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Applied Probability Lent 2025 syllabus

Notes on the Brownian snake and superprocesses (2025)

Analysis 1 Fall 2022 - Indicative syllabus

Calculus and Applications Spring 2023 - Indicative syllabus

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Imperial College First Year Project - Solitary Waves and the KdV Equation (2021)

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Part III Cambridge maths course notes 2023-2024

Imperial maths undergraduate courses 2020-2023

PDE & Probability in Interaction: Functional Inequalities, Optimal Transport and Particle Systems

October 22, 2024

Several structural equations from physics describe systems composed of a large number of interacting particles. These equations play a role in kinetic theory, population dynamics, economics, and more. The conference focuses on mathematical investigation of particle systems, functional inequalities, and optimal transport. The goal is to foster contacts between specialists in probability and PDEs to develop new methods and applications.

Probability @ Warwick Conference

July 07, 2025

The P@W Summer School - Recent Trends in Probability and Statistics is the inaugural summer school in Probability and Statistics at the University of Warwick. It engages PhD students and early-career researchers with cutting-edge topics, featuring lectures on stochastic PDEs, random planar maps, directed polymers, and Bayesian inference for time evolution PDEs. Discussions and exercises enhance interaction among participants and speakers.

Portfolio item number 1

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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 examined some of its analytical as well as numerical properties.

Imperial College Second Year Project

June 15, 2022

This is a report of our group’s study of Darboux Transformations and its role in the spectral analysis of one-dimensional Schrödinger operators. We developed the Hilbert space formalism and applied the method to three Schrödinger operator eigenvalue problems: the simple harmonic oscillator, the reflectionless potential, and the Coulomb potential. This work has broad implications in quantum mechanics and mathematical physics.

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

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 Christopher Bishop. This experience helped me refine analytical problem-solving, decomposition, and communication skills through weekly meetings with my supervisor, where feedback was incorporated into the project.

A Mathematical Analysis of Machine Learning Algorithms

August 10, 2023

This 2023 summer research project focused on the mathematical analysis of machine learning algorithms. I studied theoretical ML literature integrating statistical physics and probability, analyzed neural network approximation quality and trainability using SGD, and performed numerical experiments on toy models and datasets such as MNIST, recording practical insights.

Recommended citation: Tassopoulos, Pantelis. (2023). Imperial College research: A mathematical analysis of machine learning algorithms.

Brownian Motion and the Stochastic Behaviour of Stocks Permalink

Published in Journal of Mathematical Finance 12.1 (2021): 138-149., 2021

In this paper, we test the effectiveness of predicting the behavior of stocks utilizing stochastic calculus. We begin by exploring the intuition of Brownian motion by explaining its birth through the observations of Robert Brown and later through Bachelier’s work on its applications to the financial market and finally its rigorous and concretized form proposed by Norbert Wiener. The aforementioned motivates a stochastic differential equation to model the future price fluctuations of a stock wherein Ito integration is prominent and consequently expanded upon. The final part of this paper focuses on the accuracy of the model by back testing it with Apple stock and deriving a correlation coefficient.

New outlook on profitability of rogue mining strategies Permalink

Published in arXiv, 2022

Many of the recent works on the profitability of rogue mining strategies hinge on a parameter called gamma (γ) that measures the proportion of the honest network attracted by the attacker to mine on top of his fork. Cyril Grunspan and Ricardo P´erez-Marco, in two papers released in 2018, have surmised conclusions based on premises that erroneously treat γ to be constant. In this paper, we treat γ as a stochastic process and attempt to find its distribution through a Markov analysis. We begin by making strong assumptions on gamma’s behaviour and proceed to translate them mathematically in order to apply them in a Markov setting. The aforementioned is executed in two separate occasions for two different models. Furthermore, we model the Bitcoin network and numerically derive a limiting distribution whereby the relative accuracy of our models is tested through a likelihood analysis. Finally, we conclude that even with control of 20% of the total hashrate, honest mining is the strongly dominant strategy.

The KPZ Fixed Point and the Directed Landscape Permalink

Published in arXiv, 2024

The term “KPZ” stands for the initials of three physicists, namely Kardar, Parisi and Zhang, which, in 1986 conjectured the existence of universal scaling behaviours for many random growth processes in the plane. A process is said to belong to the KPZ universality class if one can associate to it an appropriate “height function” and show that its 3:2:1 (time : space: fluctuation) scaling limit, see 1.2, converges to a universal random process, the KPZ fixed point. Alternatively, membership is loosely characterised by having: 1. Local dynamics; 2. A smoothing mechanism; 3. Slope-dependent growth rate (lateral growth); 4. Space-time random forcing with the rapid decay of correlations. The central object that we will study is the so-called KPZ fixed point, which belongs to the KPZ universality class. Many strides have been made in the last couple of decades in this field, with constructions of the KPZ fixed point from certain processes such as the totally asymmetric simple exclusion process (with arbitrary initial condition) and Brownian last passage percolation. In this essay, we: 1. delineate the origins of KPZ universality; 2. describe and motivate canonical models; 3. give an overview of recent developments, especially those in the 2018 Dauvergne, Ortmann and Virag (DOV) paper; 4. present the strategy of and key points in the proof of the absolute continuity result of the KPZ fixed point by Sarkar and Virag; 5. conclude with remarks for future directions. The presentation is such that the content is displayed in a way that is as self-contained as possible and aimed at a motivated audience that has mastered the fundamentals of the theory of probability.

Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation Permalink

Published in arXiv, 2025

We show that the Radon-Nikodym derivative of the law of the spatial increments (with endpoints away from the origin) of inhomogeneous Brownian last passage percolation (LPP) with non-decreasing initial data against the Wiener measure (\mu) on compacts is in (L^{\infty-}(\mu)); and for any fixed (p>1), the (L^p) norm is at most of the order (O_p(\mathrm{e}^{d_pm^2\log m})) for some (p)-dependent constant (d_p>0). Furthermore, when the initial data is homogeneous, we establish optimal growth on (L^p) norms ((\asymp O(\exp(dm^2))) of the Radon-Nikodym derivative of the Brownian LPP (i.e. top line of an (m)-level Dyson Brownian motion) away from the origin, as the number of curves (m) tends to infinity, for all (p>1) sufficiently large. As an application of our framework, we show that the Radon-Nikodym derivative of certain toy models for the KPZ fixed point lies in (L^{\infty-}(\mu)), inspired by its variational characterisation in terms of the directed landscape.

Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data Permalink

Published in arXiv, 2025

We show that the spatial increments of the KPZ fixed point starting from arbitrary (finitary) initial data, exhibit strong quantitative comparison against rate two Brownian motion on compacts.

Topics in random matrix theory

The aim of these notes is to provide a survey of some basic theory and classical results in random matrix theory, as well as to include more recent progress in the field. Throughout, we will give pointers to the relevant literature and encourage the reader to initiate their own investigations in this rich and rewarding field.

The Brownian Snake and Superprocesses

In these notes, I cover Chapter 4 of Legall’s Book on Spatial Branching Processes, Random Snakes and Partial Differential Equations, covering the construction of the Brownian snake and its relationship to superprocesses.

Undergraduate Teaching Assistant

Undergraduate courses, Imperial College London, Department of Mathematics, 2022

In Fall 2022 and Spring 2023, I was a teaching assistant for undergraduate mathematics courses, conducting demonstrations and coordinating problem-solving sessions.

Cambridge Mathematical Tripos Supervisions

Undergraduate courses, CMS, University of Cambridge, 2024

I supervised third year undergraduates in the Part II courses ‘Stochastic Financial Models’ and ‘Applied Probability’ during Michaelmas 2024 and Lent 2025. I discussed students’ problem sets and more general questions regarding the material.