Offerred: Monsoon 2020, Monsoon 2021, Mondoon 2022, Monsoon 2023, Monsoon 2024

Course overview

The Calculus is a major achievement of humanity without which none of the developments in modern science or technology would have been possible. The course is a second look at the subject. I will say it a second time — it is a second look! So it is expected that you have done Calculus in high school. 

It is an introduction to mathematics. The course is about building the intuition, knowledge and skills required for studying mathematics and other mathematical subjects. It is an `enabling’ course — a pre-requisite for every other mathematics course. It will probably help you develop the mindset for any other science too.

They say that Mathematics is learnt by doing mathematics. Be prepared for a lot of doing. In addition to lectures and discussion sessions, expect to spend 10-12 hours outside of class every week working on the material and working on problems; and more, if you have not done well in math in class 12. 

Offered: Spring 2022, Spring 2023, Spring 2024

Course overview

Linear algebra is one of the most important parts of mathematics, perhaps as important or more than the Calculus. Peter Lax (NYU) has remarked: “If you can reduce a mathematical problem to linear algebra, you can most likely solve it”. The introductory concepts (that we will study in this course) are easy to understand, but are powerful and apply in many contexts. 

The applications to various fields are too numerous to mention. Lets  just say that the modern world would not be so modern without the application of linear algebra,  nor would our understanding of the universe would be where it is now. And neither would mathematics be as developed if all kinds of mathematical objects are not studied using linear algebra.  

However, the course is mostly mathematical. Applications will be to understand some mathematical objects.The subject unfolds slowly over the semester. As you work through the semester, you learn how to extend your intuition of two and three dimensions to work in an n-dimensional space. At the same time, as the title of the main textbook reflects,  this course may be regarded as an introduction to abstract mathematics. So you learn how to apply logical arguments to prove statements.

Its a busy course requiring a lot of effort and enthusiasm. You have to be young and energetic and want to learn a lot very quickly. But why am I saying this? You are Ashoka students. Of course you are young and energetic and want to learn a lot very quickly!

Offered in Spring 2021

Course overview

The word dimension means the number of pieces of information required to identify a point in the space under consideration. For example, your eyes have three types of cones, sensitive to red, green and blue lights, and these three combine in various intensities to make the colors that we see. Thus the world of colors is three-dimensional. In the first course on Calculus, we consider one or two dimensional objects,  with functions of one variable that can be visualized as graphs in the plane. 

We continue the study of Calculus by considering functions that are defined parametrically and then functions of 2 or more variables and vector valued functions. The themes remain the same (quadrature, optimisation, linearization), but visualising what is going on is more complicated.

We will make use of the computer algebra system Sage to assist us in gaining intuition and for mechanical computations. This will get you started on a handy tool to use in understanding the new mathematics that you learn in the future. 

The course is expected to be extremely useful if you study Physics, Chemistry, Computer Science, and of course, Mathematics.

Offered in Monsoon 2025

Discrete mathematics, also known as Combinatorics, is a central field of study in mathematics. It is also very useful in the study of Computer Science, and lays the foundation for computational thinking required for the study of Algorithms and Data Structures, Compilers, Principles of
Programming Languages and Theory of Computation, and even Security. The word discrete is used as opposed to continuous. A better word is concrete (as opposed to abstract) because it combines continuous with discrete.

The theme is calculation and computation. We learn to compute various items, by hand, and by using a computer. There is a large variety of combinatorial objects to study, including graphs, permutations, trees, sets, paths, partitions of sets, partitions of numbers, and more. And then there are techniques: inductive thinking, recurrence relations, generating functions, probability calculations, big Oh calculations, summing series and solving equations of various kinds. These are called Manipulatorics, sometimes with affection, and at other times, derision.

The content is taken from the mathematical preliminaries section of Knuth’s The art of computer programming, who collated the classical material and made it accessible. The focus in this course is on techniques to solve problems. These are techniques of Euler, Gauss, Jacobi, Ramanujan — all gifted human computers. One expected outcome of the course is that you will become a better problem solver.
In fact, if you work hard, you will feel yourself becoming clever!

Offered: Monsoon 2020, 2021, 2022

Course overview

This course is an optional course for the Math program. It was slightly different each time it was taught, and it was offerred with different names each time. The following is the course overview of the previous version (2022).

Combinatorics is a central field of study in mathematics, with applications to Computer Science, Physics and Chemistry and of course to other parts of mathematics. We will examine many combinatorial objects and the techniques used to study them. The central question one can ask about the objects studied here is whether one can find how many they are (enumeration); and if two sets of objects have the same number of elements, then can one find a way to prove this by pairing an object from one set to the other (i.e., give an explicit bijection). We can also seek partial solutions, or solutions in terms of generating functions. There is a large variety of objects, including graphs, permutations, trees, sets, paths, partitions of sets, partitions of numbers, and many more. We will study a large sample.

The focus of the course is to learn combinatorial techniques for solving combinatorial problems. One expected benefit of the course is help you become a better problem solver.

Monsoon 2023

Course overview

Complex analysis is one of the highlights of a mathematics education. They say that “the shortest path between two truths in the real domain passes through the complex domain”. Theorems and proofs in complex analysis, when you first encounter them, appear to be indistinguishable from magic! 

However, all this only happens when one becomes adept at combining three different skills — analytical skills, algebraic manipulations (of series and products), and geometrical thinking. The goal of the course is to learn some of the basic moves so that you can begin to learn how to make magic yourself. 

This course is mandatory for Mathematics students. It may be useful for Physics students too as background material and for Computer Science students interested in the analysis of algorithms. That said, we may only be able to cover the theoretical background, and may only hint at some applications.