Seminars and Colloquia by Series

A Roth type result for dense subsets of the integer lattice

Series
Additional Talks and Lectures
Time
Friday, October 11, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Akos Magyar – The University of Georgia

 Let A be a subset of the integer lattice of positive upper density. Roth' theorem in this setting states that there are points x,x+y,x+2y in A with the length of the gap y arbitrary large. We show that the lengths of the gaps y contain an infinite arithmetic progression, as long as one measures the length in lp for p>2 even, while this not true for the Euclidean distance.

 

Such results have been previously obtained in the continuous settings for measurable subsets of Euclidean spaces using methods of time-frequency analysis, as opposed our approach is based on some ideas from additive combinatorics such as uniformity norms and arithmetic regularity lemmas. If time permits, we discuss some other results that can be obtained similarly.

The Heilbronn triangle problem

Series
Additional Talks and Lectures
Time
Friday, September 20, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cosmin Pohoata – Emory University

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and projection theory. In this talk, we will give an overview of some of these connections, and discuss some recent developments. Based on joint work with Alex Cohen and Dmitrii Zakharov.

Improved Bounds for Szemerédi’s Theorem

Series
Additional Talks and Lectures
Time
Monday, April 29, 2024 - 17:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mehtaab Sawhney – MIT –

We discuss recent improved bounds for Szemerédi’s Theorem. The talk will seek to provide a gentle introduction to higher order Fourier analysis and recent quantitative developments. In particular, the talk will provide a high level sketch for how the inverse theorem for the Gowers norm enters the picture and the starting points for the proof of the inverse theorem. Additionally, the talk (time permitting) will discuss how recent work of Leng on equidistribution of nilsequences enters the picture and is used. No background regarding nilsequences will be assumed. 

Based on joint work with James Leng and Ashwin Sah.

Sum-Product with few primes

Series
Additional Talks and Lectures
Time
Monday, November 27, 2023 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Brandon Hanson – University of Maine

This talk concerns improving sum-product exponents for sets  of integers under the condition that each element of  has no more than  prime factors. The argument combines combinatorics, harmonic analysis and number theory.

Effective bounds for Roth's theorem with shifted square common difference

Series
Additional Talks and Lectures
Time
Monday, November 13, 2023 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Ashwin Sah – MIT

Let $S$ be a subset of $1 ,…, N$ avoiding the nontrivial progressions $x, x+y^2-1, x+2(y^2-1)$. We prove that $|S| < N/\log \log \cdots \log(N)$, where we have a fixed constant number of logarithms. This answers a question of Green, and is the first effective polynomial Szemerédi result over the integers where the polynomials involved are not homogeneous of the same degree and the underlying pattern has linear complexity. Joint work with Sarah Peluse and Mehtaab Sawhney.

 

Strong Bounds for 3-Progressions

Series
Additional Talks and Lectures
Time
Monday, October 16, 2023 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Zander Kelley – University of Illinois Urbana-Champaign –

Suppose you have a set $A$ of integers from $\{1, 2, …, N\}$ that contains at least $N / C$ elements.

Then for large enough $N$, must $A$ contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed that this is indeed the case when $C \approx \log \log N$, while Behrend in 1946 showed that $C$ can be at most $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.

Since then, the problem has been a cornerstone of the area of additive combinatorics.

Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on $C$ to $C = (\log N)^{1 + c}$, for some constant $c > 0$.

This talk will describe a new work which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.

Based on a joint work with Raghu Meka.

(Joint Combinatorics and Geometry Topology seminar) Combinatorics of Surface Deformations

Series
Additional Talks and Lectures
Time
Thursday, January 26, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Satyan Devadoss – Williams college –
We consider the moduli space of surfaces with boundary and marked points. Such spaces appear in algebraic geometry and topology, playing a strong role in holomorphic curves and open-closed string theory. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. This leads to a classification of all such spaces that can be realized as polytopes, capturing elegant hidden algebraic structure from homotopy theory. This talk is accessible to strong undergraduates, based heavily on pictures and concrete examples.