two people together (using two classes to talk about this).
- Incidences between points and lines on a two-dimensional variety / Sharir and Solomon. More about point-line incidences in three dimensions.
- Incidence bounds on multijoints and generic joints / Iliopoulou. A more recent joints result.
~~Generalizations of the Szemerédi-Trotter Theorem / Kalia and Yang. Incidences with "flags".~~- Polynomials vanishing on grids: The Elekes-Rónyai problem revisited / Raz, Sharir, and Solymosi. Studies polynomials that vanish on a large number of points of a three-dimensional lattice. Two people may do this one together (using two classes to talk about this).
~~Distinct distances on algebraic curves in the plane / Pach and de Zeeuw. Distinct distances when the point set is fully contained in a given curve.~~- On the number of incidences between planes and points in three dimensions / Rudnev. Deriving an incidence bound over a finite field.

## HOMEWORK

- You are encouraged to work in groups on the assignments. However, you are required to write your solutions on your own.
- You are supposed to rely only on material that was studied in this course. If you wish to use material from a different place, it is recommended to first ask me about it. References to a book/webpage that contains a solution to a problem will receive zero points.
- Assignments are to be submitted to the Ma191 box on the second floor of the Sloane building (even though there is no section number on the box, it belongs to us).

Due Date | Homework |
---|---|

April 28th | Homework 01 |

May 15th | Homework 02 |

June 3rd | Homework 03 |