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AN OPEN NOTEBOOK BY Tian X · @tianmathmath ↗

Starting with mathematics,
writing thoughts down.

Here I share my mathematical notes, reflections, and essays: from the intuition behind a concept to the details of a proof, from questions found in books to thoughts beyond mathematics. Understanding and writing, a little at a time.

Browse mathematical notes ↓

Noncommutative Fourier transform: a group, its matrix-valued Fourier coefficients, and reconstruction.

Mathematical Notes

Exploring the connections between different branches of mathematics.

01 · HARMONIC ANALYSIS

Harmonic Analysis

Exploring the structure of functions and the geometric intuition of Fourier analysis through frequency, oscillation, and scale.

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02 · MARTINGALE THEORY

Martingale Theory

Notes on conditional expectation, martingales, and stopping times, covering basic concepts, convergence, and inequalities in stochastic processes.

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03 · OPERATOR ALGEBRAS

Operator Algebras

Exploring the language and examples of C* algebras and von Neumann algebras, starting with operators on Hilbert spaces.

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04 · FUNCTIONAL ANALYSIS

Functional Analysis

Notes on normed spaces, linear operators, and duality, with an emphasis on concrete understanding behind abstract structures.

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BEYOND MATHEMATICS

Reflections & Essays

A place for thoughts that need not become theorems: reflections on reading, everyday observations, and questions still taking shape.

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An open notebook with mathematical ideas and a path from questions to understanding.

Latest Updates

A little understanding, one entry at a time.

Finding My Way Back to Mathematics

Reflections & Essays

On AI, the joy of doing mathematics, and finding a way to hold on to both a vocation and an inner life.

2026-10-12

We Should Stand Together

Reflections & Essays

On the disruption of mathematics by AI, what it may mean for other fields, and why people should stand together.

2026-10-12

The Basic Geometric Meaning of Frequency in Fourier Analysis (1)

Mathematics
Harmonic Analysis

Fourier atoms, directional variation, and the reciprocal relationship between spatial scale and frequency.

2026-10-10
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