Exercises, Set 2 (Homotopy)

The second set of Exercises can be found at this link. Solutions will be posted next week.

Solutions to Exercise Set 1

The solutions to Exercise Set 1 are available at this link. Many thanks to Renjan John for working them out and typing them!

Exercises, Set 1 (Topology)

The first set of Exercises can be found at this link. Solutions will be posted next week.

Clarification about definition of continuity

 A question that came up in the lecture of Friday June 16 was: why do we say functions are continuous if the inverse image of every open set is open? In particular why do we consider the inverse, rather than the forward image?

To gain intuition, consider functions f: R -> R with the usual topology on R. It turns out that even for a continuous function (in the normal calculus definition), the forward image of an open set need not be open. To see this, look at the function f(x)=x^2. Take the open interval (-1,1) in the domain of the function. Its image under f is the interval [0,1). This contains the point 0 and therefore is not an open set! The same can be true for the forward image of a discontinuous function such as f(x) = x+1, x < 0 and f(x)=x+2, x>=0, which is the example considered in class. The forward image of (-1,1), for example, is (0,1) U [2,3). Again this is not open (I may have said something wrong about this case on Friday). The message is that looking at the forward image of an open set tells us nothing useful about the continuity of a function R -> R.

The inverse image works much better. For any function R -> R that is continuous in the sense of calculus (both limits are equal at every point and are equal to the value of the function), the inverse image of an open set in the usual topology on R is always an open set. This is a theorem, the proof can be found in Section 1.4 of the book by Singer and Thorpe. You are encouraged to work it out.

Given the above results on continuous functions from R -> R, we abstract the property that the inverse image of open sets is always open, and use this to define continuity on an arbitrary topological space.

Lecture format and some suggestions to students

I would like students to continue asking questions, that is an essential part of a live lecture. At the same time, it would be best to spend a few minutes thinking about your question and trying to answer it yourself before you ask it. That is what makes one into a real scientist. It is a good idea to write down the question on a piece of paper and think about it patiently. You can still ask it after some time if you are unable to make progress with it.

You are welcome to post your questions in the chat box, but from the next lecture (July 14) onwards, I will keep that window closed while I lecture. At suitable break points in the lecture I will pause, open the chat box and answer a few questions.  

After every lecture, interested students are expected to look at the relevant portion of the textbook, work out the definitions carefully and solve the exercises. Sometimes additional exercises will be posted on this page. A significant participation from you will help you benefit from the lectures!

Reference Material

The basic reference for the course is Part I of the textbook: 

• Lectures On Advanced Mathematical Methods for Physicists by Sunil Mukhi and N. Mukunda. 

The lectures will follow the plan and style of the book quite closely.

Some exercises are provided in this book. Dr Sachin Jain and Dr Renjan John have kindly agreed to help out by providing exercises (with solutions) that will be posted on this page. 

Other textbooks aimed at physicists are:

• Topology and Geometry for Physicists" by Charles Nash and Siddhartha Sen,
• "Geometry, Topology and Physics" by Mikio Nakahara.

They are at a slightly higher level than this course.

A very useful textbook aimed at mathematicians is:

• "Lecture Notes on Elementary Topology and Geometry" by Itzhak Singer and John Thorpe. 

I used this textbook extensively when preparing my own book. It contains proofs of all the main results that will be introduced in the present course. Proofs are of course essential to mathematicians, but for physicists who want a conceptual/practical grasp of the subject, they will often be skipped in the course.

Welcome to the course

 

Welcome to this course! This page will have all the academic material related to the course. Note that administrative details like the Zoom link etc will be handled by the host institution, IIT Gandhinagar.

Here are the course details. 

Dates and times for lectures: Starts from July 12th, Three lectures per week, Monday, Wednesday and Friday, 4 PM - 5:30 PM 

Venue: Online (Zoom)

About the course:

This is an informal introduction to Topology and Differential Geometry for physicists. It will start by presenting a brief survey of the key ideas in elementary topology and will then go on to survey the basics of homotopy. Thereafter, differentiable manifolds will be introduced together with a survey of calculus on manifolds. We will then move on to Riemannian geometry, curvature and differential forms. In the last part we will discuss aspects of fibre bundles. Homology and cohomology may be discussed on the way if there is interest.

Day-wise lecture plans:

(Last edit - 2/8/2021.)

Lecture 1: Motivation and definition of topological space. 

Lecture 2: Metric topology, basis for topology, limit points and closure, connected spaces. 

Lecture 3: Compact spaces, continuity, homeomorphisms. 

Lecture 4: Loops and homotopies, fundamental group.

Lecture 5: Homotopy type, contractibility, examples.

Lecture 6: Definition of differentiable manifolds, differentiation of functions.  

Lecture 7: Calculus on manifolds: vector fields. 

Lecture 8: Differential forms, properties of vector fields and forms.

Lecture 9: Riemannian metric, frames.

Lecture 10: Connections, curvature, torsion, isometry.

Lecture 11: Integration on manifolds, Stoke's theorem, Laplacian.