**Random Quote:** *Freedom of thought is the only guarantee against an infection of mankind by mass myths, which, in the hands of treacherous hypocrites and demagogues, can be transformed into bloody dictatorships* – Andrei Sakharov

# Mathematics¶

## Interests¶

My interests are in probability, functional analysis and statistics. At the moment, most of my research centers on numerical analysis, stochastic processes and simulation.

## Books¶

If you are interested in analysis and probability with a view to mathematical economics then here is my reading list. I am extremely fussy, and I think that all of these books are great.

### Introductory¶

Introduction to Real Analysis, Bartle and Sherbet

Abstract Analysis, Will Light

### Intermediate and Graduate texts¶

An Introduction to Measure and Probability, J. C. Taylor

Introductory Real Analysis, Kolmogorov and Fomin

Probability with Martingales, D. Williams

A Users Guide to Measure-Theoretic Probability, Pollard

Principles of Real Analysis, Aliprantis and Burkinshaw

Real Analysis and Probability, R. M. Dudley

Measures, Integrals and Martingales by R. L. Schilling

Analysis for Applied Mathematics, W. Cheney

Probability and Stochastics, E. Cinlar

### Markov Chains¶

Chaos, Fractals and Noise: Stochastic Aspects of Dynmaics, Lasota and Mackey

Markov Chains and Stochastic Stability, Meyn and Tweedie.

Discrete-Time Markov Control Processes, Hernandez-Lerma and Lassere

Lectures on the Coupling Method, Lindvall

### Monographs, Other Topics¶

Approximate Solutions of Operator Equations, Mark Krasnosel’skii

Optimization by Vector Space Methods, Luenberger.

Infinite Dimensional Analysis, Aliprantis and Border

Lectures on the Coupling Method, Lindvall

The book by Bartle and Sherbert is where you should start. It is the best introduction to analysis that I know. Kolmogorov and Fomin is a nice read on functional analysis. Pollard has a great style and interesting topics. He uses a non-standard notation that takes a while to get your head around, but it’s been kind of revolutionary for me in a way. Aliprantis and Burkinshaw is an excellent graduate text with lots of exercises. Dudley is great, but he frightens me a little bit. Schilling is beautiful and useful. Cheney is beautiful too, and a nice balance between theory and applications.

For background reading, try the wonderful book called “In Search of Infinity” by N. Ya. Vilenkin.

### Writing Advice¶

Great advice on

`how to write mathematics`

by Ward Cheney (thanks Akshay)

### On-line Resources¶

Collection of free maths books: Collection of free maths books as PDF files

## Other Bits and Pieces¶

I love the work of Escher, especially this one. And here’s an interesting little equation about girls and evil.

Incidentally, the man generally recognized as the founder of modern functional analysis is Stefan Banach. Many fundamental results in normed linear space stem from his 1922 thesis. Another important contributor to the theory of abstract spaces was Maurice Frechet. Frechet also helped generalize the Lebesgue integral to operate on functions of arbitrary domain. Within the field of Markov processes, a bright and shining light was created by the short career of Wolfgang Doeblin. It’s been a huge privilege to understand and even contribute a tiny amount to his wonderful ideas.