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Foundational Courses

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The Math Club will be organizing a set of Foundational Courses starting from the Fall 2021 semester, aimed towards first year students, where some topics which are considered fundamental and pre-requisites of quite a few talks / projects yet are not covered in the first few semesters of present B. Math Curriculum will be covered. Every first year student is urged to attend them. Future math club talks and activities as well as projects which students might undertake might have these topics as pre-requisites. They will also provide some intuition (beyond the usual textbook content) behind some basic ideas involved in each topic.

Note, however, that these talks will be open to each and every member of ISI Bangalore and beyond, so everybody is welcome to attend them!

Fo231: The Probabilistic Method: Moments and Tails - A very brief introduction #

Number of Talks: 1

  • Speaker: Saraswata Sensarma (B. Math, 2024)
  • Abstract: The first half of the talk will be on moment methods. We would introduce the first and second moment methods and demonstrate some of its applications in combinatorics. These are elememtary techniques, often instrumental in proving the (non-)existence of certain combinatorial objects.
    The second part of the lecture will be focused on estimating the tail probabilities using Markov’s and Chebychev’s inequalities. Using this, we would give a short proof (due to Tutte) of a theorem of Hardy and Ramanujan involving the number of divisors of a number. We would end with an extension of this result - the Erdos-Kac Theorem.
  • Pre-requisites: There are no prequesites as such. Anybody with a working knowledge of elementary discrete probability will have no difficulty following the presentation.
  • Video: Available here.
  • Schedule: Sunday, 3rd December 2023, 3:00 PM

Fo223: Rings and Modules #

Number of Talks: 5

Target #

We will start out with the basic definitions and properties of rings and branch out to important concepts such as ideals and domains. Following this, we will see various applications of rings regarding primes and irreducibility in polynomials. Further ahead, we will see the concept of modules and conclude by discussing canonical forms of matrices.

Schedule #

  • Talk 1: An Introduction To Rings
    Speaker: Sanchayan Bhowal (B. Math, 2023)
    Abstract: A ring is a commutative group under addition that has a second operation: multiplication. These generalize a wide variety of mathematical objects like the integers, polynomials, matrices, modular arithmetic, and more. In this lecture we will take an in depth look at the motivation for rings and its properties.
    Date and Time: Saturday, 12 February 2022, 10:00-11:00 AM.

  • Talk 2: Further Examples and Factoring
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: We build on the notions and ideas of rings and ideals developed in the previous talk to introduce two special families of ideals: The maximal and prime ideals and study some of their important properties. We then take a look at three important families of rings: ED’s, PID’s and UFD’s and study their properties in an attempt to generalize a lot of elementary number theoretic concepts such as primes and divisibility to a more abstract setting.
    Date and Time: Sunday, 13 February 2022, 10:00-11:30 AM.

  • Talk 3: Maximal Ideals and Integral Domains
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: Continuing from where we left off in the previous talk, we dive further into maximal ideals and integral domains.
    Date and Time: Sunday, 20 February 2022, 10:00-11:30 AM

  • Talk 4: Gaussian Primes and Irreducibility of Polynomials
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: Having seen the notions of primes and divisibility within rings in the previous talks, we will take a further look into the Gaussian Primes and irreducibility of polynomials.
    Date and Time: TBD

  • Talk 5: Introduction to Modules and Fundamental Theorem of Finitely Generated Modules
    Speaker: Aprameya Girish Hebbar (B. Math, 2023)
    Abstract: In this talk, we will introduce Modules. To begin, we examine and characterize the modules that have a basis. After that, we will discuss finitely generated modules and the fundamental theorem of finitely generated modules. Along the way, we will learn how to solve systems of linear equations over integers.
    Date and Time: Sunday, 27 March 2022, 10:00-11:30 AM

Fo222: Metric Space and Topology #

Number of Talks: 2 (May have extra sessions if need be.)

Target #

We will start out with a generalized sense of distance and lift some notions from analysis to this setting. We will study sequences, limits, continuity and other concepts. This will lead to the definition of Open and Close sets. Followed by this, we will try to see why these notions need to be abstracted further and define what is called a “topology” on a set. The rest of this course would focus on an elementary consideration of topologies and constructions on them.

Pre-requisites: Basics of Real Analysis and Set Theory.

Schedule #

  • Talk 1: An Introduction To Metric Spaces And Continuous Functions
    Speaker: Srigyan Nandi (B. Math, 2023)
    Abstract: Elementary real analysis has introduced us to several properties of \(\mathbb R\) which are largely independent of the algebra imposed on it, such as the notion of convergence, that of a continuous function and the properties of its subsets that are preserved under such functions. In this talk, we will first see how the notion of a natural ‘distance’ on \(\mathbb R\) largely allows one to study the aforementioned properties. We will also try to extend this notion to more general and arbitrary sets in an attempt to do “analysis” on them. We will see how one can extend and generalize the notion of a continuous function and convergence on arbitrary sets and study the properties preserved by these functions on its subsets, by extending the notion of a ‘distance’ to arbitrary sets.
    Date and Time: Saturday, 11 December 2021, 3:00-5:00 PM.

  • Talk 2: An Introduction to Topology
    Speaker: Harshul Khanna (B. Math, 2022)
    Abstract: Topics to be covered - Introduction to Topological spaces: broadest regime on which continuity notions hold, continuous functions, connectedness, other useful notions of a topo space, notion of homotopy and studying spaces using group invariants: fundamental group.
    Date and Time: Sunday, 12 December 2021, 3:00-5:00 PM.

Fo221: Introduction To Group Theory #

Number of Talks: 5

Target #

To introduce the concept, motivation and basic structure of groups. A formal treatment, mostly without proofs, will also be done so that, by the end, attendees are familiar with most of the group theoretic concepts usually deemed as pre-requisites in some talks. It will also serve as a foundation for self reading/projects later on.

Pre-requisites: Naive Set Theory. Working knowledge of Linear Algebra and Number Theory will also be helpful.

Schedule #

  • Talk 1: Motivation for Group Theory
    Speaker: Sanchayan Bhowal (B. Math, 2023)
    Abstarct: Intuitive introduction to Group Theory and a few underlying concepts. Will serve as a foundation for the following talks.
    Date and Time: Sunday, 21 November 2021, 3:00-4:10 PM.

  • Talk 2: A Formal Introduction to Groups
    Speaker: Md Rahil Miraj (B. Math, 2023)
    Abstarct: Following the foundation laid by the previous talk, this talk would deal with the formal definition and treatment of groups. Starting with a few definitions, certain remarkable yet elementary facts will be proved.
    Date and Time: Friday, 26 November 2021, 5:00-6:10 PM.

  • Talk 3: Further Examples and Isomorphism Theorems
    Speaker: Pratichi Paramita (B. Math, 2023)
    Abstarct: Working on the foundations laid in the last class, we will revise the concept of quotients, look at some examples of classes of groups and develop the theory of group mappings. Our study would culminate in a brief consideration of Isomorphism Theorems.
    Date and Time: Friday, 3 December 2021, 6:00-7:10 PM

  • Talk 4: Conjugacy and Group Action
    Speaker: Saheb Mohapatra (B. Math, 2023)
    Abstarct: The target of this talk will be to discuss conjugacy in groups and motivate the class equation of a group. This would be followed by a very basic yet very useful theory of Group Actions, something which was introduced in the first class along with applications.
    Date and Time: Sunday, 5 December 2021, 3:00-4:40 PM

  • Talk 5: Structural Results in Group Theory
    Speaker: Snehinh Sen (B. Math, 2022)
    Abstarct: In this final talk, we will look at some structural results in group theory. We would try to develop some partial converses of Lagrange’s Theorem and look at what simple groups are. This would be followed by a brief look at composition series of a group.
    Date and Time: Tuesday, 7 December 2021, 5:30-7:10 PM