Instructor:   Andrew Belmonte
Contact info:   322 McAllister Building, telephone: 814-865-2491, email: alb18.AT.psu.edu
Office hrs:   Tues/Thurs 2-4pm (on weeks when I am present)

Penn State Dept of Mathematics - Summer REU Program

The concept of a manifold will be introduced for the specific case of curves and surfaces embedded in Euclidean 3D space. We will start with the parameterization and geometry of a curve, and eventually surfaces. There will be some focus on curves and surfaces that satisfy variational equations, as these often correspond to things in the physical world.

• Differential Geometry: Curves - Surfaces - Manifolds, by Wolfgang Kuhnel (AMS Library, 2002)
• Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo

• The equations for large vibrations of strings, by Stuart Antman, American Mathematical Monthly 87 (1980), pp. 359–370

Lecture	Date	Topics
1	Wed 6/29	Introduction, overview, rapid tour of the course
2'	Thurs 6/30 (1 hr)	Curves and their properties
3	Tues 7/5	Plane curves, Serret-Frenet eqs; piecewise circular curves.
4	Wed 7/6	More on evolutes and involutes; curves in R3; Intro to physical curves - tension
5	Thurs 7/6	Hanging strings - catenaries, the Bernoulli problem, shaking strings
6	Mon 7/11	Bending, buckling, supercoiling; Calculus of variations
7	Tues 7/12	Hamilton's principle, Minimal surfaces,   soap films
8'	Wed 7/13 2pm	Intro to 2D surfaces in R3;   Tour of the Pritchard Lab
9	Thurs 7/14