Seminars in Discrete Geometry

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Zeev Dvir: High-Dimensional Variants of the Finite Field Kakeya Problem

23rd October, 2023

The finite field Kakeya problem asks about the size of the smallest set in (𝔽q)n containing a line in every direction. Raised by Wolff in 1999 as a 'toy' version of the Euclidean Kakeya conjecture, this problem is now completely resolved using the polynomial method. In this talk I will describe recent progress on its higher-dimensional variant in which lines are replaced with k-dimensional flats. It turns out that, unlike in the 1-dimensional case, when k ≥ 2, one can prove that there are no 'interesting' constructions (with size smaller than trivial) even if one asks for sets that only have large intersection with a flat in every direction. This theorem turns out to have surprising applications in questions involving lattice coverings and linear hash functions.

Alex Iosevich: Some number-theoretic aspects of finite point configurations

29th June, 2023

We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centred around the Erdős/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area.

Amir Shpilka: Points, lines and polynomial identities

29th March, 2023

The Sylvester-Gallai (SG) theorem in discrete geometry asserts that if a finite set of points P has the property that every line through any two of its points intersects the set at a third point, then P must lie on a line. Surprisingly, this theorem, and some variants of it, appear in the analysis of locally correctable codes and, more noticeably, in polynomial identity testing. For these questions one often has to study extensions of the original SG problem: the case where there are several sets, or with a robust version of the condition (many "special" lines through each point) or with a higher degree analogue of the problem, etc.

In this talk I will present the SG theorem and some of its variants, show its relation to the above-mentioned problems and discuss recent developments regarding higher degree analogues and their applications.

Artem Chernikov: Recognizing Groups in Erdős Geometry and Model Theory

28th March, 2023

Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups.  Techniques from model theory - Hrushovski's group configuration and its variants - are very useful in recognizing these groups, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. In fact, all of this is not just about polynomials and works in the larger setting of definable sets in o-minimal structures.

Bjorn Poonen: Tetrahedra with rational dihedral angles

6th August, 2020

In 1895, Hill discovered a 1-parameter family of tetrahedra whose dihedral angles are all rational multiples of π. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity. Many previous authors have solved polynomial equations in roots of unity, but never with more than 12 monomials, and the Conway-Jones polynomial has 105 monomials! I will explain the method we use to solve it and our discovery that the full classification consists of two 1-parameter families and an explicit finite list of sporadic tetrahedra. Building on this work, we classify all configurations of vectors in ℝ3 such that the angle between each pair is a rational multiple of π. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than nine vectors is contained in a particular 15-vector configuration.