Wednesday 2 November, 5 pm, Mathematical Institute, Oxford
A calculator processes numbers without caring that these numbers refer to items in our shopping, or the calculations involved in designing an airplane. Number without context is a remarkable abstraction that we learn as infants and which has profoundly affected our world.
Consider an environment with two vehicles/platforms moving at a relative velocity (v). The objective is to predict the Closest Point of Approach (CPA) between the two platforms as defined by the parameters: CPA time (t0), CPA bearing (θ0), CPA distance (r0)[†].The challenge is to identify mathematical operations - either using geometric methods, or by use of tracking algorithms such as Kalman Filters (EKF, UKF), or a combination of both - to estimate the CPA parameters. The statistical errors in estimation of CPA parameters also need to be quantified with each observations at time ti. The signals to be employed are acoustic in nature and the receiver platform has one sensor. The parameters that can extracted from acoustic signals are current relative bearing (θ) and current doppler or range rate (S)
[†]Defined currently using polar coordinate system.
According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's $K$-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the $K$-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.
