Projects and Summer Research

A mathematical analysis of machine learning algorithms

August 10, 2023

This was a 2023 summer research project I undertook under the supervision of professor Greg. Pavliotis themed in the mathematical analysis of machine learning learning algorithms. In brief,

• I studied the existing literature on recent developments in the context of theoretical machine learning that integrate tools from statistical physics and probability theory, i.e., the theory of interacting particle systems.
• I analysed the approximation quality and trainability of neural networks using algorithms, such as Stochastic Gradient Descent (SGD), informed by such ideas on toy models and examples with real life examples such as the MNIST digit classification dataset.
• I performed numerical experiments by training neural networks under various circumstances, thereby graining practical insights and recorded my observations.

Recommended citation: Tassopoulos, Pantelis. (2023). "Imperial College research. A mathmematical analysis of machine learning algorithms.
Download Paper

Random constructions in the plane

August 05, 2022

I studied the notion of harmonic measure in the plane, its various formulations involving conformal maps and Brownian motion, culminating in the study of the so-called conformally balanced trees following work from Professor of mathematics at Stony Brook Christopher Bishop. This was a very profitable experience as it helped me further refine my analytical problem-solving and decomposition skills due to the nature of the work in the project. In conjunction with the above, my my communication and organisational skills were invariably improved as I engaged in weekly meetings with my supervisor Dr. Cheraghi, wherein I discussed the progress of the project and received feedback on approaches to obstacles, incorporating said suggestions into the project. I obtained a lot of insight into the world of academia and the way research is conducted.

Imperial College Second year project

June 15, 2022

This is a report of our group’s (Pantelis Tassopoulos, Michael Pristin, Ronkgai Zhang, Yuhao Liu, Ana Ciupala) study of Darboux Transformaions and its role in the spectral analysis of one dimensional Schr¨odinger operators. This method allows us to find the eigenvalues and eigenfunctions of such an operator via iteration in an algebraic manner.

We begin with some preliminary theory about the Hilbert spaces and linear operators of functions thereof. After developing this formalism, we will discuss successful applications of this approach for three Schr¨odinger equations (Schr"{o}dinger operator eigenvalue problems) with the method of Darboux Transformaions, which are as follows: the simple harmonic oscillator equation, the equation with reflectionless potential equation, and the equation with Coulomb potential. These problems have far reaching implications in the field of Quantum Mechancis and mathematical physics more broadly and are thus of fundamental physical interest.

Recommended citation: Pantelis Tassopoulos, Michael Pristin, Ronkgai Zhang, Yuhao Liu, Ana Ciupala. (2022). The Darboux Transformation.
Download Paper

Imperial College First year project - Solitary waves and the KdV equation

May 10, 2021

In this poster project, I discussed a mathematical model that accounts for the phenomenon of solitons in shallow water I examine some of its analytical as well as numerical properties.

Download Paper