Jason Karpeles · The mathematical companion

See what the
equations mean.

Explore a picture. Change an assumption. Work through a small example. Bring the mathematics to a question of your own.

17 available chapters · 136 guided illustrations · Free in your browser

How does AI use the prompt?Chapter 8 · Attention and context →What does 90% confidence mean?Chapter 14 · Uncertainty →Can I rely on this AI answer?Chapter 17 · Truth and verification →Is this a trend or a short-lived spike?Chapter 6 · Signals at different scales →Does repeated updating eventually settle?Chapter 12 · Dynamics and equilibrium →How similar are these two patterns?Chapter 1 · Measure and compare →

Start with what you want to understand

Each chapter combines readable equations, adjustable illustrations, and a worked calculation. The companion's AI skills can help you select an idea and apply it to your own inputs.

“Help me understand how attention works.”
“How similar are these two patterns?”
“Can I rely on this AI answer?”

Use the mathllms-guide master skill to find the relevant chapter. Its answer should connect the mathematics to your question and explain the assumptions.

The chapter library


CHAPTER 01

Foundations and Notation

Measure a profile, compress a table, score a forecast, trace a change, and know how much data is enough.
Explore four illustrations →
CHAPTER 02

Approximation and Expressivity

Construct curves, count pieces, and separate three obstacles.
Explore four illustrations →
CHAPTER 03

Optimization and Training Dynamics

Inspect stable steps, noisy gradients, adaptive moments, schedules, curvature limits and batch size.
Explore four illustrations →
CHAPTER 04

Generalization and Implicit Bias

Separate repeated-fit uncertainty, bounds, interpolation, and solution choice.
Explore four illustrations →
CHAPTER 05

Symmetry and Geometric Inductive Bias

Specify what should stay fixed and what should transform.
Explore four illustrations →
CHAPTER 06

Convolutional and Multiscale Models

Inspect filter conventions, aliases, localized changes, residual maps, and the cost of scale.
Explore four illustrations →
CHAPTER 07

Recurrent and State-Space Models

Track decay, Jacobian products, gated retention, and convolutional memory.
Explore four illustrations →
CHAPTER 08

Attention and Transformers

Follow information through attention, sampling, position, and growing context.
Explore four illustrations →
CHAPTER 09

Graph and Geometric Deep Learning

Follow relationships, audit bridges and test what local labels miss.
Explore four illustrations →
CHAPTER 10

Latent, Adversarial, and Flow Models

Integrate hidden variation, inspect bounds and measure missing modes.
Explore four illustrations →
CHAPTER 11

Diffusion, Score, and Transport-Based Generation

Track noise, read scores and compare two reverse journeys.
Explore four illustrations →
CHAPTER 12

Continuous-Depth and Implicit Models

Separate equilibria, convergence and numerical artifacts.
Explore four illustrations →
CHAPTER 13

Scientific ML and Operator Learning

Test laws, conditions and trajectory evidence separately.
Explore four illustrations →
CHAPTER 14

Bayesian Deep Learning and Uncertainty

Update a belief, separate uncertainties, and check a forecast.
Explore four illustrations →
CHAPTER 15

Robustness, OOD, and Causality

Check how a conclusion changes with inputs, populations and interventions.
Explore four illustrations →
CHAPTER 16

Scaling Laws, In-Context Learning, and Abstract Foundations

Distinguish resources, examples and the objective being rewarded.
Explore four illustrations →
CHAPTER 17

The Mathematics of Truth, Error, and Abstention

Separate what was seen, what is supported, and what can responsibly be answered.
Explore four illustrations →
CHAPTER 18

The Mathematics of Reasoning, Post-Training, and Test-Time Compute

Planned draft supplement

Three ways to use the companion

Explore in your browser

Open a chapter and choose among the illustrated values. Equations, figures, and calculations travel together. Everything the page needs is included.

Read a saved notebook

The notebook includes its figures and results. Optional calculation cells make each example reproducible. No code editing is required to study the mathematics.

Ask the master skill

The skill package contains mathllms-guide and the available chapter skills. Give the guide a learning goal or a practical question. It selects the relevant mathematics and helps interpret it.

The skills run in an assistant that supports SKILL.md packages. This page is the independent visual companion, with no account or AI connection required.