A more precise title for this course would be "Elements of Geometry and Topology for Theoretical Physicists." It is an introduction to areas of mathematics which find use in modern theoretical high-energy physics but are usually not taught to physics students. The topics to be covered in the course are

- Basic Topology: General Topology, CW spaces, Homotopy, Homology.
- Smooth Manifolds: Calculus on Manifolds, De Rham theory, Fiber Bundles, Connection and Curvature.
- Basic Riemannian Geometry: Parallel Transport, Levi-Civita Connection, Riemann Tensor.
- Characteristic Classes, Index Theorems, Some Notions of K-theory.
- Complex Geometry: Calculus on Complex Manifolds, Sheaf Theory, Holomorphic Vector Bundles, Kahler Manifolds.

same material and therefore is the recommended textbook. I will use other sources as well, and students

are advised to consult them.

**A rough outline of the course:**

*Fall Term:* Basic Topology, Vector Bundles, Characteristic
Classes.

*Winter Term:* Smooth Manifolds, Basic Riemannian Geometry.

*Spring Term:* Index Theorems, Complex Geometry.

There will be homework problems, but no exams. Grading will be done
on the basis of the turned in homework.

The homework posted on week N is due by Friday of week N+1. The TA
for this course is Takuya Okuda;

he is located in 422 Downs. Graded homeworks together with the
solutions
sets will be placed outside my office

(451 Lauritsen).

I will have an office hour in 451 Lauritsen each Friday, from 4:30
p.m.
to 5:30 p.m.

A more detailed outline of the fall term is here
. Besides Nakahara, the most useful books for the fall term are

Massey and Vassiliev.
A copy of Massey's book is on reserve in Millikan Library.

**Homework problems (fall term).**

*Week 1 Week
2 Week 3
Week 4 Week
5 Week 6
Week7 Week
8 Week 9*

A detailed outline of the term can be found here.
Nakahara contains almost all the material for this term,

but those wishing for a more pedagogical exposition may want to consult
Warner.
In fact, during the first half

of this term I will follow Warner pretty closely. Warner does not cover
Riemannian geometry, the theory of

connections on vector bundles, or symplectic geometry. For the first
two topics, one may consult Bishop and

Crittenden (it can also serve as an
alternative
to Warner), or, if one wants an exposition geared towards

physics applications, Curtis and Miller. These
topics are also discussed in detail in a classic monograph by

Kobayashi and Nomizu. As for
symplectic
geometry, I recommend Arnold, Guillemin
and Sternberg, or

Cannas da Silva and Weinstein. I
particularly
like the last of these: it is up-to-date, and contains lots of

fascinating mathematics which could be relevant for string
theory.

**Homework problems (winter term).**

__Week 1__ (due Jan. 17) Week
2 (due Jan. 24) Week 3 (due
Jan.
31) Week 4 (due Feb. 7) Week
5 (due Feb. 14)

*Week 6 (due Feb. 21) Week
7 (due Feb. 28) Week 8 (due March 7)
Week
9 (due March 14)*

I will discuss characteristic classes of vector bundles, index
theorems
for elliptic operators, and their

applications to physics. The rest of the spring term will be devoted
to complex manifolds. I plan to

discuss calculus on complex manifolds, Dolbeault cohomology,
holomorphic
vector bundles, coherent

sheaves and sheaf cohomology, Riemann surfaces, Kahler manifolds, and
in particular Calabi-Yau

manifolds. A useful summary of characteristic classes and
index theorems can be found in

a review by Eguchi, Gilkey, and Hanson, Physics Reports 66, p.
213.
The original papers by Atiyah, Singer,

and collaborators are also quite readable. A good textbook on complex
manifolds is Wells. Chapter 0 of

Griffiths and Harris is a compressed
account
of the first few chapters of Wells. A good new book on

complex manifolds is Zheng. For
Riemann
surfaces, you may consult a book by Miranda,
or a review

by Shokurov. A good textbook on
complex
algebraic geometry (I will barely mention this subject in this course)

is Shafarevich. There is also an
excellent review of modern algebraic geometry by Danilov.

**Homework problems (spring term).**

__Week 1__*(due
April 11) Week
2 (due April 18) Week 3
(due April
25) Week 4 (due May 2) Week
5 (due May 9)*

*Week 6 (due May 16) Week
7 (due May 23) Week 8 (due May
30)*