Applied mathematician and AI practitioner. Founder of MathLumen, exploring mathematics behind machine learning and scientific AI.
The computational engine behind topological data analysis, built up from geometric intuition
Persistent homology tracks how loops, clusters, and voids in data survive across scales, revealing shape that means and...
A rigorous accounting of proven theorems, computational evidence, and 2026's record-breaking bound — not another 'is it finally solved?' piece
What is rigorously proven about the Riemann Hypothesis, what recent record-breaking results actually show, and why the...
A first-principles primer on the function that ties prime numbers to the geometry of the complex plane
Starting from a simple infinite sum, we build up the Riemann zeta function, its analytic continuation, and the critical...
Superhuman engines have mastered practical chess. Mathematics has not yet proven what perfect chess is.
Stockfish and AlphaZero have mastered practical chess, but its true value under perfect play remains formally unproven...
When a computer checks every step, what does it mean to know something is true?
Mathematicians are now encoding entire research papers into Lean, a proof assistant that verifies logic at the level of...
The German mathematician's proof of the Mordell conjecture — and decades of structural insight — earn mathematics' highest honour
Gerd Faltings has been awarded the 2026 Abel Prize for introducing powerful tools in arithmetic geometry and resolving...
The greatest unsolved problem in mathematics
The Riemann Hypothesis, proposed in 1859, remains unproven. We survey recent progress, computational verification to...
Inside the equation that powers modern AI
The Transformer architecture revolutionized AI with a single mechanism: self-attention. We break down the linear...
A meditation on e^{iπ} + 1 = 0
Euler's identity unites five fundamental constants in a single, breathtaking equation. We explore why mathematicians...
Why optimization in deep learning is a geometric problem
Gradient descent navigates high-dimensional loss landscapes shaped by curvature and saddle points. We explore the...
The extraordinary formulas that arrived in letters from Madras
Srinivasa Ramanujan produced formulas for 1/π of staggering complexity and beauty — with no formal proofs. We trace the...
When Fourier meets computation
Spectral methods transform PDEs into algebraic systems using global basis functions. We survey how Fourier and...
Persistent homology and the topology of point clouds
Topological data analysis uses algebraic topology to extract structural features from complex datasets. We explain...
How ratios, frequencies, and transforms create harmony
From Pythagorean tuning to the Fast Fourier Transform, mathematics and music have been intertwined for millennia. We...
Random walks on Wall Street — literally
Monte Carlo simulation is the backbone of derivatives pricing. We explore the mathematics of random sampling, variance...
Why your online banking depends on abstract algebra
Elliptic curve cryptography provides the strongest security per bit of any known public-key system. We explore the...
How an algebraist revealed the deepest structure of physical law
Emmy Noether's theorem connects symmetries to conservation laws, uniting mathematics and physics in a profound way. We...
The Riemannian manifold of probability distributions
Information geometry equips the space of probability distributions with a Riemannian metric — the Fisher information....