Mathematics

A classical problem in combinatorial geometry, posed by Erdös in 1946, asks to determine the maximum number of unit segments in a set of n points in the plane. Since then a great variety of extremal problems in finite point sets have been studied. Here, we look at generalizations of this question concerning regular simplices. Among others we answer the following question asked by Erdös: Given n points in R6, how many triangles can be equilateral triangles? For our proofs we use hypergraph Turán theory. This is joint work with Dumitrescu and Liu.

As an academic mathematician with a few decades of experience working with industry, the speaker has encountered many challenging problems that required the knowledge and development of a diverse collection of mathematical tools to effectively meet these challenges. This talk will present the mathematics arising in these collaborations, discussing both some technical details and why these skills might be useful to you as a young mathematician interested in an industrial career.

The talk will include work in the oil and gas sector (mathematics of imaging, partial differential equations, inverse problems, numerical methods), psychology and acoustics (Fourier transforms, digital signal processing), smart buildings (mathematical modeling and data science) and K-12 math education (mathematical visualizations and more data science).

There will be a few videos and animations to lighten up the gory technical details!

Fix a positive integer $X$ and multi-sets of complex numbers $\mathcal{I}$ and $\mathcal{J}$. We study the shifted convolution sum \[ D_{\mathcal{I},\mathcal{J}}(X,1) = \sum_{n\leq X} \tau_{\mathcal{I}}(n)\tau_{\mathcal{J}}(n+1), \] where $\tau_{\mathcal{I}}$ and $\tau_{\mathcal{J}}$ are shifted divisor functions. These sums naturally appear in the study of higher moments of the Riemann zeta function and additive problems in number theory. We review known results on $2k$-th moment of the Riemann zeta function and correlation sums associated with generalized divisor function. Assuming a conjectural bound on the averaged level of distribution of $\tau_{\mathcal{J}}(n)$ in arithmetic progressions, we present an asymptotic formula for $D_{\mathcal{I},\mathcal{J}}(X,1)$ with explicit main terms and power-saving error estimates.

We use a U-Net to make baseline power forecasts and train a diffusion model on its residuals to capture uncertainty. The diffusion samples naturally show low ensemble spread during stable atmospheric conditions and much wider spread when the atmosphere is more turbulent. This improves both reliability and interpretability compared to using the U-Net alone. The method provides a practical alternative to running full WRF simulations for uncertainty-aware wind farm power modelling.

Industrial mathematics is a field that spans a broad spectrum of activity ranging from applied R&D performed by mathematicians employed in industry, to purely academic research projects undertaken by university mathematics professors. In this talk, I will survey several research projects I have been involved with that fall under the heading of what I'll call "mathematics *for* industry", which relates specifically to direct collaborations between university mathematicians and non-academic partner organizations. These projects encompass a diverse collection of mathematical techniques (ranging from simple algebra to partial differential equations, finite volume methods, inverse problems and homogenization theory) as well as applications from many scientific disciplines (such as fluid mechanics, image processing, atmospheric science and plant biology). In the process, I will attempt to characterize the job of an industrial mathematician and to identify the qualities and skills that are most desirable for anyone interested in making significant contributions to research at the interface between university and industry. I also hope to convince you that industrial collaborations can be a rich source of challenging and novel mathematical problems for academic mathematicians.