## 24.06.2024 15:00 Dan Hill: 'Approximating fully localised planar patterns: A radial spatial dynamics approach'

Fully-localised planar patterns with dihedral symmetry, including cellular hexagons and squares, have been found experimentally and numerically in various continuum models; for example, in nonlinear optics, semi-arid vegetation, and on the surface of a ferrofluid (a magnetic fluid). However, there is currently no mathematical theory for the emergence of these types of patterns. In this talk, I will present recent progress regarding the existence of localised dihedral patterns (not necessarily hexagon or square) emerging from a Hamiltonian--Hopf bifurcation for a general class of two-component reaction-diffusion systems.
The planar problem is approximated through a Galerkin scheme, where a finite-mode Fourier decomposition in polar coordinates yields a large, but finite, system of coupled radial differential equations. We then apply techniques from radial spatial dynamics to prove the existence of a zoo of localised dihedral patterns in the finite-mode reduction, subject to solving an (N+1)-dimensional algebraic matching condition. We conclude by studying this matching condition for various finite-mode reductions, and present a computer-assisted proof for the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions.
This work is in collaboration with Jason Bramburger (Concordia University) and David Lloyd (University of Surrey).

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## 24.06.2024 16:30 Partha Pratim Gosh: Extremal Process of Last Progeny Modified Branching Random Walks

In this work, we consider a modification of the usual Branching Random Walk (BRW), where the position of each particle at the last generation 𝑛 is modified by an i.i.d. copy of a random variable 𝑌, which may differ from the driving increment distribution. This model was introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Depending on the asymptotic properties of the tail of 𝑌, we describe the asymptotic behaviour of the extremal process of this model as 𝑛 → ∞.

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## 25.06.2024 14:00 Thomas Richardson (University of Washington, Seattle): Short Course on “Graphical causal modeling” (Lecture 1/3)

This short course covers recent developments in graphical and causal modeling in Statistics/Machine Learning. It is comprised of the following three lectures, each two hours long.
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June 25, 2024; Lecture 1: “Learning from conditional independence when not all variables are measured: Ancestral graphs and the FCI algorithm”
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June 27, 2024; Lecture 2: “Identification of causal effects: A reformulation of the ID algorithm via the fixing operation”
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July 2, 2024; Lecture 3: “Nested Markov models”
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The course targets an audience with exposure to basic concepts in graphical and causal modeling (e.g., conditional independence, DAGs, d-separation, Markov equivalence, definition of causal effects/the do-operator).

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## 25.06.2024 16:00 Regina Bühler: DER operation in residential areas: Modelling coordination strategies via Linear Programming

Ongoing changes in the residential energy sector, along with the emergence of Distributed Energy Resources (DER) such as photovoltaic (PV) energy generation and battery storage, as well as newer (and potentially more flexible) energy demands like heat pumps and electric vehicle charging, have significantly increased the importance of demand response management to ensure grid stability.
The goal of this thesis is to formulate a linear program to model the operation of DER technologies such as electric vehicle charging, battery energy storage, heat pumps, and thermal storages in residential households. The objective is to minimize customers' energy costs based on given energy prices and demands.
Different use-case scenarios are established. In these scenarios, households can either be considered individually or as a community where households have the option to share excess energy with their neighbors in a coordinated manner. Furthermore, the scenarios are distinguished based on whether they enforce network limitations, i.e., constrain the available power a transformer substation can supply, or not.
After defining the corresponding LPs for these scenarios, some necessary conditions for solutions to the problems are analyzed and proven. Numerical experiments on exemplary data sets provided by Siemens are performed using the Pyomo modeling framework and Gurobi. Key performance indicators such as substation and household load profiles, substation peak loads, and energy costs for households are evaluated.
This thesis was done in cooperation with the Siemens AG.

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## 26.06.2024 13:00 Hermann Eberl (University of Guelph, Canada): A Spatio-Temporal Model of Fire Blight During Bloom

Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder’s fix point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed.

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