Modern cellular radio systems such as 4G and 5G use antennas with multiple elements, a technique known as MIMO, and the intention is to increase the capacity of the radio channel.  5G allows even more possibilities, such as massive MIMO, where there can be hundreds of elements in the transmit antenna, and beam-forming (or beam-steering), where the phase of the signals fed to the antenna elements is adjusted to focus the signal energy in the direction of the receivers.  However, this technology poses some difficult optimization problems, and here mathematicians can contribute.   In this talk I will explain the background, and then look at questions such as: what is an appropriate objective function?; what constraints are there?; are any problems of this type convex (or quasi-convex, or difference-of-convex)?; and, can big problems of this type be solved in real time?

Abstract: 

In this talk I will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill posed SDEs. 

In a recent work together with   Nicolas Perkowski we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE. 

As an example, we show that a certain type of Gaussian processes has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. I will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​

We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).

We discuss hyperkahler implosion spaces, their relevance to group actions and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.
 

I will talk about a novel idea on the Swampland program that uses consistency of what lives on the string probes in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the supergravity theories based on unitarity of the worldsheet CFT. I will show some of the theories with 8 or 16 supersymmetries, which are otherwise consistent looking, belong to the Swampland.