Mathematics

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Review by Navdha

Motivation for undertaking minor

I’d enjoyed all of the first year math courses (with the exception of 108) - particularly MA109 (now 105) - and wanted to study more of such math in the 2nd year. Also felt that a math minor would complement the physics courses that I was going to be taking (it did).
If you liked the first year math courses (esp. 105), and/or are interested in theory, it may be helpful, though often not directly. It teaches a lot of abstract concepts, and if you think that's something you'd like, go for it surely (but make sure to be sincere and devote time to it).

Overview of Minor

Unlike many other minors, there aren't many choices in terms of courses. There are a total of five courses of eight credits each, of which you've to complete four to get the minor degree. Real Analysis is a prerequisite for two of the other courses - General Topology and Fourier Analysis & Applications; and General Top. and Complex Analysis run alternately every spring semester.
All courses run in slot 5, added to which there’s a tutorial slot on Wednesday evenings, which is customarily converted to an hour (and a half) long lecture slot by the prof.
(Note: Since the courses are 8 credits, you might find yourself short of 2 credits from taking an extra course more often than you’d like. So while your friends might be using up all their 54 credits in some sem, you might be stuck at 50 because of the 2 extra credits. )
CPI cutoffs are usually not much, I think, the highest being for RA (somewhere around 8.5, refer Course Info Booklet), which should be negotiable if the prof is willing. The class strength starts off at close to 20, which is the official limit, but drops down to below 15 by the end of most courses. Low class strength has its own advantages, personal interaction with the prof being one, and allows for better and more engaging class discussions.
All the minor courses require a fair bit of regularity, for skipping one class means that you'd find yourself dozing off in the next if you go without preparation - each class builds up on the next.
In any case, it’ll help you develop a rigorous and abstract way of thinking, which might prove helpful in some of your core courses or theory readings (it did for me). You’ll also be practising writing proofs for a major part of the minor, and it’ll be helpful to quickly identify what constitutes a good proof and what does not, which is mostly overlooked in the core courses (sometimes even in core MA courses that your department requires you to take), but is integral to theoretical research.

Courses Undertaken

• MA403 : Real Analysis
  Taken by Prof.Sandip Singh (2022-23 Autumn)

  Prerequisite

  None

  Course Content

  Review of basic concepts of real numbers: Archimedean property, Completeness.Metric spaces, compactness, connectedness, (with emphasis on Rn).Continuity and uniform continuity.Monotonic functions, Functions of bounded variation; Absolutely continuous functions. Derivatives of functions and Taylor`s theorem. Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions.Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration. Fourier series, pointwise convergence, Fejer`s theorem, Weierstrass approximation theorem.

  Grading Scheme

  2x10 for quizzes
  30 for midsem
  50 for endsem

  Overview

  Real Analysis is possibly the most relevant from the perspective of an engineering student, and it’s better to take it first since it’s also a prerequisite for two others. The prerequisites might not be strict, but it is always helpful to start with something that is closely related to what you've already studied, especially when it comes to deciding if you would want to continue with a minor like math. We primarily followed Rudin for the course, and most of the problems in the tutorial sheets were from the book itself.Class notes and tutorials were sufficient for exams, many problems in the midsem and endsem were a direct repetition of tut problems.

• MA406 : General Topology
  Taken by Prof.Sandip Singh (2022-23 Spring)

  Prerequisite

  Real Analysis

  Course Content

  Prerequisites: MA 403 (Real Analysis) Topological Spaces: open sets, closed sets, neighbourhoods, bases, sub bases, limit points, closures, interiors, continuous functions, homeomorphisms. Examples of topological spaces: subspace topology, product topology, metric topology, order topology. Quotient Topology: Construction of cylinder, cone, Moebius band, torus, etc. Connectedness and Compactness: Connected spaces, Connected subspaces of the real line, Components and local connectedness, Compact spaces, Heine-Borel Theorem, Local -compactness. Separation Axioms: Hausdorff spaces, Regularity, Complete Regularity, Normality, Urysohn Lemma, Tychonoff embedding and Urysohn Metrization Theorem, Tietze Extension Theorem. Tychnoff Theorem, One-point Compactification.Complete metric spaces and function spaces, Characterization of compact metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Category Theorem. Applications: space filling curve, nowhere differentiable continuous function. Optional Topics: Topological Groups and orbit spaces, Paracompactness and partition of unity, Stone-Cech Compactification, Nets and filters

  Grading Scheme

  2x10 for quizzes
  30 for midsem
  50 for endsem

  Overview

  we followed Munkres really closely throughout. The content covered was a little less than what BS Math people covered. GT requires you to remember quite a few definitions and results, but it is not something very difficult if you understand the broad idea and the motivation behind them, and this is where having a good prof really helps. Exams were again based on tutorials and the content covered in the class.

• MA419 : Basic Algebra
  Taken by Prof.Preeti Raman (2023-24 Autumn)

  Prerequisite

  Real Analysis

  Course Content

  Review of basics: Equivalence relations and partitions, Division algorithm for integers, primes, unique factorization, congruences, Chinese Remainder Theorem, Euler ϕ-function.Permutations, sign of a permutation, inversions, cycles and transpositions. Rudiments of rings and fields, elementary properties, polynomials in one and several variables, divisibility, irreducible polynomials, Division algorithm, Remainder Theorem, Factor Theorem, Rational Zeros Theorem, Relation between the roots and coefficients, Newton`s Theorem on symmetric functions, Newton`s identities, Fundamental Theorem of AlgebraRational functions, partial fraction decomposition, unique factorization of polynomials in several variables, Resultants and discriminants. Groups, subgroups and factor groups, Lagrange`s Theorem, homomorphisms, normal subgroups. Quotients of groups, Basic examples of groups: symmetric groups, matrix groups, group of rigid motions of the plane and finite groups of motions. Cyclic groups, generators and relations, Cayley`s Theorem, group actions, Sylow Theorems. Direct products, Structure Theorem for finite abelian groups.Simple groups and solvable groups, nilpotent groups, simplicity of alternating groups, composition series, Jordan-Holder Theorem. Semidirect products. Free groups, free abelian groups. Rings, Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains. Polynomial rings over UFD`s

  Grading Scheme

  The course total was 50
  7 for a group presentation
  8 for a take-home quiz
  15 for midsem
  20 for endsem

  Overview

  Probably my favourite course of the four in terms of its content. The prof was rather easy-going in terms of evaluations.Only the endsem had a few new and challenging problems, almost all midsem problems were done in class. The lectures were interesting and easy to follow. The pace was much slower as compared to the other 3 courses - and we solved a lot of problems in class which was fun.

• MA412 : Complex Analysis
  Taken by Prof.Sandeep Singh (2023-24 Spring)

  Prerequisite

  None

  Course Content

  Complex numbers and the point at infinity. Analytic functions. Cauchy-Riemann conditions. Mappings by elementary functions. Riemann surfaces. Conformal mappings.Contour integrals, Cauchy-Goursat Theorem. Uniform convergence of sequences and series. Taylor and Laurent series. Isolated singularities and residues. Evaluation of real integrals. Zeroes and poles, Maximum Modulus Principle, Argument Principle, Rouche`s theorem

  Grading Scheme

  Overview

  This was my 3rd course with Prof. Sandip (and the last quarter of the three of my minor which he taught) and so almost all the evaluations were easily predictable. Attendance was usually really low in the lectures due to which he gave some problems in the exams which we solved in class for the benefit of those who were regular. Same pattern as that of RA and GT. Content-wise you cover more in depth and width compared to what the EP core course covers (which for us was evaluating contour integrals for the major part). This time the lectures were more interactive and fun to attend.

Final Takeaways

Note that unlike minors like CMInDS or CS, where your friends might be doing projects under profs, it is not very common (but certainly possible!) in a math minor (there are a few people from our batch doing this, so check out with them).
Imp: As you would have seen in your first year, the math that you’ll be doing in college is vastly different from what you did back in school and so take up the math minor only if you’re willing to devote time to the subject, and not study just when a panicky frantic bit strikes in the night before the exam. I personally learned a lot from it, and I think it has helped me become mathematically mature. Finally, talk to a lot of seniors and make a sound, calculated decision.

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