Undergraduate Research Project:
Title: Topology of Surfaces (Final Draft)
Supervisor: Dr. Danish Ali (Assistant professor at IBA)
About the project: This project aims to study 2-dimensional manifolds, better known as surfaces. The first few chapters lay a combinatorial foundation to discuss the intuition leading to the famous classification theorems for compact, connected surfaces with and without boundary. The latter part is dedicated to develop algebraic tools to further distinguish between well-known surfaces. Firstly, we develop the Euler characteristic and use it to define the genus of a surface. Then, we define and compute Homology groups for different surfaces and show that its invariant for any complex chosen to represent the surface. The final chapter lays a detailed introduction to the Fundamental Group and ends with its computation for several surfaces using the Seifert–van Kampen Theorem.
Primary Reference: L. C. Kinsey, Topology of Surfaces, Springer, 1993
International Mathematics Master's Research Project:
Title: The Geometry and Statistics of Self-Similar and Self-Affine Sets (Final Draft)
Supervisor: Dr. Thomas Jordan (Senior lecturer at University of Bristol) and Dr. Hani Shaker (Associate Professor at CUI Lahore campus)
About the Project: This thesis aims to study a specific class of fractal sets i.e sets that can have non-integer dimension. Although this has been a vibrant field over the last 40 years, it is still difficult to find a universal definition of a fractal set. This is because of the lack of a universally acceptable definition of the dimension of a set. Nevertheless, this thesis outlines some well-known definitions of dimension that are applicable to any subset of the n-dimensional real Euclidean space. Using these definitions, the class of self-similar (SS) and self-affine (SA) sets is defined along with some classical examples and their respective dimensions are calculated. This thesis also attempts to provide methods of computing dimensions that can be generalized to several SS sets and SA sets.
Primary References: (1) K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2014 (2) C.J. Bishop, Y. Peres, Fractals in Probability and Analysis, Cambridge University Press, 2016