Concept
Explorations
Four connected modules
Follow a connected sequence of interactive investigations into how mathematics helps us observe, understand, predict, and test complex systems.
Begin with a scientific scene. Carry one mathematical idea into an interactive experiment. Then return to the Earth system and ask what the model can, and cannot, establish.
Many explorations pair with a short video from Applied Mathematics in Geosciences, so you can watch the idea first and then test it yourself.
Browse the video series →The Planet We Never See Directly
How can we reason about a planet whose most important states and mechanisms are never observed in full?
Seeing a Dynamic Planet
Chaos, rhythms, scales, randomness, and structure
The Butterfly and the Forecast
If a system obeys deterministic laws, why can its future become unpredictable?
Exploration 2A Planet of Rhythms
How can a system return without keeping perfect time?
Exploration 3Many Clocks, One Earth
How can processes too small or fast to resolve still shape the evolution we care about?
Exploration 4When Randomness Becomes a Force
Can rapid random fluctuations create slow, structured climate variability?
Exploration 5Finding Structure in the Pacific
What does a reduced climate representation preserve, and what can it leave out?
Learning from an Incomplete Planet
Hidden states, inverse problems, data assimilation, and uncertainty
The Earth Between the Measurements
Can you reconstruct a world you cannot observe directly?
Exploration 7When Models Meet Data
How should an uncertain model forecast and an uncertain observation be combined to estimate a changing state?
Exploration 8How Uncertainty Travels
Where does uncertainty enter, how does it reach a chosen target, and which observations can reduce it?
Predicting a Changing Planet
Probability, extremes, tipping, and AI under change
A Forecast Is a Distribution
What makes a probabilistic forecast useful, trustworthy, and better than a single best guess?
Exploration 10Near the Edge
How can we distinguish a rare excursion, nonlinear amplification, and a transition to another regime?
Exploration 11Can AI Learn the Earth?
When a learned model predicts well, what has it learned, and where might that knowledge fail?
Building and Trusting Models
Model hierarchy, mechanisms, response tests, and trust
The Model Ladder
Why do scientists use a hierarchy of imperfect models instead of searching for one final model of the Earth?
Exploration 13The Next Observation
When evidence is costly, how should we choose the next observation, experiment, or high-fidelity simulation?
Exploration 14From Pattern to Trust
How does a model earn bounded confidence by surviving questions it was allowed to fail?
What Mathematics Can Promise
What can mathematics make visible without pretending that any model contains the whole planet?