Course Information

Lecturer:

Associate Professor, Julie Rowlett, Chalmers University of Technology

Course Synopsis:

1. Lecture 1

Pre-Lecture Notes

We’ll begin with the very basics.

Suggested reading for this lecture is The Atiyah-Patodi-Singer Index Theorem by Richard Melrose, Chapter 7.

  1. The heat kernel: what is it?
  2. Explicit computation of the heat kernel in Rn.
  3. Heat spaces: what are they?
  4. The heat space for Rn.
  5. Properties of the heat kernel on Rn.

Post-Lecture Notes

2. Lecture 2

Pre-Lecture Notes

Moving right along, we will next consider compact smooth manifolds.

Suggested reading is The Laplacian on a Riemannian manifold, by Steve Rosenburg, Chapter 3.

  1. Duhamel construction of the heat kernel.
  2. The short time asymptotic expansion of the heat trace.
  3. Geometry captured by the heat trace.
  4. Spectral invariants and “hearing things.”

Post-Lecture Notes

3. Lecture 3

Pre-Lecture Notes

We will continue with the notion of “hearing things” as well as increasing the geometric complexity. Now we will consider smoothly bounded domains in Rn and we may consider heat kernels associated to Schrodinger operators on Rn with compactly supported L potentials.

Suggested reading is Can one hear the shape of a drum, by Mark Kac.

  1. Locality principles (generalizations of Kac’s principle of “not feeling the boundary”)
  2. Kac’s “hole.”

Post-Lecture Notes

4. Lecture 4

Pre-Lecture Notes

Today we will finally arrive at non-smooth geometric settings: domains in Rn which have non-smooth boundary. We may also look at manifolds with singularities. In this context we will investigate

  1. Locality principles
  2. Hearing singularities

Post Lecture Notes

5. Lecture 5

Pre-Lecture Notes

In conclusion we shall learn about the dangers of heat, specifically

  1. Infinite speed of propagation
  2. Randomness

Post-Lecture Notes

Lecturer Biography

Heat Flow and Geometry

Associate Professor, Julie Rowlett, Chalmers University of Technology

Julie Rowlett is an American mathematician working and living in Gothenburg, Sweden (or Göteborg, Sverige in Swedish).  She studied mathematics at the University of Washington and continued to earn her Ph.D. in mathematics under the supervision of Rafe Mazzeo at Stanford University in 2006.  Her research interests include: geometric, microlocal, and harmonic analysis; spectral theory; differential geometry; partial differential equations; dynamical systems; mathematical physics; game theory; and applications.  Julie spent five years in Germany before moving to accept a position as associate professor in Gothenburg, Sweden.  She works in the joint mathematics department of Chalmers University of Technology and the University of Gothenburg, the largest mathematics department in Sweden.  In 2017, she was nominated to serve on Sweden’s National Committee for Mathematics.  Julie also keeps busy with a large international network of research collaborations, in North and South America, Europe, the Middle East, Asia, and Oceania.  When not busy with mathematical activities, her hobbies include studying languages (French, German, Mandarin, Korean, Russian, and of course Swedish), practicing the Korean martial art of Tang Soo Do, scuba diving, making and listening to music, dancing, and cooking.