L4

Everything you have been taught about Turing patterns is wrong! (Well, not everything, but qualifying statements tend to weaken a punchy first sentence). Turing patterns are universally used to generate and understand patterns across a wide range of biological phenomena. They are wonderful to work with from a theoretical, simulation and application point of view. However, they have a paradoxical problem of being too easy to produce generally, whilst simultaneously being heavily dependent on the details. In this talk I demonstrate how to fix known problems such as small parameter regions and sensitivity, but then highlight a new set of issues that arise from usually overlooked issues, such as boundary conditions, initial conditions, and domain shape. Although we’ve been exploring Turing’s theory for longer than I’ve been alive, there’s still life in the old (spotty) dog yet.

Certain models of collective dynamics exhibit deceptively simple patterns that are surprisingly difficult to explain. These patterns often arise from phase transitions within the underlying dynamics. However, these phase transitions can be explained only when one derives continuum equations from the corresponding individual-based models. In this talk, I will explore this subtle yet rich phenomenon and discuss advances and open problems.

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.