A ``dangerous idea seminar" framework has been a MIT tradition for decades, where the speaker
has to structure the talk around five questions based on Fear, Joy, Mom, Cool, New.
The questions can be adapted also to each lecture. Lets do that:
1. Why should I fear the topic? Partial differential equation ideas are used in any technology,
this includes face recognition, building weapons etc. But this can be said about essentially any science.
But you know what is probably the most scary thing about PDE's: the topic is not easy!
It is a quite technical area of mathematics.
Also when studying the topic with a computer, one has to deal with complicated numerical frame works.
One has to work hard in order to make numerical approximations which are robust and for which
the numerical solution is close to the actual solution one sees when one makes the experiment.
2. Why should I rejoice it to be done? Partial differential equations are the fabric we are made
of. Understanding the Schrödinger equation allows us to understand elementary particles.
Just an example. If one looks at the Energy operator L of a Hydrogen atom, then the structure of the
eigenvalues describes the periodic system of elements.
Partial differential equations are used to predict the weather, the paths of hurricanes, the impact
of a tsunami, the flight of an aeroplane. They are used to understand complex stochastic processes.
Partial differential equations appear everywhere in engineering, also in machine learning or
statistics. A generalization of the transport
is the gradient flow which is used to get good solutions to problems. As mentioned at the beginning, all
fundamental laws in nature (classical mechanics can be described by variational problems leading to partial
differential equations, the Euler equations, general relativity is described by the Einstein equations,
gravitational waves emerging from a black hole merger recently measured out allows us to see what happened
billions of years ago in an other part of the universe, diffusion equations can help to track the spreading
of a virus or environmental disaster like an oil spill).
3. What should I tell my mom about it?
Partial differential equations allows us to look into the future and allows us to take action in order
to avoid difficult situations. Similarly as ordinary differential equations allow us to predict
how far an asteroid zooms by the earth, we can build and use models to predict how the climate changes,
we can take measures to soften the impact of a storm, or use it even for rather mundane things like
how to make money (or lose some ...) on the stock market.
4. What is a cool discovery in that field?
One of the fundamental results is the theorem of CauchyKovalevski which assures a system of partial
differential equations with analytic functions as coefficients has a unique solution. This is quite
subtle, as analyticity is stronger than just smoothness.
Analytic functions are functions which have a Taylor series which converges.
5. What is a recent discovery in the subject?
A rather recent discovery is a result of Gregory Perelmann which tells that a simply connected
bounded three dimensional space must be a three dimensional sphere. The theorem was proven using
some sort of heat equation acting on a curvature functions.
Given the space, the space is deformed by applying the heat flow. The flow smooths out the space, making
it round. The limiting shape is then a sphere, like the bubbles seen in the lava lamp placed on the
table during the lecture.
