# Scientific

## Multivariate forecasting in energy systems with a large share of renewables

Forecasts of renewable power production and electricity demand for multiple time periods and/or spatial expanses are required to operate modern power systems. Furthermore, probabilistic forecasts are necessary to facilitate economic decision-making and risk management. This gives rise to the challenge of producing forecasts that capture dependency between variables, over time, and between multiple locations. The Gaussian Copula has been widely used for multivariate energy forecasting, including for wind power, and is readily scalable given that the entire dependency structure is described by a single covariance matrix; however, estimating this covariance matrix in high dimensional problems remains a research challenge. Furthermore, it has been found empirically that this covariance matrix is often non-stationary and evolves over time. Two methods are presented for parameterising covariance matrices to enable conditioning on explanatory variables and as a step towards more robust estimation.

We consider two approaches, one based on modelling the parameters of covariance functions using additive models, and the second modelling individual elements of the modified Cholesky decomposition, again using additive models. We show how this gives rise to a wide range of possible parametric structures and discuss model selection and estimation strategies. Finally, we demonstrate though two case studies the improvement in forecast quality that these methods yield, and the importance and value of capturing the dynamics of dependency structures in wind power forecasting and net-load forecasting in the presence of embedded renewables.

## A Concise Overview on State-of-the-Art Solar Resources and Forecasting

The ability to forecast solar irradiance plays an indispensable role in solar power forecasting, which constitutes an essential step in planning and operating power systems under high penetration of solar power generation. Since solar radiation is an atmospheric process, solar irradiance forecasting, and thus solar power forecasting, can benefit from the participation of atmospheric scientists. In this talk, the two fields, namely, atmospheric science and power system engineering are jointly discussed with respect to how solar forecasting plays a part. Firstly, the state of affairs in solar forecasting is elaborated; some common misconceptions are clarified; and salient features of solar irradiance are explained. Next, five technical aspects of solar forecasting: (1) base forecasting methods, (2) post-processing, (3) irradiance-to-power conversion, (4) verification, and (5) grid-side implications, are reviewed. Following that, ten research topics moving into the future are enumerated; they are related to (1) data and tools, (2) numerical weather prediction, (3) forecast downscaling, (4) large eddy simulation, (5) dimming and brightening, (6) aerosols, (7) spatial forecast verification, (8) multivariate probabilistic forecast verification, (9) predictability, and (10) extreme weather events. Last but not least, a pathway towards ultra-high PV penetration is laid out, based on a recently proposed concept of firm generation and forecasting.

## Free boundary problems in optimal transportation

In this talk, we introduce some recent regularity results of free boundary in

optimal transportation. Particularly for higher order regularity, when

densities are Hölder continuous and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$ smooth. We also consider another mode

case that the target consists of two disjoint convex sets, in which

singularities of optimal transport mapping arise. Under similar assumptions,

we show that the singular set of the optimal mapping is an $(n-1)$-dimensional

$C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a

series of joint work with Shibing Chen and Xu-Jia Wang.

## A Nonsmooth Approach to Einstein's Theory of Gravity

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under curvature and dimension bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We highlight how the null energy condition of Penrose admits a nonsmooth formulation as a variable lower bound on timelike Ricci curvature.

## Paths and Pathways

We talk about how some simple sounding problems about straight line paths on surfaces require many different kinds of mathematical thinking to solve, focusing on the example of understanding straight line paths on Platonic solids. We'll use this to start a discussion of how we can emphasize teaching different ways of thinking, and why geometry is an important resource for students. There will be lots and lots of fun pictures and hopefully interesting and provocative ideas!

The following resources are referenced during this talk:

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## Linearised Optimal Transport Distances

Optimal transport is a powerful tool for measuring the distances between signals and images. A common choice is to use the Wasserstein distance where one is required to treat the signal as a probability measure. This places restrictive conditions on the signals and although ad-hoc renormalisation can be applied to sets of unnormalised measures this can often dampen features of the signal. The second disadvantage is that despite recent advances, computing optimal transport distances for large sets is still difficult. In this talk I will extend the linearisation of optimal transport distances to the Hellinger–Kantorovich distance, which can be applied between any pair of non-negative measures, and the TLp distance, a version of optimal transport applicable to functions. Linearisation provides an embedding into a Euclidean space where the Euclidean distance in the embedded space is approximately the optimal transport distance in the original space. This method, in particular, allows for the application of off-the-shelf data analysis tools such as principal component analysis as well as reducing the number of optimal transport calculations from $O(n^2)$ to $O(n)$ in a data set of size n.

## Multivariate Symmetry: Distribution-Free Testing via Optimal Transport

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures — based on signs, (absolute) ranks and signed-ranks — yield distribution-free tests for symmetry in one-dimension. However, multivariate distribution-free generalizations of these tests are not known in the literature. In this talk we propose a novel framework — based on the theory of optimal transport — which leads to distribution-free generalized multivariate signs, ranks and signed-ranks, and, as a consequence to analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples, with an asymptotic normal distribution, and adapt to various notions of multivariate symmetry such as central symmetry, sign symmetry, and spherical symmetry. We study the consistency of the proposed tests and their behaviors under local alternatives, and show that the proposed generalized Wilcoxon signed-rank test is particularly powerful against location shift alternatives. We show that in a large class of models, our generalized Wilcoxon signed-rank test suffers from no loss in (asymptotic) efficiency, when compared to the Hotelling’s T^2 test, despite being nonparametric and exactly distribution-free. These ideas can also be used to construct distribution-free confidence sets for the location parameter for multivariate distributions.

This is joint work with Zhen Huang at Columbia University.

## Discussion on the history of the Prime Number Theorem and bounds of $\psi(x)$

In this talk I will be discussing the history of the Prime Number Theorem following the works of Legendre, Gauss, Riemann, Hadamard, and de la Vallée Poussin, followed by a survey on explicit bounds for $\psi(x)$ beginning with the work of Rosser in 1941. I will go over various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, and Büthe. I will finally briefly discuss my work on the survey of a paper by Büthe and its significance.

## On the quality of the ABC-solutions

Let the triple $(a, b, c)$ of integers be such that $\gcd{\left(a, b, c\right)} = 1$ and $a + b = c$. We call such a triple an ABC-solution. The quality of an ABC-solution $(a, b, c)$ is defined as

$$

q(a, b, c) = \frac{\max\left\{\log |a|, \log |b|, \log |c|\right\}}{\log \operatorname{rad}\left({|abc|}\right)}

$$

where $\operatorname{rad} (|abc|)$ is the product of distinct prime factors of $|abc|$. The ABC-conjecture states that given $\epsilon >0$ the number of the ABC-solutions $(a, b, c)$ with $q(a, b, c)\geq 1 + \epsilon$ is finite. In the first part of this talk, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of ABC-solutions that has quality $> 1$. In the remainder of the talk, we provide a result on a conjecture of Erdõs on the solutions of the Brocard-Ramanujan equation

$$

n! + 1 = m^2

$$

by assuming an explicit version of the ABC-conjecture proposed by Baker.

## Twelve on the twelfth

The under representation of women, especially women of color, has been persistently well documented (see for example the data dashboard on www.womendomath.org). One reason that this is a problem is that it can be difficult for women to identify role models - this can in turn make it harder for women to envision their own success. During my career, I found it very helpful to learn the stories of women in STEM and to draw on aspects of their success to try to invent my own path. In this talk, I will retell twelve stories of women in STEM that influenced me. I can’t promise that the stories will be historically accurate, but I will try to say what I learned from the stories as I heard them and what lessons I hope others might take from them as well.

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