What is this course about?
A mathematical model is more than an equation: it is a deliberately simplified and testable claim about a system. This course develops the habits needed to move from a scientific question to a useful model - by making assumptions explicit, choosing an appropriate scale, connecting models with evidence, and recognizing the limits of a conclusion.
Three ways mathematical models help us
The course is organized around three central questions. Each theme introduces a family of models, the real systems that motivate them, and the kinds of insight they make possible.
Modeling
Starting from cell cultures, interacting populations, chemical reactions, and transport processes, we choose state variables and derive laws of change at the right level of description.
Models and applications
- Growth and population models
- Reaction networks and enzyme kinetics
- Ecological interactions, stability, and bifurcation
- Transport, diffusion, and biological pattern formation
- Collective behavior, agent-based models, and mean-field limits
What they reveal
We ask whether a state is stable, when a threshold or qualitative change occurs, how waves and patterns emerge, and how individual interactions can create collective behavior.
Modeling
We make the meaning of “better” explicit through an objective, represent real limitations as constraints, and distinguish a fixed plan from a strategy that adapts to what a system does.
Models and applications
- Allocation of limited resources
- Shortest-time paths and variational models
- Energy, interfaces, and gradient flows
- Drug dosage and differential-equation-constrained optimization
- Optimal control, dynamic programming, and feedback
What they reveal
We learn how objectives and constraints shape a solution, and how mathematics helps weigh competing priorities such as benefit and risk, cost and performance, or present action and future consequences.
Modeling
We identify what is observed, what remains hidden, and how random events occur. The goal is not to remove uncertainty, but to describe it well enough to reason and decide responsibly.
Models and applications
- Random walks and first-passage questions
- Markov chains and hidden-state models
- Continuous-time event models and master equations
- Stochastic paths and distribution equations
- Stochastic control, rare events, and risk
What they reveal
We examine how noise changes predictions, how to update conclusions as evidence arrives, and how to make decisions when outcomes are variable rather than certain.
By the end of the course
- Translate a scientific question into state variables, assumptions, and mathematical relations.
- Choose and analyze models of dynamics, decisions, and uncertainty.
- Connect a model with data or other evidence and interpret its predictions carefully.
- Recognize when a model is useful, when it needs revision, and what its limits mean.
If you are interested in using mathematics to understand, predict, and shape real-world systems, you are warmly welcome to join us in Fall 2026.