Please Note: We will start with a presentation by Daniel Hwang and Juliet Whidden and continue with a free discussion.
We will start with a presentation by Elanor Moore and continue with a free discussion.
In this talk, I will show how tools from VC-dimension theory can be used to address questions from incidence geometry, enumerating intersection graphs, and graph drawing.
We will start with a presentation by Griffin Edwards and continue with a free discussion.
We will start with a 15-minute presentation by Noah Solomon and continue with a free discussion.
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 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.
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.
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.
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.
