Pantelis Tassopoulos

Pantelis Tassopoulos

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classes

Fourier analysis and the Theory of Distributions, Imperial College London 2022

Coursework 1

Introduction to SDEs and diffusion processes, Imperial College London 2022

Coursework 1

Part III Approximation Theory Michaelmas 2023

conferences

PDE & Probability in interaction: functional inequalities, optimal transport and particle systems

October 22, 2024

A brief description from the conference website

Several structural equations from physics describe systems composed of a large number of interacting particles. These equations play a role in a wide number of fields (kinetic theory of gases, population dynamics, economics, etc…). To understand their large scale behavior by taking some suitable scaling limit is a well-known scientific challenge which has generated an intense research activity in the past decades, laying at the intersection between probability theory and PDEs.

This conference will focus on the mathematical investigation of such particle systems and on fundamental tools which are currently developed for their study, in particular: functional inequalities, and optimal transport, which both find applications in several domains of applied mathematics. Our aim is to foster contacts between specialists of the varied areas of probability and PDEs that are connected to these topics, in order to develop new methods and applications.

Some notes based on Michael Goldman’s minicourse on the optimal matching problem

portfolio

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projects

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

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

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.

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

publications

Brownian Motion and the Stochastic Behaviour of Stocks

Published in Journal of Mathematical Finance, 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 It\hat{o} 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.

Recommended citation: Tassopoulos, Pantelis, and Yorgos Protonotarios. "Brownian Motion & the Stochastic Behavior of Stocks." Journal of Mathematical Finance 12.1 (2021): 138-149.
Download Paper

New outlook on profitability of rogue mining strategies

Published in Preprint, 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.

Recommended citation: Pantelis Tassopoulos and Yorgos Protonotarios. (2022). New outlook on profitability of rogue mining strategies.
Download Paper

The KPZ Fixed Point and the Directed Landscape

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.

Recommended citation: Pantelis Tassopoulos. (2024). The KPZ fixed point and the Directed landscape.
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talks

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.

teaching

Undergraduate Teaching Assistant

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

I was a teaching assistant in the first-year course in analysis. I had the responsibility of conducting demonstrations and coordinating problem-solving sessions.

Indicative syllabus

  1. Real Numbers: The archimedean property, and density of Q in R; The completeness axiom; Sup and Inf, and basic properties; Decimal Expansions; Countability and uncountability.
  2. Real and complex sequences: Convergence and Divergence; the sandwich test; Sub-sequences, monotonic sequences, [limsup and liminf,] Bolzano-Weierstrass Theorem, Cauchy sequences and the general principle of convergence.
  3. Real and complex series: Convergent and absolutely convergent series; Comparison test for non-negative series and for absolutely convergent series; Alternating test series; Rearranging absolutely convergent series Radius of convergence of power series; Exponential series.

Undergraduate Teaching Assistant

Undergraduate course, Imperial College London, Department of Mathematics, 2023

I was a teaching assistant in the first-year calculus and applications course. I had the responsibility of conducting demonstrations and coordinating problem-solving sessions.

Indicative syllabus

  1. Fourier Transforms: Exponential, cosine and sine transforms; Elementary properties; Convolution theorem; Energy theorem.
  2. Ordinary Differential Equations: Introduction to ODEs: definitions and notations; Solutions for 1st and some 2nd order ODEs, linear ODEs; Separable, homogeneous and linear equations; Special cases; Linear higher order equations with constant coefficients; Systems of constant-coefficient linear ODEs; Qualitative Analysis of linear ODEs: Phase plane Analysis, stability of systems; Qualitative Analysis of nonlinear ODEs: Bifurcation Analysis. Including numerous examples from Newtonian dynamics such as motion point particle in an external potential and oscillatory motion.
  3. Introduction to Multivariable Calculus: General properties of functions of several variables; Partial derivatives and total derivatives; Second order derivatives and statement of condition for equality of mixed partial derivatives; Taylor expansions; Chain rule, change of variables, including planar polar coordinates.

Cambridge Mathematical Tripos Supervisions

Undergraduate course, CMS, University of Cambridge, 2024

I supervised five third year undergraduates in the Part II course ‘Stochastic Financial Models’. I had the responsibility of discussing students’ submissions to problem sets (can be found here) and more general questions regarding the material.

Indicative syllabus

  • Standing Assumptions and Notation: Assumptions about financial markets and asset prices; definitions of market setup.
  • One-Period Model: Investor wealth and random asset prices; mean-variance analysis and portfolio optimization.
  • Market Portfolio and Mutual Fund Theorem: Market portfolio definition; efficient frontier formulation.
  • Capital Asset Pricing Model (CAPM): Assumptions and derivation; risk-return relationship.
  • Expected Utility Hypothesis: Utility functions, preferences, and risk aversion measures.
  • State Price Densities: Definition, derivation, and relationship to utility maximization.
  • Risk Neutral Measures: Concept, applications in pricing, and equivalent measures.
  • Arbitrage and Fundamental Theorem of Asset Pricing: Arbitrage definition and implications; fundamental theorem proof.
  • Utility Maximization in Binomial Models: Optimal portfolios and contingent claim pricing.
  • Multi-Period Models: Filtrations, adapted processes, and conditional expectations.
  • Stopping Times and Optional Stopping Theorem: Definitions, applications, and proofs.
  • Martingale Theory: Martingale transforms, examples, and properties.
  • Pricing and Hedging of Derivatives: Attainable claims, no-arbitrage prices, and examples.
  • Advanced Topics: Filtrations, sigma-algebras, and conditional probability.