Math
Large Deviations and Weak Convergence of Measures, with applications to Monte Carlo Estimators
(M.Sc. thesis) slides
- Awarded Mittag-Leffler Prize for mathematical thesis work of outstanding scientific value.
- Develops the theory of weak convergence of measures and introduces the \(\tau\)-topology on the space of finite signed measures. Proves that the strong law of large numbers for the empirical distributions of the importance sampling estimator.
- Develops the central theory for large deviations of i.i.d. random variables in topological linear spaces. Proves Cramér's theorem in Polish spcaces and shows how Sanov's Theorem in the \(\tau\)-topology can be proved using projective systems, using results due to de Acosta.
Regularity of semilinear elliptic partial differential equations with critical Sobolev exponents
(B.Sc. thesis)
- Weak solutions of semilinear elliptic equations and their regularity properties. Where the main theorem is the following
Let \(\Omega\) be of class \(C^2\), \(v:\Omega \to \mathbb{R}\), and \(\mathbf{B}(x,v):\Omega\times\mathbb{R} \to \mathbb{R}\). If \(|\mathbf{B}(x,v)| \leq |v(x)|^\gamma\), for some \(\gamma < \frac{2}{n-2}\) and \(g\in W^{2,2}(\Omega)\), then there exists a solution \(u \in W^{1,2}(\Omega)\) to the equation
Weak Solutions of PDEs A motivation/introduction to weak solutions of partial differential equations, only neccessary background is basic real analysis and knowledge of Hilbert spaces.
Topological Dynamics notes Notes written for a lecture I gave on topological dynamics and Van der Waerden's Theorem in a PhD course. They are a trimmed down version of some lecture notes posted by Terrence Tao on his blog
Complex Analysis notes Notes containing the essential results in complex analysis:
I have quite a lot of notes/summaries containing the most important results from a variety of graduate and advanced undergraduate courses I have read, which I'm merging into one large reference text at the moment.