My supervisor has asked for a list of my university modules so that he can put my experiance and skills to their best use. I have written this list for his benefit but the list does give a brief idea of some of the work that I am interested in and going to be doing. The most relevant modules are Bayesian Statistics and Decision Theory, Applied Statistical Modelling, Multivariate Statistics and Mathematical Programming. These have given me experience with multi-dimensional models, plotting regressions, using mathematical computing software, optimising a wide range of problems and interpreting data.
This module was an introduction to university mathematics building up knowledge needed for studying further modules. The main topics covered were number systems, set theory, polynomials.
This module covered interpreting problems on the form of first and second order differential equations, classifying equations by various types and solving these equations through various methods.
This module was an introduction to the use of linear algebra including the study of properties of vectors, matrices, determinants, eigenvectors and eigenvalues.
Analysis I, II, III:
These three modules covered sequences, series, convergence (in various forms), continuity, differentiation, integration, normal vector spaces.
This module covered differentiation of functions mapping from and into multiple dimensions and its use for optimisation leading onto manifolds. The module was largely based on the inverse and implicit function theorems.
This module covered metric spaces, topological spaces and various properties of these spaces including connectedness, compactness and completeness.
Measure theory covered properties of measures (involving product, signed and radon measures), measure spaces, measurable spaces, integration over measures, convergence theorems and function spaces.
Markov Processes and Percolation Theory:
This module covered Markov processes (in both discrete and continuous time) and percolation theory as the title suggests. It mainly focused on random walks, explosions and Poisson processes.
Applications of Algebra and Analysis:
This module applied techniques learnt in first year mathematics modules to population dynamics and some approximation algorithms.
Probability A, B:
These modules introduced basic probability including Bayes’ Theorem, probability and moment generating functions as well as an introduction to common distributions and their properties.
Games and Decisions:
This module introduced the optimisation of a decision maker’s utility including eliciting probabilities from a client and solving Nash equilibria in competitive and co-operative games.
This module was an introduction to the use the computer technology for mathematics using the software Mathematica and LaTeX.
Mathematical Statistics A, B:
These modules introduce hypothesis testing, confidence intervals and transformations of distributions.
Mathematics of Random Events:
This module was an introduction to basic probability theory including probability measures, product measures, expectation and integration.
Forecasting and Control:
This module introduced the mathematical programming software SPLUS which was used to determine regressions given large quantities of data by using transformations and statistical tests. The module continued onto time series involving weighted regressions.
In this module we covered discrete time stochastic processes involving useful properties such as recurrence and periodicity. Their use for genetics was briefly covered as were branching processes and renewal processes.
Applied Stochastic Processes:
This module covered stochastic processes in continuous time especially explosions and birth-death processes. This progressed onto queuing theory and reversibility of Markov chains.
Applied Statistical Modelling:
This is an extension of the module Forecasting and Control for multiple variable models. In this module I was given five problems involving determining regressions for multiple variable models, determining logistic regressions, simulating values from distributions and the use variance reduction techniques.
Bayesian Statistics and Decision Theory with Advanced Topics:
In this module we studied decision making through decision trees and maximising utility functions. This involved eliciting probabilities from clients using various methods. This progressed onto testing the optimal decisions sensitivity to changes in probabilities and calibrated forecasting. We then studied graph theory and its use for Bayesian networks, irrelevance statements, causality and the updating of probabilities using propagation and sampling methods. The advanced topics were casual manipulation and casual networks which used to test circumstances in which we can test a system by manipulating variables in simulations.
This was an extension of measure theory and mathematics of random events mainly concentrating on conditional expectation, martingales and convergence theorems.
This module was an extension of mathematical statistics but for multiple variable models.
Mathematical Programming I, II, III:
In these three modules we mainly covered optimising equations. These ranged from linear, non-linear, integer, stochastic and various other types of problems involving graphs and matrices. To solve these we used various techniques including problem specific algorithms, simplex method, approximation algorithms and heuristics.
Introduction to Quantitative Economics:
This module was mainly an introduction to economics but it did involve some game theory and optimisation methods.
Mathematical Economics 1a, 1b:
This first of these modules was purely game theory and decision theory with the games extended to involve uncertainty and possibly infinite iterations. The second half focused on simple economies and optimising consumer, producer and multiple entity systems.