Archive of undergraduate term papers, seminars, projects and presentations
This is an archive of term papers, seminars, projects and presentations I worked on during my undergraduate years at IISERM. The section headings are clickable.
Abstract. Nim is one of the most important games in combinatorial game theory. The ideas and concepts from Nim can be extended to many other combinatorial games. Here, we introduce Nim as a two-player game played on any finite number of heaps and solve it.
Abstract. Function spaces in the category Set can be generalized to exponential objects in more general categories. This gives rise to Cartesian closed categories in which any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed $\lambda$-calculus.
Abstract. A Turing machine is an abstract computational model with a finite control and an unbounded memory. Despite its simplicity, a Turing machine is capable of implementing any algorithm. This gives rise to the idea of recognizability and decidability of languages. We also look at the famous Church-Turing conjecture and Rice’s theorem for undecidability.
Abstract. Integration on manifolds is made possible using differential forms and orientability. In this term paper, we discuss differential k-forms and orientability, and how they naturally generalize integration on Euclidean spaces to integration on manifolds. Then we look at manifolds with boundary and discuss two classic theorems in differential geometry, generalized Stokes’ theorem and Green’s theorem.
Abstract. Spectral graph theory can be used to analyze the topological properties (e.g., connectivity) of graphs. Each graph has a Laplacian matrix whose eigenvalues and eigenvectors reveal many properties of the graph. We look at discrete mathematical (graph theoretical) models for biological networks, then study some mathematics of spectral graph theory and in particular, properties of the Laplacian.