**Instructor:** Tobias Holck Colding

**Email:** colding [at] math.mit.edu

**Office:** Room 2-369

**Lectures:** TR 9:30 - 11:00 (2-142)

**Text Book:** Notes from forthcoming:
Tobias Holck Colding & William P. Minicozzi II: "Heat equations in
analysis, geometry and probability".

## Course Information

This course will cover the heat equation and equations modelled by the heat equation. The exact topics covered will depend on student interest.

In its most basic form, the heat equation is a partial differential equation that describes the evolution of temperature in space over time. Physically, the equation can be deduced from the number of heat particles at a given point and time.

This and similar equations play a central role in diverse fields, such as physics, economics, information theory, engineering and mathematics. We will study both the classical continuous heat equation, as well as discrete heat equations on lattices or more general graphs.

The goal is to provide an introduction to the heat equation and its generalizations, including connections to geometry, probability, game theory, material science, and mathematical finance.

One of the fundamental phenomena of the heat equation is diffusion: heat spreads out over time as the temperature becomes more and more constant. On a finer level the distribution of heat particles approaches a scaled Gaussian distribution. This is the most basic instance of the central limit theorem and already has implications for functional inequalities.

In the classical heat equation, the heat particles move independently and do not interact. If the particles interact with each other, then the situation becomes much more complicated and the evolution is governed by a nonlinear PDE. This is seen, for instance, in the evolution of interfaces where due to interactions the governing principle is a nonlinear version of the heat equation.

Our focus will be all of these equations, linear as well as nonlinear, discrete as well as continuous, and their basic properties.