Tue, 18 Feb 2025
14:00
L6

On a geometric dimension growth conjecture

Yotam Hendel
(Ben Gurion University of the Negev)
Abstract

Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown, and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X. 

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally has the structure of an algebraic variety, which we denote by X(B). In ongoing work with Tijs Buggenhout and Floris Vermeulen, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components  of X(B) of largest possible dimension​ analogously to dimension growth bounds. We do this by developing a geometric determinant method, and by using results on rational points on curves over function fields. 

Joint with Tijs Buggenhout and Floris Vermeulen.

Thu, 20 May 2021
11:30
Virtual

Chromatic numbers of Stable Graphs

Yatir Halevi
(Ben Gurion University of the Negev)
Abstract
This is joint work with Itay Kaplan and Saharon Shelah.
Given a graph $(G,E)$, its chromatic number is the smallest cardinal $\kappa$ of a legal coloring of the vertices. We will mainly concentrate on the following strong form of Taylor's conjecture:
If $G$ is an infinite graph with chromatic number$\geq \aleph_1$ then it contains all finite subgraphs of $Sh_n(\omega)$ for some $n$, where $Sh_n(\omega)$ is the $n$-shift graph (which we will introduce).
 
The conjecture was disproved by Hajnal-Komjath. However, we will sketch a proof for a variant of this conjecture for $\omega$-stable\superstable\stable graphs. The proof uses a generalization of  Ehrenfeucht-Mostowski models, which we will (hopefully) introduce.
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