Ma 191c:  The Polynomial Method (Spring 2014-15)


  • 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.


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.


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.).


The class will meet Mondays, Wednesdays, and Fridays 2:00 – 2:55 PM, in 151 Sloan.


Adam Sheffer
276 Sloan, 626-395-4347,


Email me to schedule an appointment.


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.


Date 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 24thDistinct 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.


  • 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