This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.

The class of $t$-perfect graphs consists of graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. These were first studied by Chvátal in 1975, motivated by the related and well-studied class of perfect graphs. While perfect graphs are easy to colour, the same is not true for $t$-perfect graphs; numerous questions and conjectures have been posed, and even the most basic, on whether there exists some $k$ such that every $t$-perfect graph is $k$-colourable, has remained open since 1994. I will talk about joint work with Maria Chudnovsky, Linda Cook, James Davies, and Sang-il Oum in which we establish the first finite bound and show that a little less than 200 000 colours suffice.

In string theory, elementary particles correspond to the various oscillation modes of fundamental strings, whose dynamics in spacetime is described by a two-dimensional conformal field theory on the worldsheet of the propagating strings. While the theory enjoys several desirable features - UV-finiteness, the presence of the graviton in the closed-string spectrum, a pathway to unification - several aspects remain elusive or unsatisfactory, including the on-shell and perturbative nature of string scattering amplitudes and the presence of infrared divergences. String field theory - the formulation of string theory as a quantum field theory - provides a unique and complete framework for describing string dynamics, allowing for example to compute off-shell amplitudes and non-perturbative contributions, to regulate infrared divergences and to approach background independence. This talk will be concerned with the construction of string field theories. Following a brief review of string theory, I will introduce the string fields, and discuss the construction of a string field action and the associated Feynman diagrams. Finally, I will mention some applications before concluding.

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.