How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.

 Hyper-Kähler manifolds are rigid geometric structures. They have three different symplectic and complex structures, in direct analogy with the quaternions. Being Ricci-flat, they solve the vacuum Einstein equations, and so there has been considerable interest among physicists to explicitly construct such spaces. We will discuss in detail the examples arising from the Gibbons-Hawking ansatz. These give concrete descriptions of the metric, giving many examples to work with. They also lead to the generalised classification as hyper-Kähler quotients by P.B. Kronheimer, with one such space for each finite subgroup of SU(2). Finally, we will look at the McKay correspondence, relating the finite subgroups of SU(2) with the simple Lie algebras of type A,D,E.

TBA

TBA

The path signature is a powerful tool for solving regression problems on path space, i.e., for computing conditional expectations $\mathbb{E}[Y | X]$ when the random variable $X$ is a stochastic process -- or a time-series. We provide new theoretical convergence guarantees for two different, complementary approaches to regression using signature methods. In the context of global regression, we show that linear functionals of the robust signature are universal in the $L^p$ sense in a wide class of examples. In addition, we present a local regression method based on signature semi-metrics, and show universality as well as rates of convergence. 

Based on joint works with Davit Gogolashvili, Luca Pelizzari, and John Schoenmakers.

Please note: The MCF seminar usually takes place on Thursdays from 16:00 to 17:00 in L5. However, for this week, the timing will be changed to 15:00 to 16:00.