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In the latest run of the Early Career Mathematicians Seminar Series, we will be joined by the winners of the Tomorrow’s Mathematician Today 2024 prizes. Mingjia Yan (Imperial) won the IMA Popular Vote Prize and Andi Hani (Warwick) won the IMA Prize for the best presentation.
Mingjia and Andi will be speaking for 30 minutes each, see below for their abstracts.
Mingjia’s talk: Lagrangian Reduction by Symmetry and Semidirect Product Structures in Compound Dynamical Systems
We present a coordinate-independent and operational geometric formulation to study the dynamics of a variety of compound dynamical systems using geometric mechanics. Compound dynamical systems are ordered so that each degree of freedom evolves in the frame of motion of the previous one. The complex dynamics of these mechanical systems is considered to take place on a composition of two or more Lie group actions known as a composition of maps or a sum of Lie group actions on corresponding vector spaces as a semidirect product action. The former applies to systems with symmetries described by Lie groups in mathematics while the latter refers to physical systems with broken symmetries due to the presence of certain order parameters. This structure capturing the symmetry-breaking behaviours in the compound systems is called semidirect product action. By the identification between the tangent bundle (configuration space) and the Lie algebra the invariance of the Lagrangian reduces the dynamics from the tangent bundle to the Lie algebra.
We will first discuss the Lagrangian reduction by symmetry as a methodology using the symmetries in different Lie groups stage by stage. Following a reduced Legendre transform the Hamiltonian equations written on the dual of of the composed semidirect product Lie algebra can be obtained which are called the Lie-Poisson equations. They contain a special non-canonical Poisson bracket called the Lie-Poisson bracket. Then we consider the dynamics of 2D Alfvén wave turbulence as a concrete example and obtain its equations of motion on the Lagrangian side (known as the Euler-Poincaré equations) using the Lagrangian reduction by symmetry scheme and later show that the derived Hamiltonian equations possess a semidirect product structure. As we shall see the finite-dimensional cases contribute useful insight and analogues for the infinite-dimensional versions due to the advantages in the nature of geometric mechanics.
Andi’s talk: The complex relationship between deprivation and disease
Following the past years the knowledge of epidemiological models has become increasingly widespread. They are used to predict the severity of disease outbreaks estimate the dreaded R0 value and contribute to policy decisions. With COVID-19 the risk factor of age is well documented with the heaviest burden of disease falls on the most elderly. As a result these models are adapted to include this and spread of disease amongst the elderly are of particular interest. However it is also well known and there are a growing number of empirical studies that show the most deprived are disproportionally impacted. Despite this very few models included this risk factor and the potential impact it could have on spread and outcome of infection.
Within this talk we explore the difficulties of incorporating deprivation stratification and social determinants in general into models and the advancements being made to tackle this problem.
Bios:
Mingjia Yan, Imperial College London: http://linkedin.com/in/mingjia-yan
Mingjia recently graduated with a MSci Mathematics degree from Imperial College London. Her interests in mathematics broadly and widely spread across fluid dynamics geometric mechanics dynamical systems soft condensed matters and electromagnetism. In particular, she is very interested in studying chaotic and active dynamical systems with dissipation in maths and physics using geometric mechanics and differential geometry such as many various pendulum dynamics active matters in biomathematics liquid crystals and a variety of MHD theory and models.
Mingjia’s fun fact: she is obsessed with collecting different types of CDs and albums in pop music. She has a whole three-layer shelf dedicated to all of her CDs and she probably has over 100 now. She also loves to review different places she has been to (especially cafes and restaurants) and write reviews on Google Map. There’re almost 4000 followers on her Google Map account which is really unbelievable!
Andi Hani, Warwick University: https://www.linkedin.com/in/andi-hani-1108aa243/
Andi is a recent masters graduate from the University of Warwick. He has a particular interest in applied mathematics including epidemiology and dynamical systems. Outside of maths, he is a diehard Arsenal fan and plays football whenever he can!
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