$$ \newcommand{\fk}[1]{\mathfrak{#1}}\newcommand{\wh}[1]{\widehat{#1}} \newcommand{\br}[1]{\left\langle#1\right\rangle} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\qp}[1]{\left(#1\right)}\newcommand{\qb}[1]{\left[#1\right]} \renewcommand{\Im}{\mathbf{Im}}\newcommand{\I}{\mathbf{Id}}\newcommand{\Id}{\mathbf{I}}\renewcommand{\ker}{\mathbf{ker}}\newcommand{\supp}[1]{\mathbf{supp}(#1)}\renewcommand{\tr}[1]{\mathrm{tr}\left(#1\right)} \renewcommand{\norm}[1]{\left\lVert #1 \right\rVert}\renewcommand{\abs}[1]{\left| #1 \right|} \newcommand{\lb}{\left\{} \newcommand{\rb}{\right\}}\newcommand{\zl}{\left[}\newcommand{\zr}{\right]}\newcommand{\U}{_}\renewcommand{\star}{*} \newcommand{\A}{\mathbb{A}}\newcommand{\C}{\mathbb{C}}\newcommand{\E}{\mathbb{E}}\newcommand{\F}{\mathbb{F}}\newcommand{\II}{\mathbb{I}}\newcommand{\K}{\mathbb{K}}\newcommand{\LL}{\mathbb{L}}\newcommand{\M}{\mathbb{M}}\newcommand{\N}{\mathbb{N}}\newcommand{\PP}{\mathbb{P}}\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\newcommand{\T}{\mathbb{T}}\newcommand{\W}{\mathbb{W}}\newcommand{\Z}{\mathbb{Z}} \newcommand{\Aa}{\mathcal{A}}\newcommand{\Bb}{\mathcal{B}}\newcommand{\Cc}{\mathcal{C}}\newcommand{\Dd}{\mathcal{D}}\newcommand{\Ee}{\mathcal{E}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathcal{H}}\newcommand{\Kk}{\mathcal{K}}\newcommand{\Ll}{\mathcal{L}}\newcommand{\Mm}{\mathcal{M}}\newcommand{\Nn}{\mathcal{N}}\newcommand{\Pp}{\mathcal{P}}\newcommand{\Qq}{\mathcal{Q}}\newcommand{\Rr}{\mathcal{R}}\newcommand{\Ss}{\mathcal{S}}\newcommand{\Tt}{\mathcal{T}}\newcommand{\Uu}{\mathcal{U}}\newcommand{\Ww}{\mathcal{W}}\newcommand{\XX}{\mathcal {X}}\newcommand{\Zz}{\mathcal{Z}} \renewcommand{\d}{\,\mathrm{d}} \newcommand\restr[2]{\left.#1\right|_{#2}} $$

About this blog

Welcome traveler! It’s pouring outside. Would you please step inside? Take a chair by the fire and make yourself comfortable. Ah! Where are my manners? Let me briefly introduce myself. My name is Liam, and I studied Mathematics at the University of Salamanca. I recently completed my Ph.D. at the University of Edinburgh and am currently a postdoctoral researcher at the University of Lancaster. My academic interests grew from analysis and PDEs to their applications in stochastics, such as SPDEs and Bayesian inference, and, more recently, optimal transport and sampling.

This blog is my attempt to map this intersection. If you’re still interested, welcome to the team!

I’m a firm believer that to understand anything, you have to make it your own, distil it to its essence, and be capable of re-explaining it in your own words. This blog provides a framework for this process.

What to expect

  • When active, I will post roughly once a month.

  • Posts will contain a three-sentence summary where I hope to reduce the topic of the post to its essence.

  • Posts will have a why should I care section? (always a good question to ask) to place the information in context.

  • Exercises with hints that I’ve used to build my own understanding, and that you can test yours on.

  • Finally, the finer workings of the topic (definitions, propositions, theorems, and examples) will be discussed with a focus on the key ideas that make everything fit together. When proofs are very technical, I will present the main details and cite primary sources for the details.

Ok, but where do I, as the reader, fit in?

Good question!

  1. First of all, I hope the presentation of the material is such that it provides a quick overview of topics while preserving the crucial information. As if you had an ally for an expedition through the jungle, helping you cut any vines, brambles, and snares that you may encounter.

  2. Secondly, and perhaps most importantly, I hope that, if this proposition has sparked your interest, you can engage with the blog. Insightful comments, questions, and different perspectives are always welcome, and feel free to point out any mistakes you may find. If you want, you can contribute your own posts on related topics. If you have something to teach, we all have something to learn.

Please feel free to contact me through the comments or my email.