* $$ v\bigm|_=u\bigm|_ \qquad \implies \qquad \|\nabla u\|_\leq \|\nabla v\|_.$$! is absolutely minimizing if and only if it is the solution of the infinity Laplacian, which is the (highly degenerate) Euler-Lagrange equation for the prototypical problem in the calculus of variations in ! Recommended modules: Functional Analysis, Partial Differential Equations References: ! *

$$\begin \tag* P (E^S) \leq P(E) \quad \text E \subset \mathbb^n.

, and (see, for instance, Theorem 14.4 in [3]) is satisfied: !

For more information on any of these projects, please contact the project supervisor.

Recommended modules: Measure and Integration, Perturbation theory and calculus of variations, Functional Analysis.

We will first give a rigorous proof of (S), and then investigate the family of sets such that equality holds true in Steiner inequality. De Giorgi: Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.

Maggi: Sets of Finite Perimeter and Geometric Variational Problems: an Introduction to Geometric Measure Theory. Several seminars will include presentations by members of the Department of Mathematics on proposed topics for dissertations that they would be interested in supervising.Additional seminars will cover important aspects of writing a dissertation in mathematics, including: what plagiarism is and how to avoid it, the use of libraries for research, electronic research, general aspects of writing mathematics, and preparing a manuscript using mathematical text processing software (in particular, La Te X). Assessment is primarily based entirely on the Dissertation, three hard copies and one electronic copy of which must be submitted by early September (exact date to be specified later). is a plane curve with the increasing chord property with length ! We will examine how to improve the above constant "! 2, 123--139 Hausdorff dimension is the principal notion of dimension in the context of fractal sets in ! We will examine some of these and their inter-relationship. However, other definitions are in widespread use, for example, packing dimension, upper and lower box-counting dimension, upper and lower Minkowski dimension, ... Students will be given support in the seminars on how to prepare, how to present and what is expected.This page contains details for the topics available for final year dissertations for MMath students, and for projects for BSc students. The Dissertation in Mathematics is an individual, substantial project as an introduction to mathematical research methods.The student will investigate and study an area of mathematical research or application of advanced mathematical techniques, and to write a report on their findings. The dissertation topic will normally be proposed by the Department. The seminars include a mixture of lecture time, presentations and discussion.

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