L6

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Non-shearing congruences of null geodesics on four-dimensional Lorentzian manifolds are fundamental objects of mathematical relativity. Their prominence in exact solutions to the Einstein field equations is supported by major results such as the Robinson, Goldberg-Sachs and Kerr theorems. Conceptually, they lie at the crossroad between Lorentzian conformal geometry and Cauchy-Riemann geometry, and are one of the original ingredients of twistor theory.
 
Identified as involutive totally null complex distributions of maximal rank, such congruences generalise to any even dimensions, under the name of Robinson structures. Nurowski and Trautman aptly described them as Lorentzian analogues of Hermitian structures. In this talk, I will give a survey of old and new results in the field.

In this talk I will discuss my extension of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the more general setting of quasi-abelian categories. In particular, I will define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders' embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of derived categories for certain categories of modules over Koszul monoids and their duals. The key examples of categories for which this theory works are the categories of complete bornological spaces and the categories of inductive limits of Banach spaces. These categories frequently appear in derived analytic geometry.

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

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

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