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**Email:**`tyson@illinois.edu`**Office Hours:**by arrangement**Lecture Times:**TueThu 9:30-10:50, starting October 24**Lecture Location:**445 Altgeld Hall**Course web site:**https://faculty.math.illinois.edu/~tyson/595f17.html.**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**:- Information from UIUC Police Department regarding emergency situations
- Altgeld Hall Emergency Action Plan
- Information regarding emergency situations

**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.