Projects 2021

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1 - Probability and Statistics : From Classical to Bayesian

Mentors: Aniket Kumar (aniki#8797), Pranay Agarwal (Pranzer#5431), Munish Gupta (Alexios#2464)
Description : This Project will guide you through the basics of Probability Theory, data generation , decision theory , Bayesian Analysis and then application of Bayesian to real world problems. The project would be of immense help to many as probability theory has wide applications ranging from its use in the corporate world to the realm of research in almost any field. Familiarity with probability theory helps theoreticians and experimentalists alike.
Expected Duration : 2 months
Expected Weekly Commitment: 4-5 hours
Showcase : here

2 - Communicating Mathematics

Mentors: Harshit Bisht (Harshit Bisht#7766)
Description : Disclaimer: This project will very likely be useless for both jobs and MS/PhD admissions. You have been warned. Mathematical concepts are often disconnected from people’s reality, people don’t relate to statistics shared by politicians, economists, public organizations, and other domains that are relevant to everyday life. The project would aim to create examples, counterexamples, simulations, and games to connect math concepts to their relevance to the world at large. Some arbitrary examples:
- Law of large numbers and population statistics : A biased coin that lands H 99% of the time feels like it would never land tails, but if you flip it enough the sample distribution would converge to the true distribution. This is similar to if a country/society takes a 1% individual risk together, in a country of 1B more than 10M are affected.
- Basic reproduction number (R0) and how to understand contagiousness of a disease.
- Why do people talk about dummy candidates in elections? Are there election systems where no one has incentive to start dummy candidates (and what are the flaws of such systems?)
Since this will a project of interest/love more than technical depth, the participating group can be decided based on how many people want to do this. Please note that unfortunately, this will be a project where your communication skills might be barrier, but please don’t shy away from applying if you feel interested.
Expected Duration : 2-3 months
Expected Weekly Commitment: 5 hours

3 - The Dynamics of Life

Mentors: Mohit Kulkarni (returnofhairy#1384)
Description : In this course we wish to discover aspects of mathematics widely used in dynamical systems with application to biology. Nature has beautiful patterns as well as chaos, we wish to understand how dynamical systems work particularly in the brain. Initial weeks would cover basic biology and basic mathematics required thereafter. Here we will try to get an intuituon on why mathematics and biology aren’t completely different areas. Later, we will understand what a network, what chaos is and why dynamics is important. We will look into one/two dimensional dynamical systems(would also include the dynamics of a neuron). Later topics will depend on you and may include RNN’s, large-scale neural models, attractor dynamics etc.

Maths may be difficult to understand, so we will take the help of cool visualisations. Programming knowledge(any language) is a plus but not required.

Some books we may refer to (Tentative)
1. Mathematics of Life, Ian Stewart,
2. Nonlinear dynamics and chaos, Steven Strogatz
3. Dynamical Systems in Neuroscience, Eugene Izhikevich
Expected Duration : 8 weeks
Expected Weekly Commitment: 4
Showcase : here

4 - Number Theory and Algebra

Mentors: Ajay Prajapati (Ajay Prajapati#6448), Hargun Preet Singh Bhatia (Hargun#7217)
Description : The goal of this project is to give a gentle introduction to Modern Number theory. We will cover basic topics like Divisibility, modular arithmetic, Euler’s theorem, fundamental theorem of arithmetic, quadratic residues and Gauss’s famous Quadratic reciprocity law. Along the way, we will introduce several key algebraic objects like groups, rings and fields. If time permits, we will talk about some well known problems in the field like Dirichlet’s theorem for primes in Arithmetic progressions and the Riemann hypothesis.
Expected Duration : 2-3 months
Expected Weekly Commitment: 3-4 hours
Showcase : here

5 - Algorithms based on maths

Mentors: Utkarsh Gupta (Pegasus#3765), Nitin Garg (ZeroDeaths#3886), Yash Gidwani (yashg#3927)
Description : This project is intended to analyse algorithms that are based on maths and discover their applications. We will mainly cover the following algorithms/ techniques:
1. Hashing: It is mainly used to store big arrays/ strings compactly and is thus used for pattern matching/sub-array equality, etc. in an optimal way.
2. Linear Diophantine Equations: We will discuss the conditions for solutions to exist and look for optimal ways to find them explicitly.
3. Gaussian Elimination: It is mainly used for solving linear equations but it has many other tricky applications like finding all xor subsets, etc. which will be discussed explicitly.
4. Probability Theory: We will look at some applications of Linearity of Expectation and formulating recursive relations for calculation of various parameters.
5. Generating Functions: We will learn to solve recursive relations in an optimal way like Fibonacci, etc.
6. FF: A very widely used algorithm to multiply polynomials/n-bit numbers optimally.
These are the algorithms which will be discussed in the project. More algorithms may be introduced depending on the timeline. This project expects you to be familiar with some programming language (preferable C++) and a basic knowledge of algorithms.
Expected Duration : 2 months
Expected Weekly Commitment: 5 hrs
Showcase : here

6 - Mathematical Finance & Stochastic Processes

Mentors: Aditya Prakash Singh (adityapi314#1095), Shaurya Jain (shaurya#9116)
Description : We will first study some basic financial concepts such as time value of money, arbitrage and risk & return. Further, we will study the most basic derivatives prevalent in the market and finally we will learn how these derivates are priced and hopefully derive the celebrated Black-Scholes equation.

Disclaimer - This project will NOT teach you how to trade in the market. It will definitely help you to understand the market and its instruments better.

Pre-reqs: None.
Expected Duration : 3 months
Expected Weekly Commitment: 4-5 hours
Showcase : here

7 - Computer Vision: A Classic and Modern approach

Mentors: Shivanshu Tyagi (spazewalker#5877), Akshay Gupta (gakshay#2382)
Description : This project will cover the basics of computer Vision, then dive into some classic computer vision algorithms and then finally will cover the modern Deep Learning approaches in CV.

The proposed project would roughly have the following structure:
1. Introduction: an introduction of Computer Vision and some basic Linear Algebra and numpy required for the later parts of the project.
2. Image processing: This portion will start with color spaces and conversions,Image Transformations(rotation, scaling, translation, etc.), then cover filters and kernels (noise removal, blurring, image thresholding) and finally conclude with hough transform, Edge detection(Canny Edge Detection) and contour detection algorithms(Watershed). Can also cover feature detection, depending upon the response.
3. ML for Vision: Here, we’ll deal with some basics of Deep Neural Networks and talk about convolutions(also covered in image processing part), various optimizers, backpropagation algorithm, Gradient Descent, SGD various Loss functions and their usages in CV context, Regularizations and Bias-Variance trade off.
4. Applications: In this part we’ll talk about some famous algorithms for some basic computer vision tasks.
  - Image Classification: Inception, VGG and Resnet.
  - Object detection: SSD, YOLO
  - Face Detection: HAAR cascades
We’ll accompany each lecture with a fun CTF type assignment along with some further readings.

For implementation, we’ll use opencv, numpy in the starting. At later stages we’ll include pytorch.
Expected Duration : 3 Months
Expected Weekly Commitment: 6-7 hrs
Showcase : here

8 - Introduction to Fourier Analysis of Boolean Functions

Mentors: Mohd Talib Siddiqui (arrow#4554)
Description : We will start with discussing various kinds of boolean functions and their examples. Boolean functions are involved in various domains such as combinatorics and voting theories. However, instead of their applications we will analyse the properties boolean functions in themselves exhibit. We start with a very relaxed domain of functions on binary strings and will look at the properties that change when we restrict the domain to boolean values. We will be going over Fourier expansion of boolean functions in detail. We will touch upon property testing and introduce the procedure of linearity testing of boolean functions in short. We will mainly discuss on the Fourier support and dimension of boolean functions and see various bounds involving the same. We may include discussions on Parity Decision Trees as well depending on the interest shown by the mentees.
Expected Duration : 5-8 weeks
Expected Weekly Commitment : 2-4
Showcase : here

9 - Counting is fun

Mentors: Farzan Byramji (geekotechy#7312)
Description : Enumeration of combinatorial structures will mainly be what we will be doing. This can involve some very elegant arguments (like nice bijections) as well as some relatively mundane tools (like manipulating algebraic expressions, ‘manipulatorics’ in the words of Zeilberger).

Depending on the interests and background of people, we can start by looking at standard things. If algebra is of interest, ideas from algebraic combinatorics can be explored. For those who wish to focus on some specific sub-topic, one of various classes of objects like pattern-avoiding permutations, lattice paths, plane partitions, hyperplane arrangements, etc. can be picked up. Computations using computer algebra have been very useful in this area, so that can also be a component.
Expected Duration : 1-1.5 months
Expected Weekly Commitment: 2-4

10 - Cross Validated

Mentors: Kushagra Gupta (kushagra#9379), Ayush Agarwal (mclaren#1275), Medha Agarwal (Medha#6652)
Description : We plan on covering fundamental statistical algorithms that will serve as a bridge to advanced statistical concepts. We will focus on hands-on implementation with real-life problems. The project will expect students to be familiar with basic linear algebra and have adequate coding skills. The end goal of the project is to equip students with a basic understanding of statistical science while offering flexibility of choice in sub-areas like regression, optimization, sampling, etc.
Expected Duration : 2 months
Expected Weekly Commitment: 5-6 hours

11 - Physically-Based Rendering

Mentors: Mayant Mukul (wabadump#4634)
Description : Realistic rendering techniques in computer graphics usually boil down to solving an integral equation called “the rendering equation”, which takes into account various physical properties of light. We’ll present the rendering equation in its original form and introduce radiometry to see how the rendering equation captures most physical phenomena under the assumptions of geometric optics. We’ll then see why finding an analytical solution can be difficult, develop a naive algorithm and finally introduce Monte Carlo methods that modern renderers like Pixar’s RenderMan or Blender’s Cycles are based on.
Expected Duration : 6-8 weeks
Expected Weekly Commitment: 3-4 hours
Showcase : here

12 - Introduction to Formal Logic and Computation

Mentors: Udit Narayan Pandey (UNP#0155), Dawood Bin Mansoor (rockstar2514#0635), Ayush Kumar
Description : The goal of the project would be to introduce the mentees to formal logic first order and propositional logic and basic ideas of Computation. We would be covering axiomatisaion, rules of inference, syntactic and semantic consequence, Models of a Theory, Consistency Soundness, Completeness, First Order Theories and after covering Computation a peek into Godel’s Incompletness Theorem, for Computation part we will be covering the notion of computability, formal languages, Models of computation(turing machines), Church turing hypothesis, decidability, and brief overview of complexity.
Expected Duration : 7 weeks
Expected Weekly Commitment: 4-6 Hrs
Showcase : here

13 - Numbers Made Dumber

Mentors: Yatharth Goswami (alpha_hacker#3464), Vivek Kumar Singh (bb8#2778), Rohan Baijal (rohanblueboybaijal#5838)
Description : This project will serve as an introduction to Number Theory and some of its cool real world application like in Cryptography!

One component (the obvious one) is theoretical where the idea would be to cover theory (some Olympiad related and some algebra related) and tickle the brain cells by solving some problems.

The other component would be implementing some of the algorithms in code as we study them. This would give a more hands on feel and an idea about using it in applications.
Expected Duration : 3-4 months
Expected Weekly Commitment: 6-8 hrs
Showcase : here

14 - Generating rational distance sets on a parabola

Mentors: Sayak Bhattacharjee (saybha_00#0654)
Description : Introduction: A rational distance set is a set of points with each of its pairwise distances to be rational. We study the problem of finding N-point rational distance sets on the parabola y = x^2. An open problem is to find the largest N for which a rational distance set exists on the parabola. While we know that N is finite, we do not yet know if even a 6-point rational distance set on the parabola exists!

Reference: G. Campbell, Points on y = x^2 at rational distance, Math. Comp., 73 (2004), 2093–2108.

Outline of project:
1. Introduction to existing literature; explore a correspondence of the problem with Pythagorean triplets, and study existence and nature of the solutions for general N from a linear algebraic point-of-view.
2. Primary goal: Attempt to obtain solutions computationally for N=4 and N=5 by building on techniques available for N=3. This will involve building time/space efficient algorithms to efficiently search for solutions.
3. Possible extension: Attempt to understand density of the solutions on the parabola for small N using real analysis.
Necessary pre-requisite knowledge: Linear algebra, basic combinatorics, Python/C. Not strictly necessary pre-requisite knowledge: Parallel computing, real analysis.
Expected Duration : about 2 months (min.)
Expected Weekly Commitment: 5+
Showcase : here

15 - Quantum Algorithms

Mentors: Rishabh Kothary (rishk76#7995)
Description : Quantum Algorithms is a hot research area, and quantum algorithms like Shor’s factorization algorithm have threatened the current state-of-the-art cryptosystems like the RSA. Shor’s algorithm has brought an exponential speedup to the integer factorization problem compared to its classical counterpart. It has forced the cryptography community to build cryptosystems resilient to attacks from quantum computers. Our main aim is to understand the quantum computation model and learn about Shor’s factorization algorithm in the project. If time permits, we could explore algorithms like Grover’s Search algorithm and Simon’s Algorithm and their applications in breaking cryptosystems. Not many pre-requisites are assumed; a basic understanding of Linear Algebra should suffice. However, the pre-requisites are not very strict and can be covered during the project if interested.
Expected Duration : 2-3 months
Expected Weekly Commitment: 8-10 hrs
Showcase : here

16 - Introduction to Metric Spaces and Topology

Mentors: Kushlam Srivastava (Kuchlam#9851)
Description : This project is intended to be a brief introduction to Metric Spaces and Topology, an important tool which is very useful in Pure Mathematics and a bit of theoretical Physics as well. If you want a bit of motivation for this topic you can watch the following VSauce video which addresses the famous question ““How many holes does a straw have?”” through topology link
We will first cover Metric Spaces which will be based on a lot of the first part of MTH101 and we can then move on topological concepts based on the comfort and understanding of the mentees. Since most students are not able to do the course offered on Topology by MTH department before the 5th semester, this can be a great opportunity for you to learn this topic. This project is mainly intended for Y20 students.
Expected Duration : 2 months
Expected Weekly Commitment: 5 hours
Showcase : here

17 - Understanding Graphs with Analysis

Mentors: Ayush Basu (abasu#5088)
Description : What are some nice ways to understand the handshake lemma? How many triangles does a tripartite graph contain? How can we count the number of subgraphs of a particular kind in a complete graph (or a random graph)? How can we define quasirandomness in graphs? How can we extend these notions to hypergraphs?

We will try and answer some of these questions in the project exploring counting techniques one can use for graphs and look at some of the famous conjectures of extremal graph theory, a field primarily developed by Paul Erdös, one of the most prolific problem solvers to grace mathematics.

We will also focus on the theme of the dichotomy between structure and randomness and look at examples throughout our project that lead to a better understanding of these ideas.

Later on we may explore some advanced results in this field like the complete proof of Szemeredi’s theorem and regularity Lemma.
Expected Duration : 2.5 months
Expected Weekly Commitment: 6

18 - Topics in Analysis: A Tour of Measure Theory and Functional Analysis

Mentors: Aniket Das (Aniket#9998)
Description : This is a project on intermediate-level mathematical analysis intended for highly motivated Y18s and Y19s (and exceptional Y20s) who are interested in analysis, probability theory and machine learning theory. I intend to cover a broad range of topics in measure theory and functional analysis and discuss their applications in areas such as probability theory, machine learning theory and quantum mechanics

Pre-requisites: Mathematical maturity and a high level of comfort with basic calculus (MTH101) and Linear Algebra (MTH102).

Topics: The exact syllabus will depend on the mathematical maturity of the group. Ideally, with a passionate and proficient group, I will be able to cover the following:
1. Topology and Analysis Preliminaries: Metric Spaces, Open and Closed Sets, Completeness, Connectedness and Compactness
2. A Fast Paced Linear Algebra Refresher on Normed Spaces, Inner Product Spaces and Linear Operators. Introduction to Matrix Norms and Singular Value Decomposition
3. Basics of Hilbert Space Theory: Basic properties, Jordan von Neumann Theorem, Orthonormality, Applications to Fourier Analysis and Approximation Theory
4. Bounded Linear Operators and Functionals on Hilbert Spaces: Definitions, Reisz Representation Theorem, Hilbert Adjoint
5. Self Adjoint Linear Operators, Compact Operators, Spectral Theorem. Applications to Polar Decomposition and Singular Value Decomposition in Hilbert Spaces
6. (Optional) Basics of Banach Space Theory: Hahn Banach Theorem, Open Mapping and Closed Graph Theorem, Banach Fixed Point Theorem and its Applications
7. (Optional) Application of Functional Analysis in Basic Quantum Mechanics
8. Measure Theory: Definition and Basic Results on Measures, Outer Measures, Borel Regularity with a special focus on Lebesgue Measure
9. Measure Theory II: Simple Functions, Measurable Functions and Measure Theoretic Notions of Convergence
10. Integration Theory I: Measurable Functions, Definition and Properties of Integral with special focus on Lebesgue Integrals
11. Integration Theory II: Fundamental Theorems: Monotone Convergence Theorem, Fatou’s Lemma, Dominated Convergence Theorem, Multiple Integrals and Fubini’s Theorem
12. Lp spaces: Development of functional analysis on Lp spaces, integral operators, kernels.
13. (Optional) Applications to Probability Theory: Measure Theoeretic Probability, Expectation and Convergence Theorems in Probability. Weak Law of Large Numbers and Central Limit Theorem
14. (Optional) Advanced Calculus on Banach Spaces: Metric Derivatives, Frechet and Gateaux Derivatives
15. (Optional) Connections to Kernel Methods in Machine Learning: Reproducing Kernel Hilbert Spaces and some more Fourier Analysis (Bochner’s Theorem and related results)
Expected Duration : 3-4 months on full syllabus. 1-2 months on a reduced syllabus
Expected Weekly Commitment: 8-10 hours (involves significant self study and possible exercises)