Larry Guth's homepage

Contact information.

Email: lguth@math.mit.edu

Office: 2-278

Teaching.

Fall 2017, I am teaching 18.118, a topics course in Fourier analysis about decoupling theory.

Spring 2016, I am teaching 18.156, Real Analysis. See also the Open Courseware website for the course 18.156 on OCW.

I am also teaching a unit on decoupling in Math 158: Topics in analysis. There is a little webpage for the unit here: Decoupling unit.

Fall 2015, I am teaching 18.103, Fourier Analysis.

Spring 2015, I am teaching 18.156, Real Analysis.

Fall 2014, I am teaching 18.103, Fourier Analysis.

Spring 2014, I am teaching 18.156, Real Analysis, and 18.979, Geometry Seminar (a reading course on classic papers in differential geometry).

Fall 2013, I am teaching 18.821, Math Project Lab. If you are in the course, you can access the course info on Stellar.

Spring 2013, I am teaching 18.966 Geometry of manifolds.

Fall 2012, I am teaching a topics class on the polynomial method. See also the Open Courseware website for the course Polynomial methods in combinatorics on OCW. I wrote a book developing the notes for this course, which is available through the AMS here Polynomial methods in combinatorics .

Seminars.

In Fall 2014, I am teaching a mini-course / seminar on decoupling and multilinear estimates in harmonic analysis.

Here is a short webpage related to the seminar: Decoupling seminar .

Research.

My research interests are in metric geometry, combinatorial geometry, and harmonic analysis.

Metric geometry refers to estimates about lengths, areas, volumes, distances, etc. It includes

the study of isoperimetric inequalities and systolic inequalities. I'm particularly interested in

quantitative geometric estimates that are related to topology. In combinatorial geometry, I'm

interested in estimates about the intersection patterns of simple shapes like lines and circles.

This area includes the Szemeredi-Trotter theorem, and the unit distance problem. In harmonic

analysis, I'm interested in Kakeya-type inequalities about how simple shapes like cylinders can

overlap in Euclidean space, and in related estimates in Fourier analysis or PDE.

My papers are available on the arxiv here.

Expository writing.

Metric geometry

Metaphors in systolic geometry This is a survey paper on systolic geometry that I gave at the 2010 ICM. There's also a slightly longer version which you can read here.

The waist inequality in Gromov's work This is a short expository essay that I contributed to a survey about Gromov's work in an Abel prize volume. It discusses some of his recent work. The waist inequality is a fundamental geometric inequality that I think deserves to be more popular.

Recent progress in quantitative topology This paper discusses the problem of understanding the most efficient way of contracting a contractible map, where efficiency is measured using the Lipschitz constant of a map. It describes a recent breakthrough by Chambers, Dotterrer, Ferry, Manin, and Weinberger.

Polynomial method

Unexpected applications of polynomials in combinatorics This is an expository essay about recent applications of the `polynomial method' in combinatorics. I wrote it as a contribution to the new edition of the book The Mathematics of Paul Erdos, edited by R. Graham and J. Nesetril.

Namboodiri Lectures. I am giving the 2015 Namboodiri lectures at the University of Chicago on ``Polynomial methods in combinatorics and Fourier analysis''. Here are notes for the lectures. There are separate files with the figures for each lectures.

Lecture 1: Introduction to the polynomial method and incidence geometry. Figures for Lecture 1.

Lecture 2: The distinct distance problem . Figures for Lecture 2.

Lecture 3: Incidence geometry and polynomials in Fourier analysis. Figures for Lecture 3.