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Master Supervisions
2023/24

Olapeju Enitan Arowobusoye
MSc Mathematics, University of Hull, UK
Thesis title: A Complex Analysis approach to the isoperimetric inequality
Mathematicians consider the isoperimetric inequality a fundamental principle which states that among simple closed curves of a given length, circle has the largest possible area \citePathak2019. This work proves the isoperimetric inequality through the application of complex analysis, specifically the Riemann mapping theorem. Each idea is inspired by Peter L. Duren’s monograph, Univalent Functions and Robert Osserman’s paper. The thesis covers a brief history of the inequality and presents a survey of classic proofs of the inequality while giving a constructive proof based on Riemann mapping theorem from the complex analysis. Practical applications will also be explored.

Lucky Ekeshili
MSc Mathematics, University of Hull, UK
Thesis title: The EulerLagrange equation
This dissertation inquires into the EulerLagrange equation and the method of Lagrange multipliers, with special emphasis on how they apply to variational problems. In particular we are illustrating through some example problems such as the brachistochrone problem that both of these mathematical tools are necessary when seeking an optimal solution for particular field tasks.A marriage of the EulerLagrange equation with Lagrange multipliers can turn an obejctive into manageable problems and remove all of their complications. The dissertation tests and verifies the cycloid as the curve of quickest descent in practice, thus confirming the practical usefulness of these methods. More broadly, the dissertation highlights the wideranging application and importance of these tools for tackling complex optimization problems.
2022/23

David Awuku
MSc Mathematics, University of Hull, UK
Thesis title: The Isoperimetric Problem
A closed curve’s perimeter and the area it encloses are related by the isoperimetric inequality. It claims that a circle is the closed curve with the biggest area enclosed for all closed curves with a specified perimeter.
A circle with the same perimeter as the curve will enclose the maximum feasible area for any closed curve enclosing a region. No other form can include as much space around a set boundary.
This thesis presents a comprehensive proof of the isoperimetric inequality in the Euclidean plane \mathbbR^2.
The proof relies on the Fourier analysisbased Wirtinger inequality. By reparameterizing an arbitrary curve on [1,1] and applying Wirtinger, the inequality is shown to hold for smooth curves. A density argument extends the conclusion to all finitelength W^1,1 curves. The equality case precisely corresponds to circular curves. Applications like deriving explicit a priori PDE estimates and bounding ellipse parameters are explored, demonstrating the inequality’s utility beyond circles. It highlights the inequality’s theoretical significance while exploring practical uses in mathematics and physics. The journey elucidates deep connections between geometry and analysis via an elegant isoperimetric constraint.
Undergraduate Supervisions
2024/25

Sam Fowler
BSc Mathematics, University of Hull, UK
Thesis title: Optimal transport

Declan Hodges
BSc Mathematics, University of Hull, UK
Thesis title: The isoperimetric inequality

Joe Varley
BSc Mathematics, University of Hull, UK
Thesis title: The Hausdorff measures