Talking maths on YouTube is a lot of fun. Your audience will contain maths enthusiasts, young people, and the general public. These are people who are interested in what you have to say, and want to learn something new. Maths videos on YouTube can be used to teach maths, or to just show people something interesting. Making videos doesn't have to be technically difficult, but is good practice in explaining difficult concepts in clear and succinct ways. In this session we will discuss how to make your first YouTube video, including questions about content, presentation and video making.

Dr James Grime started making his first maths YouTube videos while working as a postdoc in 2008. James has made maths videos with Cambridge University, the Royal Institution, and MathsWorldUK, and is also a presenter on the popular YouTube channel Numberphile, which now has over 3 million subscribers worldwide.

We propose Swampland constraints on consistent 5d N=1 supergravity theories. In particular, we focus on a special class of BPS monopole strings which arise only in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be computed using the anomaly inflow mechanism and provide constraints for the 5d supergravity using unitarity of the worldsheet CFT. In M-theory, where these theories can be realised by compactification on Calabi-Yau threefolds, the special monopole strings arise from M5 branes wrapping “semi-ample” 4-cycles in the threefolds. We further identify necessary geometric conditions that such cycles need to satisfy and translate them into constraints for the low-energy gravity theory.

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality (CoD) in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an  approximation precision $\varepsilon >0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed  and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the CoD in the sense that  the number of real parameters used to describe the approximating DNNs  grows at most polynomially in both the PDE dimension $d \in  \N$ and the reciprocal of the prescribed approximation accuracy $\varepsilon >0$. There are now also a few rigorous mathematical results in the scientific literature which  substantiate this conjecture by proving that  DNNs overcome the CoD in approximating solutions of PDEs.  Each of these results establishes that DNNs overcome the CoD in approximating suitable PDE solutions  at a fixed time point $T >0$ and on a compact cube $[a, b]^d$ but none of these results provides an answer to the question whether the entire PDE solution on $[0, T] \times [a, b]^d$ can be approximated by DNNs without the CoD. 
In this talk we show that for every $a \in \R$, $ b \in (a, \infty)$ solutions of  suitable  Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0, T] \times [a, b]^d$ without the CoD.