# Mathematics

## Understanding arithmetic and geometry through cutting and pasting

Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker biography:

Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

## Vertex-transitive graphs with large automorphism groups

**Gabriel Verret (University of Auckland, New Zealand)**

Many results in algebraic graph theory can be viewed as upper bounds on the size of the automorphism group of graphs satisfying various hypotheses. These kinds of results have many applications. For example, Tutte's classical theorem on 3-valent arc-transitive graphs led to many other important results about these graphs, including enumeration, both of small order and in the asymptotical sense. This naturally leads to trying to understand barriers to this type of results, namely graphs with large automorphism groups. We will discuss this, especially in the context of vertex-transitive graphs of fixed valency. We will highlight the apparent dichotomy between graphs with automorphism group of polynomial (with respect to the order of the graph) size, versus ones with exponential size.

## Turing Patterns on Growing Domains

Turing patterns have been suggested as an explanation for morphogenesis in a variety of organisms. Despite the fact that morphogenesis occurs during growth, most studies of Turing patterns have been conducted on static domains. We present experimental and computational studies of Turing patterns in a chemical reaction-diffusion system on growing two-dimensional domains. We also investigate the effect of inert obstacles on pattern evolution. We find that the rate of domain growth significantly affects both how the patterns are laid down and their ultimate morphology.

## Existence, Stability and Slow Dynamics of Spikes in a 1D Minimal Keller–Segel Model with Logistic Growth

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller-Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity of the chemoattractant concentration field. In the limit, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state N-spike patterns, we analyze the spectral properties associated with both the “large” O(1) and the “small” o(1) eigenvalues associated with the linearization of the Keller-Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that N-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an N-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of cellular diffusion rate where the N-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an O(1) time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis of the KS model with logistic growth in the small chemical diffusion rate regime is rather closely related to the analysis of spike patterns for the Gierer-Meinhardt RD system.

This paper is a joint work with Professor Michael J. Ward and Juncheng Wei.

## PIMS/FACTS Panel: Tackling Climate Change and the Just Transition to Renewable Energy

This panel was organized as part of the PIMS Workshop on Forecasting and Mathematical Modeling for Renewable Energy and Public Panel Discussion on Climate Change. Ahead of the panel, the panelists were invited to give a short talk on their areas of expertise.

- Réne Aïd: More electricity demand response for less carbon emissions
- Gaël Giraud: Macroeconomics and Climate
- Seth Klein: Mobilizing Canada for the Climate Emergency
- Judith Sayers: First Nation Leadership in Clean Energy and Climate Action
- Andrew Weaver: Privilege, agency, and the climate scientist’s role in the global warming debate

## The Canadian regional climate model

High resolution climate simulations (horizontal resolutions of 25km or smaller) are a desired product for policy makers, the public, and renewable energy applications. Since running climate models at high resolution is not feasible, dynamical and statistical downscaling methods are applied. The latest version of the Canadian regional climate model (CanRCM5) is specifically designed to dynamically downscale future climate projections of its parent Canadian Earth System Model (CanESM5). The close relationship between these two Canadian models allows for improved RCM driving relative to independent RCM modelling centres, as all required prognostic variables are available on CanRCM5's lateral boundaries from its parent global model. Coupling different scale size models is challenging as regional climate models can easily develop their own climate. To keep the features of CanRCM5 consistent with CanESM5, the large-scale dynamics of CanRCM5 are typically nudged towards its parent model. We present a framework that identifies appreciable differences between the regional and global model and apply it to near-surface wind and precipitation fields showing that particularly the influence of better resolved topography yields substantial differences in the climate projections of CanRCM and CanESM. Finally, we discuss latest developments on research on bias correcting CanESM and CanRCM which allows for more accurate representations of the climate state. (Joint work with John Scinocca and Slava Kharin)

## Operational Numerical Weather Prediction (NWP) of Hub-height Winds for Mountainous British Columbia

Boundary-layer wind and turbulence-profile theories as described in most textbooks apply to flat prairies, not to the rugged terrain of British Columbia (BC). For the mountainous terrain of BC, different turbines at the same wind farm experience different winds and turbulence associated with their locations relative to small-scale (unresolved) terrain features. Convolution of the resulting wind-speed distribution with the wind-turbine power curve for an individual turbine yields a wind-farm power curve that differs from the theoretical power curve. For several wind farms in BC, the farm-average power curve does not achieve the cube of wind speed even between the cut-in and rated-power speeds.

To partially compensate for these wind variations, we run an ensemble of up to 51 NWP model runs each day, with fewer ensemble members covering the more distant wind farms. These runs are based on a variety of initial/boundary conditions (from gov’t centers in Canada, USA, France, Germany), a variety of model cores (WRF-ARW, WRF-NMM, MM5, MPAS), a variety of horizontal grid spacings, and a variety of physics parameterizations. Each forecast is individually bias corrected based on recent-past observations at any wind farm, and then the separate runs are combined to yield ensemble-average and calibrated-probabilistic forecasts.

UBC has been making operational limited-area NWP forecasts of hub-height winds for all the active wind farms in BC for the past decade. BC Hydro uses our ensemble forecasts to better manage the integration of wind power with their much-greater hydro-power generation. BC Hydro also uses our forecasts of Bonneville Power Administration (Columbia River region) wind-farm hub-height winds to optimize their energy-trading to the USA.

Based on our experience, we created a course ATSC 313 “Renewable Energy Meteorology”, which covers meteorology for hydro, wind, and solar power.

## Simulating Long-Distance Wind Farm Wake Propagation Using Numerical Weather Prediction Models

Growing evidence is demonstrating the startling reach of wind farm wakes, which can often exceed 100 kilometers under frequently occurring stable atmospheric conditions. These "long wakes" can significantly affect the annual energy yield of neighboring wind farms. Unfortunately, this impact is not accounted for in industry-standard wind resource assessment methods, leading to consistent overestimations of energy production and the suboptimal placement of new wind farms. As the global wind farm fleet continues to rapidly grow, the development of tools to assess these impacts is crucial prior to making any major investments in new project sites.

In this presentation, we will elucidate how numerical weather prediction (NWP) modeling is emerging as the preferred tool for simulating long-wake propagation. We will initially discuss the physical mechanisms underlying long-wake propagation, and provide observational evidence substantiating their occurrence. Subsequently, we will delve into the suitability of the Weather Research and Forecasting (WRF) NWP model for long-wake modeling, specifically its ability to accurately simulate diurnal and seasonal atmospheric stability fluctuations and its capacity to model wake propagation through its wind farm parameterization option. In conclusion, we will present encouraging validation results from five onshore wind farms in the U.S., reinforcing the potential and accuracy of the WRF model in tackling this growing issue.

## Turbulence, wakes and wind farm control

The dynamics of the atmospheric boundary layer (ABL) play a fundamental role in wind farm power production, governing the velocity field that enters the farm as well as the turbulent mixing that regenerates energy for extraction at downstream rows. Understanding the dynamic interactions between turbines, wind farms, and the ABL can therefore be beneficial in improving the efficiency of wind farm design and control approaches. This talk introduces a suite of models that exploit this knowledge to improve predictions of both static and dynamic conditions in the wind farm. We first introduce the area localized coupled (ALC) model, which couples the steady state solution of a dynamic wake model with a localized top-down model that focuses on the effect of the farm on the ABL. The ALC model improves the accuracy of power output and local velocity predictions over both conventional wake models and top-down models, while extending the applicability of this type of coupled model to arbitrary wind farm layouts. In the second part of the talk, we focus on using attributes of the turbulent ABL to provide improved models for power production and wake behavior under turbine yawing, which has been shown to increase turbine power output potential. Finally, we demonstrate how these ideas can be extended to a control setting.

## Privilege, agency, and the climate scientist’s role in the global warming debate

Andrew Weaver is a professor of climate science at the University of Victoria. He is also a lead author for the IPCC and a former BC MLA and leader of BC Green Party. This presentation was given ahead of his participation in a panel discussion on Tackling Climate Change and the Just Transition to Renewable Energy.