Instructor: Christophe (Chris) Golé

Office: Burton 316

Telephone: (585) 3875

Email & www: cgole@math.smith.edu & www.math.smith.edu/~cgole

Office hours (tentative): Mondays & Wednesdays, 10 -11, Tuesdays & Thursdays, 3-4

Class meeting: T-TH 1 - 2:50, Burton 307

Text: Geometry from a differentiable viewpoint, J. McCleary, Cambridge University Press.
 

About the material: Geometry is the field where the abstract, axiomatic-deductive approach to mathematics was developed for the first time (by Euclid). At the same time, geometry has been driven by applications: cartography is a good example. Nowadays, the language of differential geometry is central to physics, from general relativity to quantum field theory. This class  will attempt to bring together these two aspects, as they played out in the development of modern differential geometry.  We will raise fundamental questions such as: What is a geometry? How can we know the shape of the space we live in?  And a more technical  question that took more than 2000 years to settle: does Euclid's Postulate V (equivalent to: from a point out of line L, one can draw a parallel line) follow from his Postulates I-IV?  In their proof that this is not the case, Gauss, Bolyai and Lobachevskii formed a new, non-euclidean  geometry. However, their geometry lacked a concrete, rigorous model. This motivates the rest of the course, where we develop the tools of differential geometry, which will eventually allow us to give a concrete model of non euclidean geometry (the hyperbolic plane). In the process, we study curves in the plane and space, their arclength and curvatures, surfaces with their different notions of curvature, geodesics on surfaces, the Gauss-Bonnet theorem (which gives a beautiful connection between topology and geometry), surfaces of constant curvature, intrinsic surfaces (2 dimensional manifolds) and finally models of hyperbolic geometry. You may consult the  tentative schedule of lectures for more details.

Your  participation:  Because of the historical approach the text follows you will be asked to read some chapters (especially  Chapters 2-5), discuss them in class and write small essays about them. The exams, and more specially the final, will also comprise essays on the history of the development of differential geometry.  Apart from your discussion on the readings, your oral participation may include one short presentation on some topic of your choice (the proof of a theorem, some section we could not cover in class, some computer experiment you made). Otherwise, half of your grade will come from homework problems. You are encouraged to form groups for the homework,  some of which you may find challenging. You should make clear arrangements so that eachmember contributes to eachassignment.
 

Grading:
 

What	When	%
Oral participation	In class	10
Homework	Weekly	50
Midterm	October 24--31	20
Final	Due December 22	20

 

Attendance: Attendance to all the classes is very important. I would appreciate your writing me an email message or calling me if you cannot make it to a class.