During growth and development, tissue dynamics, such as tissue folding, cell intercalations and oriented cell divisions, are critical for shaping tissues and organs. However, less is known about how tissues regulate their dynamics during tissue homeostasis and repair, to maintain their shape after development. In this talk, we will discuss how differential growth rates can generate precise folds in tissues. We will also discuss how tissues respond to mechanical perturbations, such as stretching or wounding, by altering their actomyosin contractile structures, to change tissue dynamics, and thus preserve tissue shape and patterning. We combine genetics, biophysics and computational modelling to study these processes.

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

Last summer, there was a lot of activity regarding an old conjecture of van der Waerden, culminating in its solution by Bhargava, and including joint work by Sam Chow and myself on which I want to report in this talk: We showed that the number of irreducible monic integer polynomials of degree n, with coefficients in absolute value bounded by H, which have Galois group different from S_n and A_n, is of order of magnitude O(H^{n-1.017}), providing that n is at least 3 and is different from 7,8,10. Apart from the alternating group and excluding degrees 7,8,10, this establishes the aforementioned conjecture to the effect that irreducible non-S_n polynomials are significantly less frequent than reducible polynomials.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

For graphs $G$ and $H$, we say $G \stackrel{r}{\to} H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G \stackrel{r}{\to} H;t)$ is the minimum $n$ such that $G[n] \stackrel{r}{\to} H[t]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G \stackrel{r}{\to} H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t \in \mathbb{N}$, $B(G \stackrel{r}{\to} H;t) \leq ab^t$. They conjectured that there exist some graphs $H$ for which the constant $a$ depending on $G$ is necessary. We prove this conjecture by showing that the statement is true when $H$ is a $3$-chromatically connected, which includes all cliques on $3$ or more vertices. We are also able to show perhaps surprisingly that for any forest $F$ there is $f(F,t)$ such that  for any $G \stackrel{r}{\to} H$, $B(G \stackrel{r}{\to} H;t)\leq f(F,t)$ i.e. the function does not depend on the ground graph $G$. This is joint work with Robert Hancock.