The Balog-Szemerédi-Gowers theorem is a powerful tool in additive combinatorics, that allows one to roughly convert any “large energy” estimate into a “small sumset” estimate. This has found applications in a lot of results in additive combinatorics and other areas. In this talk, we will provide a friendly introduction and overview of this result, and then discuss some proof ideas. No hardcore additive combinatorics pre-requisites will be assumed.

Monumo is interested in computing physical properties of electric motors (torque, efficiency, back EMF) from their designs (shapes, materials, currents). This involves solving Maxwell's equations (non-linear PDEs). They currently compute the magnetic flux, and then use that to compute the other properties of interest. The main challenge they face is that they want to do this for many, many different designs. There seems to be lots of redundancy here, but exploiting it has proved difficult.

As one of the largest retailers in the world, Tesco relies on automated forecasting to help with decision making. A common issue with forecasts is that of the cold start problem; that we must make forecasts for new products that have no history to learn from. Lack of historical data becomes a real problem as it prevents us from knowing how products react to events, and if their sales react to the time of year. We might consider using similar products as a way to produce a starting forecast, but how should we define what ‘similar’ means, and how should we evolve this model as we start getting real live data? We’ll present some examples to hopefully start a fruitful discussion.

I will present two new results. The first concerns minimal surfaces of the hyperbolic space and is a relation between their renormalised area (in the sense of Graham and Witten) and the length of their ideal boundary measured in different metrics of the conformal infinity. The second result concerns minimal submanifolds of the sphere and is a relation between their volume and antipodal-ness. Both results were obtained from the same framework, which involves new monotonicity theorems and a comparison principle for them. If time permits, I will discuss how to use these to answer questions about uniqueness and non-existence of minimal surfaces.