# Summer School: Polynomial Methods

**Dates:** June 7-17, 2021

Venue: online

Organizers: Valentin Blomer, Farrell Brumley, Philip Gressman, Marina Iliopoulou, Lillian B. Pierce

Description:

Powerful progress on a wide selection of problems spanning across harmonic analysis and number theory has involved methods using auxiliary polynomials. On the more number-theoretic side, this includes the recent resolution of the dimension growth conjecture via the determinant method, as well as Stepanov’s method for proving square-root cancelation of exponential sums, and results in transcendence theory related to open conjectures in algebraic number theory. On the more analytic side, applications of polynomial methods include the resolution of the Kakeya conjecture over finite fields, progress in incidence geometry, and new methods for tackling difficult questions on restriction inequalities.This summer school gathers together all these "polynomial methods" with accessible lecture series from each of these perspectives. Graduate students and postdocs will gain intuition and technical skills for how these methods can be applied in a wide range of settings. While the polynomial methods presented in this summer school may have “evolved” independently, the lecture series will explore connections and parallels that unify the methods.

Given the current situation with COVID-19, the ongoing hygienic restrictions, difficulties in obtaining visa and further obstacles for travelers, all talks will be available for viewing live online. The lectures will take place over two weeks, concentrated within a few hours each day.

Lecture series by:

- Valentin Blomer (University of Bonn)
- Samit Dasgupta (Duke University)
- Roger Heath-Brown (University of Oxford)
- Marina Iliopoulou (University of Kent)
- Hong Wang (Institute for Advanced Study)

Click here for the abstracts.

Click here for the schedule.

If you are interested in attending the school, please click here for online registration.

## Video recordings and slides

Click here for the slides Samit Dasgupta, lecture I

Click here for the slides Samit Dasgupta, lecture II

Click here for the slides Samit Dasgupta, lecture III

Click here for the slides Samit Dasgupta, lecture IV

Click here for the slides Roger Heath-Brown, lecture I

Click here for the slides Roger Heath-Brown, lecture II

Click here for the slides Hong Wang, lecture I+II

Click here for the slides Hong Wang, lecture III+IV

Click here for the slides Valentin Blomer

**Marina Illiopoulou: Three polynomial methods for point counting**

**Lecture I**

**Lecture II**

**Lecture III**

**Lecture IV**

**Samit Dasgupta: An introduction to auxiliary polynomials in transcendence theory**

**Lecture I**

**Lecture II**

**Lecture III**

**Lecture IV**

**Roger Heath-Brown: The Determinant Method**

**Lecture I**

**Lecture II**

**Lecture III**

**Hong Wang: The restriction problem and the polynomial method**

**Lecture I**

**Lecture II**

**Lecture III**

**Lecture IV**

**Valentin Blomer: The polynomial method for point counting and exponential sums**

**Lecture I**

**Lecture II**

**Lecture III**

**Lecture IV**