Discover condensed mathematics
Eight interactive worlds take you from three objects on a table to the duality theorem developed in the lectures. You form an intuition, make a prediction, and test each idea yourself.
This course builds enough intuition to begin reading research on condensed mathematics. It assumes no mathematical background: World 1 starts with three ordinary objects on a table.
Each idea is presented as a brick with three stages:
- Concept — a direct explanation of the idea.
- Intuition — a familiar way to picture it, followed by a question you answer before seeing the explanation.
- Experiment — an interactive simulation or a short hands-on task, followed by the standard terminology and a plain-language reading of the notation.
There are no scores or wrong turns. Your progress is saved in this browser, so you can pause at any point and continue later.
- World 1Four familiar ideas
This world gives you four tools for everything ahead: collections, maps, sameness, and addition. They start simple, but their usual relationship will soon break.
5 bricks · 1 simulation
- World 2Nearness and a one-way bridge
You will build two spaces from the same numbers. Their points match perfectly, but the spaces do not. That mismatch sets up the rest of the course.
5 bricks · 3 simulations
- World 3The algebraic test that fails
You will learn kernels and cokernels, see why they usually detect reversible maps, and watch them fail on the one-way bridge.
5 bricks · 2 simulations
- World 4Study spaces with probes
You will build the subject's basic test objects from finite stages and use one to separate spaces that point-counting cannot tell apart.
5 bricks · 3 simulations
- World 5The answer table becomes the object
You will replace a space with its answers to probes, see why no detectable information is lost, and meet useful objects that have no points.
5 bricks · 1 simulation
- World 6Verify the repair
You will make the missing object visible, see the algebraic test give the right answer again, and confirm that familiar topology survives.
5 bricks · 2 simulations
- World 7Infinite sums with well-defined values
You will see why 1 + 2 + 4 + 8 + ... equals −1 in 2-adic arithmetic, how measures keep sums compatible, and what makes an object solid.
5 bricks · 3 simulations
- World 8From multiplication to duality
You will meet analytic rings, see why the key exponent stays at or below one, learn why compact support matters, and connect the new foundations to duality.
5 bricks · 1 simulation