Academic
Academic
Research, education, talks, teaching, study notes and outreach — the academic side of my work, gathered in one place.
01Research
Randomness, geometry and markets
My research sits where probability meets analysis. Stochastic analysis provides a language for systems that evolve randomly in time; optimal transport provides a geometry for comparing probability distributions. Mathematical finance and machine learning supply many of the questions that motivate the work.
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dXt
Stochastic Analysis
The calculus of randomness: how to describe, and compute with, quantities that change unpredictably over time.
In technical terms Stochastic differential equations, semimartingale theory and stochastic calculus on infinite-dimensional spaces.
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W2(μ, ν)
Optimal Transport
The most efficient way to move one distribution of mass onto another, which gives a natural way of measuring the distance between probability distributions.
In technical terms Wasserstein geometry, martingale optimal transport and connections to partial differential equations.
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𝔼ℚ[ · ]
Mathematical Finance
How to price and hedge financial contracts robustly: when the model itself is uncertain, when buying and selling prices differ, and when markets contain very many assets.
In technical terms Model-independent pricing, conic finance and large financial markets.
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∇θ ℒ
Mathematical Machine Learning
Using the same tools to understand when and why learning algorithms work.
In technical terms Applications of stochastic analysis and optimal transport to learning theory.
02Education
From Lancaster to Amsterdam and Lisbon
A mathematics degree in England, a master’s in stochastics and financial mathematics in Amsterdam, and a joint doctorate between Lisbon and Amsterdam.
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2024 – present
PhD Mathematics (joint degree)
Instituto Superior Técnico, Universidade de Lisboa (opens in a new tab) & University of Amsterdam (opens in a new tab)
Grupo de Física Matemática (GFM) (opens in a new tab) · Korteweg-de Vries Institute for Mathematics (KdVI) (opens in a new tab)
- Co-supervisors
- Sven Karbach (opens in a new tab) and Léonard Monsaingeon (opens in a new tab)
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2020 – 2023
MSc Stochastics and Financial Mathematics
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2017 – 2020
BSc (Hons) Mathematics
03Talks & Conferences
Talks and conferences
Presentations at international meetings in mathematical finance, stochastic analysis and partial differential equations.
- events
- 6
- talks
- 4
- posters
- 2
- countries
- 5
2026
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Bologna, Italy
XIII Bachelier World Congress (opens in a new tab)
The world congress of the Bachelier Finance Society, bringing together academics, practitioners and researchers in quantitative finance to discuss recent advances in financial mathematics.
Presented
Neural network empowered liquidity pricing in a two-price economy under conic finance settings
· 30 June 2026 -
Pittsburgh, USA
Topics at the Interface of PDE and Probability (opens in a new tab)
A summer school of the Center for Nonlinear Analysis research training group, with lecture series by Scott Armstrong, Ivan Corwin, Yu Gu and Lenya Ryzhik on problems where partial differential equations and probability meet.
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Soesterberg, Netherlands
23rd Winter School on Mathematical Finance (opens in a new tab)
The annual Dutch winter school in mathematical finance, meeting point for academia and industry. The 2026 edition featured minicourses on reinforcement learning and rough volatility.
2025
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Lisbon, Portugal
Stochastics & Computational Finance 2025: New Trends from Academia to Real-World Applications (opens in a new tab)
An international conference organised by ISEG and CEMAPRE on recent advances in financial mathematics, held in honour of Albert Shiryaev and Ernst Eberlein.
Presented
Neural network empowered liquidity pricing in a two-price economy under conic finance settings
· 4 September 2025 -
Vienna, Austria
Vienna Congress on Mathematical Finance (VCMF) (opens in a new tab)
The third edition of an international congress covering the breadth of mathematical finance, from stochastic control and optimal transport to machine learning in finance and insurance.
Presented
Neural network empowered liquidity pricing in a two-price economy under conic finance settings
· 11 July 2025 -
Lisbon, Portugal
Lisbon Young Mathematicians Conference (LYMC) (opens in a new tab)
A conference for students and young researchers in mathematics, statistics and their applications, aimed at fostering discussion and collaboration early in research careers.
Presented
Neural network empowered liquidity pricing in a two-price economy under conic finance settings
· 12 April 2025
04Teaching
In the classroom
Teaching assistantships at Nova School of Business and Economics and Faculdade de Ciências, Universidade de Lisboa, most recent first.
| Academic year | Term | Course | Role | Institution |
|---|---|---|---|---|
| 2025/26 | Spring | Econometrics | Teaching assistant | Nova SBE (opens in a new tab) |
| Fall | Econometrics | Teaching assistant | Nova SBE (opens in a new tab) | |
| Fall | Mathematics for Biologists | Teaching assistant | FCUL (opens in a new tab) | |
| 2024/25 | Spring | Machine Learning | Teaching assistant | Nova SBE (opens in a new tab) |
| Fall | Computer Programming | Teaching assistant | Nova SBE (opens in a new tab) |
05Notes & Learning Resources
A library of study notes
Study notes on core topics in mathematics, statistics and computing, compiled while learning and teaching. Most sets build from definitions to the main theorems, with figures, common pitfalls and references for further reading.
Please note. These are personal notes compiled for reference and learning. They may contain errors, incomplete information or personal interpretations, and they are not peer-reviewed. For academic or professional purposes, please consult textbooks, research papers or other authoritative sources.
Foundations of mathematics
The core of an undergraduate mathematics degree.
Mathematical Analysis 3
- Sequences and Series Completeness, Bolzano–Weierstrass, Cauchy sequences, convergence tests, power series. Updated August 2026
- Continuity and Limits Limits, the Intermediate and Extreme Value Theorems, uniform continuity. Updated August 2026
- Convergence Theory Pointwise and uniform convergence, interchanging limits, the Weierstrass M-test. Updated August 2026
Calculus 3
- Differentiation The chain rule, the Mean Value Theorem, L’Hôpital’s rule, Taylor’s theorem. Updated August 2026
- Integration Riemann sums, the Fundamental Theorem of Calculus, improper and numerical integration. Updated August 2026
- Multivariable Calculus Gradients, the Hessian, the Implicit Function Theorem, Lagrange multipliers. Updated August 2026
Linear Algebra 3
- Matrices and Operations Matrix multiplication, determinants, invertibility, rank, Gaussian elimination. Updated August 2026
- Vector Spaces Bases and dimension, rank–nullity, inner products, orthogonal projection, Gram–Schmidt. Updated August 2026
- Eigenvalues and Eigenvectors Diagonalisation, the spectral theorem, the power method, PCA. Updated August 2026
Discrete Mathematics 3
- Combinatorics Inclusion–exclusion, generating functions, Catalan numbers, Ramsey theory. Updated August 2026
- Graph Theory Trees, planarity and Euler’s formula, colouring, matchings and Hall’s theorem. Updated August 2026
- Mathematical Logic Propositional logic, soundness and completeness, compactness, resolution. Updated August 2026
Number Theory 3
- Prime Numbers The Fundamental Theorem of Arithmetic, the Prime Number Theorem, primality tests. Updated December 2024
- Modular Arithmetic Congruences, modular inverses, Euler’s theorem, the Chinese Remainder Theorem. Updated December 2024
- Cryptography RSA, Diffie–Hellman, elliptic-curve cryptography, digital signatures. Updated December 2024
Statistics, econometrics & learning
Inference from data, from regression to neural networks.
Econometrics 4
- Regression Methods OLS as projection, Gauss–Markov, robust standard errors, ridge regression. Updated August 2026
- Time Series Analysis Stationarity, ARMA models, unit roots, spurious regression, forecasting. Updated August 2026
- Causal Inference Potential outcomes, instrumental variables, regression discontinuity, difference-in-differences. Updated August 2026
- Econometrics: personal notes PDF A single 32-page PDF.
Machine Learning 3
- Neural Networks Universal approximation, backpropagation, vanishing gradients, initialisation. Updated August 2026
- Optimization Methods Convexity, gradient descent rates, Newton’s method, SGD, duality and KKT. Updated August 2026
- Probabilistic Models Maximum likelihood, Bayesian inference, the EM algorithm, variational inference, MCMC. Updated August 2026
Computing
What can be computed, and how to reason about programs and systems.
Theoretical Computer Science 3
- Computability and Formal Languages Automata, Turing machines, the Halting Problem, P versus NP. Updated August 2026
- Type Theory and Functional Programming The lambda calculus, type safety, strong normalisation, Curry–Howard. Updated August 2026
- Distributed Systems and Concurrency Logical and vector clocks, FLP, the CAP theorem, Byzantine fault tolerance. Updated August 2026
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06Science Communication & Outreach
Beyond the seminar room
Mathematics, AI and scientific research are also about education, collaboration and community. Alongside research, I help build initiatives that bring students into contact with these fields.
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ONIA (opens in a new tab)
Co-founder · Organising Committee
Olimpíadas Nacionais de Inteligência Artificial · Portuguese Olympiads in Artificial Intelligence
The national olympiad in artificial intelligence for students from the 7th to the 12th grade in Portugal. It combines a classroom challenge that requires no programming with an olympiad track that selects the team representing Portugal at the International Olympiad in Artificial Intelligence (IOAI).
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Gradient
Co-founder
Co-organiser of the Portuguese Olympiads in Artificial Intelligence, together with TreeTree2.
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JunctionX Lisbon (opens in a new tab)
Event lead, founding edition · 2025
The Lisbon chapter of the international JunctionX hackathon series, bringing students, developers, researchers and companies together to build working prototypes for real challenges.