Math 595 Section TF (Tube Formulas in Convex,
Riemannian and Metric Geometry)
Weyl's tube formula computes the volume of a small tubular
neighborhood of a smooth submanifold of Euclidean space. Remarkably,
the volume of the scale r neighborhood is a polynomial in r, whose
coefficients encode various curvatures of the submanifold. There are
similar formulas in convex geometry for the volumes of neighborhoods
of convex sets, or more generally, Minkowski sums of convex sets.
These sometimes go under the name quermassintegrals or mixed volumes,
and go back to the work of Steiner. In this course we will
discuss various tube formulas for neighborhoods of convex sets and
neighborhoods of smooth submanifolds in Riemannian manifolds. At the
end of the course, time permitting, we will indicate some recent
extensions of these ideas to fractals and to sub-Riemannian manifolds,
specifically, the sub-Riemannian Heisenberg group.
- Instructor: Prof. Jeremy Tyson
- Office Location: 329 Altgeld Hall
- Office Phone: 244-4132
- Office Hours: by arrangement
- Lecture Times: TueThu 9:30-10:50, starting October 24
- Lecture Location: 445 Altgeld Hall
- Course web site:
- Textbook: There is no required textbook. Material
will be drawn from a number of sources. I will be writing a set of
lecture notes for the course.
- Prerequisites: A working knowledge of real analysis and
differential geometry (at the level of Math 540 and Math 518)
will be assumed. Familiarity with Riemannian geometry will be
helpful, but we will review all necessary tools of Riemannian
geometry as needed. In the final part of the course, some prior
exposure to Lie groups will be helpful.
- Presentations: Each student is expected to give a 15 minute
presentation and submit a written summary on some topic related to
the material of the course. Some ideas for possible topics for
these presentations will appear in the lecture notes. Student
presentations will take place at the time of the scheduled final
exam for the course (Friday December 15, 1:30-4:30 pm).
- Emergency Information:
- Lecture Notes:
- Announcements: There will be no class during the week of November 6-10. I
will schedule additional times to make up for the three hours missed
during this week.