This is a PhD module which aims to provide students with strong foundations in Bayesian inference and Methods for Data Analysis. Specifically, the course aims to give a brief introduction to the Bayesian approach in decision theory. After an overview to the subjective probability concept that underlies Bayesian inference, the course moves on to the advanced topics in Bayesian theory. Some applications will also be provided.
Optimization problems arise in various fields ranging from economics to physics and in many aspects of our daily lives. For example, airlines arrange their schedules to maximize their profit subject to constraints imposed by limited resources such as the number of crew-members and planes. Ray of light follows a path to minimize the travel time. Two main ingredients of an optimization problem are: an objective function which we want to minimize or maximize, and a set of constraints which determines the set of allowable points over which the objective function must be minimized or maximized. The main purpose of this course is to learn an algorithmic approach to continuous numerical optimization, constrained and unconstrained. Emphasis on practical methods with enough practical examples and enough theory to make them work.
Functional Analysis is an abstract branch of mathematics that originated from the classical analysis. It concerned with the study of certain topological-algebraic and geometric structures and techniques by which knowledge of these structures can applied to analytic problems. It has an applications in every branch of mathematics like in mathematics finance, differential equations, computer science, probability theory, quantum mechanics, quantum physics, etc.
The course aims to give advanced knowledge of Banach spaces, Hilbert spaces, linear bounded operators, and their properties. Moreover, the course describes both applications of fundamental theorems and applications of Hilbert spaces methods.
This module aims at giving a solid background on Measure Theory and Lebesgue Integration. Measure generalizes the familiar concepts of lengths, area, and volumes in finite-dimensional Euclidean spaces. The Lebesgue integral is a generalization of the Riemann integral to a large class of functions. The resulting theory is a fundamental tool for advanced studies in many areas of Mathematics, and Mathematical Physics, which is the prime motivation for developing this module. Topics covered in this module are Construction and Properties of Measures in abstract spaces, Measurable functions, and their properties, Construction and properties of Lebesgue integral, Product measures, General set functions, and Transformations and Functions Spaces.
The module aims to describe advanced methods of solving partial Differential Equations (PDEs). It gives a thorough advanced form of representation formulas for solutions of PDEs. The important existence and uniqueness theorems are proved. The properties of solutions (e.g., regularity, dependence on data, etc ) are analysed.The module also describes the theory for Linear LPartial Differential Equations (LPDEs) and the theory for Nonlinear partial Differential Equations (NLPDEs)