Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects.

I will give an overview of the Nielsen-Thurston theory of the mapping class group and its connection to hyperbolic geometry and dynamics. Time permitting, I will discuss the surface entropy conjecture and a theorem of Hamenstadt on entropies of `generic' elements of the mapping class group. No prior knowledge of the concepts involved is required.

Since differentiation generally lowers exponents, it is straightforward that the space of Laurent polynomials $\mathbb{C}[x, x^{-1}]$ is a finitely generated module over the ring of differential operators $\mathbb{C}[x, \mathrm{d}/\mathrm{d}x]$. This innocent looking fact has been vastly generalized to a statement about holonomic D-modules, using the beautiful theory of b-functions (or Bernstein—Sato polynomials). I will give an overview of the classical theory before discussing some recent developments concerning a $p$-adic analytic analogue, which is joint work with Thomas Bitoun.

The Strominger-Yau-Zaslow (SYZ) conjecture postulates that mirror dual Calabi-Yau manifolds carry dual special Lagrangian fibrations. Within the study of Mirror Symmetry the SYZ conjecture has provided a particularly fruitful point of convergence of ideas from Riemannian, Symplectic, Tropical, and Algebraic geometry over the last twenty years. I will attempt to provide a brief overview of this aspect of Mirror Symmetry.

Polar classes are very classical objects in Algebraic Geometry. A brief introduction to the subject will be presented and ideas and preliminarily results towards generalisations will be explained. These ideas can be applied towards variety sampling and relevant applications. 
 

Massless Quantum Field Theories can be described perturbatively by chiral worldsheet models - the so-called Ambitwistor Strings. In contrast to conventional string theory, where loop amplitudes are calculated from higher genus Riemann surfaces, loop amplitudes in the ambitwistor string localise on the non-separating boundary of the moduli space. I will describe the resulting framework for QFT amplitudes from (nodal) Riemann spheres, building up from tree-level to two-loop amplitudes.
 

After considering motivations in symplectic geometry, I’ll give a summary of $C^\infty$-Algebraic Geometry and how to extend these concepts to manifolds with corners.