01/nov/2015 Information about the composition has been posted here.
3/nov/2015 The problem set 2 is available.
5/nov/2015 Solutions of the previous non-mandatory problems and statement of a new one.
12/nov/2015 A new non-mandatory problem.
17/nov/2015 Yet another non-mandatory problem.
26/nov/2015 More comments about the composition here.
Important: If somebody is willing to attend the exam instead of handling the composition, please let me know in advances because I have to teach on Thurday 21.
No lecture on Tuesday 19. To deliver problem set 3, use email or leave a copy in Secretaría de Matemáticas to be put in my mailbox.
Course detailsThe contents, according to the guía docente employed last year is:
In case you miss a lecture a brief summary of each of them wil be posted here.
It is intended to use mechanics and other topics on Physics as motivation and to provide examples and applications. No prior knowledge on Physics is assumed beyond the very basic concepts.
The lectures are scheduled on Tuesday and Thrusday 4:00pm - 5:30pm in room 320 (Building 17, Science Faculty, Mathematics).
For general reference, please check the master program website in the Department of Mathematics.
In this course we follow mainly the notes in Spanish Geometría Diferencial. They do not cover De Rham cohomology that is included Geometría IV 2008-2009. Another main source is the book by do Carmo on Riemannian geometry.
Occasionally some files will be posted in the table below, with summaries of some topics.
Corrections and comments are wellcome.
ExercisesThe following sheets are home assignments that contribute to the grading.
The grades for officially enrolled students can be checked in the Moodle server after the home assignments are corrected.
From time to time some non mandatory exercises will be proposed.
The grading is as follows:
1) Home assignments 50%.
2) Final exam or extra activities 30%.
3) Quizzes, in-class exercises, participation 20%
More details will be given along the course.
"Extra activities" in 2) means a brief composition about a topic related to the course.
About the composition: It is a short essay of 10-15 typed pages. The idea is to give an overview about a topic not covered in the course. You are not expected to give fine details. You should emphasize the general ideas showing that you know what is the topic about. The deadline is 14th January 2016.
(Added November 26th) The main guideline for the composition is to emphasize the ideas. You should try to show that you have understood the motivation of the topic, the main ideas, the achivements, the targets, the conjectures. Please do not try to be very technical. I'd prefer a composition addressed to an educated layman reader. A couple of you have chose topics so far from my knowledge that this is close to reality.
Roughly speaking, you should write theorems but do not forget to show me that you have learned something beyond the scheme lemma-theorem-proof that I can find in any book. Originality is a crucial point and it will be considered in the assessment.
You can choose freely any topic related to the course (even loosely. If it is the case, ask for confirmation). The following list will be updated from time to time.
Some of you have asked about bibliography. I include here something but, please, do not follow it blindly because my knowledge is very limited in some of the topics and finding the good references is part of your work. Your main task is not understanding a topic on a book but using different books and papers (including the one you understand) to give an overview.
1) Warner. "Foundations of differentiable manifolds and Lie groups".
2) S.-S. Chern, "On the curvatura integra in Riemannian manifold", Annals of Mathematics 46 (4): 674-684. (Understanding the full proof can be difficult. It is enough for the composition to follow a sketch of the ideas and to understand the elements appearing in the statement and its relation with the 2 dimensional case).
M. Taylor. Partial differential equations. Volume II Appendix C.
C. Zhu. The Gauss-Bonnet theorem and its applications. https://math.berkeley.edu/~alanw/240papers00/zhu.pdf
There are some brief comments in Berger. "A panoramic view of Riemannian geometry"
3) Arnold. "Mathematical methods of classical mechanics".
5) Do Carmo. "Riemannian geometry".
Cheeger & Ebin. "Comparison theorems in differential geometry".
6) Gualteri. http://www.math.toronto.edu/mgualt/MAT1300/1300%20Lecture%20notes.pdf
Hirsch. "Differential topology".
It is worthy to look Whitney's original papers.
Other topics that you have considered for your composition:
Schutz. "A first course in general relativity".
Misner, Thorne & Wheeler. "Gravitation" (this is a very thick book but it is not an encyclopedia. The explanations are good and it is readable in general).
Bott & Tu. Differential forms in algebraic topology.
Lie algebras and physics:
H. Georgi. "Lie algebras in particle physics from isospin to unified theories".
M. Hammermesh "Group Theory and its Application to Physical Problems"
These can be hard for a first encounter with physics. Just understanding the relation between SU(2), SO(3), spin and angular momentum in quantum mechanics can be the topic for a composition if you do not know too much about physics. It appears in many of the books on quantum mechanics.