L4

How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.

Given a set $A$ and a scalar $\lambda$, how large must the sum of dilate $A+\lambda\cdot A=\{a+\lambda a'\mid a,a'\in A\}$ be in terms of $|A|$? In this talk, we will discuss two different settings of this problem, and how they relate to each other.

• For transcendental $\lambda\in \mathbb{C}$ and $A\subset \mathbb{C}$, how does $|A+\lambda\cdot A|$ grow with $|A|$?
• For a fixed large $\lambda\in \mathbb{Z}$ and even larger prime $p$, with $A\subset \mathbb{Z}/p\mathbb{Z}$, how does the density of $A+\lambda\cdot A$ depend on the density of $A$?

Joint with David Conlon.

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We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

Even in a totally discrete space $X$ you need to know how to move between distinct points. A path $P_{x,y}$ between two points $x,y \in X$ is a sequence of points in $X$ that starts with $x$ and ends with $y$. A path system is a collection of paths $P_{x,y}$, one per each pair of distinct points $x, y$ in $X$. We restrict ourselves to the undirected case where $P_{y,x}$ is $P_{x,y}$ in reverse.

Strictly metrical path systems are ubiquitous. They are defined as follows: There is some spanning, connected graph $(X, E)$ with positive edge weights $w(e)$ for all $e\in E$ and $P_{x,y}$ is the unique $w$-shortest $xy$ path. A metrical path system is defined likewise, but $w$-shortest paths need not be unique. Even more generally, a path system is called consistent  (no $w$ is involved here) if it satisfies the condition that when point $z$ is in $P_{x,y}$, then $P_{x,y}$ is $P_{x,z}$ concatenated with $P_{z,y}$. These three categories of path systems are quite different from each other and in our work we find quantitative ways to capture these differences.

Joint work with Daniel Cizma.

The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.

 This is joint work with Renato Velozo Ruiz (Imperial College London).