The Oxford-Princeton Workshops on Financial Mathematics & Stochastic Analysis have been held approximately every eighteen months since 2002, alternately in Princeton and Oxford. They bring together leading groups of researchers in, primarily, mathematical and computational finance from Oxford University and Princeton University to collaborate and interact. The series is organized by the Oxford Mathematical and Computational Finance Group, and at Princeton by the Department of Operations Research and Financial Engineering and the Bendheim Center for Finance.

The classical Dirac operator is part of an osp(1|2) realisation inside the Weyl-Clifford algebra which is Pin-invariant. This leads to a multiplicity-free decomposition of the space of spinor-valued polynomials in irreducible modules for this Howe dual pair. In this talk we review an abstract generalisation A of the Weyl algebra that retains a realisation of osp(1|2) and we determine its centraliser algebra explicitly. For the special case where A is a rational Cherednik algebra, the centralizer algebra provides a refinement of the previous decomposition whose analogue was no longer irreducible in general. As an example, for the  group S3 in specific, we will examine the finite-dimensional irreducible modules of the centraliser algebra.

Fermat’s two-squares theorem is an elementary theorem in number theory that readily lends itself to a classification of the positive integers representable as the sum of two squares. Given this, a natural question is: what is the minimal number of squares needed to represent any given (positive) integer? One proof of Fermat’s result depends on essentially a buffed pigeonhole principle in the form of Minkowski’s Convex Body Theorem, and this idea can be used in a nearly identical fashion to provide 4 as an upper bound to the aforementioned question (this is Lagrange’s four-square theorem). The question of identifying the integers representable as the sum of three squares turns out to be substantially harder, however leaning on a powerful theorem of Dirichlet and a handful of tricks we can use Minkowski’s CBT to settle this final piece as well (this is Legendre’s three-square theorem).

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November).

All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November)

All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.