Courses
The following are new courses from Camp 2021 along with prerequisites for each course. If no prerequisites were mentioned, the course was meant to be accessible to all students at the camp. It is completely alright if you are not familiar with any of the content or terms here  that’s what the camp is for!

Spanning Sets and Spamming Sets: a Glimpse of Combinatorial Games
Sutanay Bhattacharya, Indian Institute of Science, Bangalore
Alice and Bob alternately pick points on the plane. Alice wants to build a congruent copy of some agreedupon set S with her chosen points, and Bob wants to prevent this. In the process of analyzing this game, we’ll visit some basic linear algebra, the method of opportunity functions, and a ubiquitous problemsolving heuristic we will call ‘spamming’, and hopefully realize the real treasure was the techniques we developed along the way.
Prerequisites: Ability to follow proofs, familiarity with highschool vector geometry. Prior familiarity with linear algebra will be helpful but not necessary.

Quantum Computation: Let’s stay Real
Pulkit Sinha, Indian Institute of Science, Bangalore
Most of the existing literature for Quantum Computation is aimed towards people with at least 23 years of undergraduate coursework, and involves a bit of quantum mechanics, but this course will take an alternate approach. We will skip over the parallel universes and cat abuse, and instead focus on trying to precisely define a model for Quantum Computation by modifying a few fundamental properties of probabilistic/randomised computation, followed by looking at some Quantum algorithms. There would be absolutely zero involvement of physics, and we will restrict ourselves to working with only real numbers in order to utilize the existing intuition from Euclidean Geometry.
Prerequisites: High school Matrices (must be comfortable with working with n dimensional matrices), 3D Geometry, Probability. Familiarity with basic graph notions (must be familiar with weighted directed graphs and adjacency matrices). Familiarity with algorithms/computation. Highly Recommended: 3Blue1Brown's Linear Algebra Lecture Series. See videos 1,2,3,4,5,13.

The Theory of Computation: Turing Machines, Incompleteness and Uncomputability
Chinmay Ingalagavi, Indian Institute of Science, Bangalore
Julian D’Costa, University of Oxford
Can you think about thinking? What does it mean for a system to be able to describe itself? Does every true statement have a proof? These are ancient philosophical questions Godel tackled when proving his Incompleteness Theorems, some of the most famous results in mathematical logic. In modern terms, Godel built a programming language out of elementary arithmetic, an incredibly complex endeavour.
The good news is that we now have much better programming languages! Thanks to Alan Turing and the concept of the Turing machine, proving Godel’s theorems becomes surprisingly easy. In this course we will try to convince you of the ChurchTuring Thesis  that any “algorithmic” task you can think of can be done by a Turing machine. We then discuss the opposite  tasks that cannot be solved by an algorithm  such as the Halting Problem  and prove this as well as Godel’s theorem. Finally we discuss the Busy Beaver numbers  a sequence of integers that grows so fast it is literally “uncomputable”.
Prerequisites: This course has no prerequisites.

Introduction to Probability
Akshay Hegde, Milind Hegde, and Vasanth Pidaparthy
There are several courses being offered on probabilistic topics, and an understanding of some more basic concepts of probability theory will be needed for them. We'll build those concepts in this course, discussing topics like random variables, distributions, independence, expectations and conditional expectations, the Markov and Chebyshev inequalities, and laws of large numbers.
Prerequisites: Familiarity with the notion of sets and functions.

How to Make No Money at the Casino
Vasanth Pidaparthy, University of Maryland (incoming math PhD)
Gambling and games of chance have historically been a driving force for studying probability. At the heart of such analysis is the powerful tool of martingales. In this course, we will learn the basics of martingales, stopping times and the optional stopping theorem, and learn how to answer fundamental questions in life such as, how many hours will a monkey at a typewriter take to type out the Hitchhiker's Guide to the Galaxy? The answer (not 42) may surprise you.
Prerequisites: The introduction to probability course at Monsoon Math Camp 2021.

Staying Connected Amidst the Noise: Percolation and Criticality
Akshay Hegde, Indian Statistical Institute, Bangalore
This course aims to understand 'phase transitions' of physical models in a mathematically rigorous way. You'll be surprised to see order even in random systems! We'll begin with lots of examples then transit to prove basic properties of the bond percolation model. In the end, we'll look at properties of the model at the criticality.
Prerequisites: First 2 lectures of the 'Intro to Probability' sequence.

Randomness and Compression: The Shannon SourceCoding Theorem
Milind Hegde, UC Berkeley
How much can a message be compressed, and what does a maximally compressed message look like? These are the type of questions that lead to the "source coding theorem", the first milestone theorem of information theory. In these two lectures we'll discuss and try to gain some intuition about entropy, the quantity that answers the first question, and see a (fairly weak) version of the source coding theorem. If time permits, we'll gesture towards stronger and more useful versions as well.
Prerequisites: The introduction to probability course at Monsoon Math Camp 2021.

Introduction to Proofs via InquiryBased Learning
Miraya poddarAgrawal, Reed College
In this workshop style class we will develop intuitions for group theory and analysis, the two pillars of collegiate mathematics. Students will be provided with short puzzles and the definitions and results that will be useful in finding solutions. While some puzzles will resemble classic problem sets, most puzzles will zoom into what is interesting about a set of definition with respect to (1) the questions that can now be posed and (2) the structure of theorems to answer them.

Cut, Fold, Paste: Homology and the Classification of Surfaces
Chinmaya Kausik
How many truly different shapes can you get by pasting polygons at their boundaries? Mathematicians often like to “classify objects”  many big research endeavours in mathematics are geared towards classification. What does it mean to classify a mathematical object?
This will be illustrated using the “classification of surfaces,” which is related to our first question. We will try to understand how a surface might be defined from our intuitive idea of it, reduce it to a combinatorial object and then classify these combinatorial objects using tools we develop on the way. We will see powerful tools like the Euler characteristic, homology and the MayerVietoris sequence.
Prerequisites: Comfort with the ideas of sets, functions and induction. Visual intuition and familiarity with the notion of a graph will be very helpful.

Geometry of Curves and Surfaces
Sidharth Soundararajan
Can an atlas be wrapped around a globe such that all of the mappings are to scale? Inspired by the task of surveying, mapping and geodesy, Carl Friedrich Gauss laid the foundations of the Differential Geometry of Curves and Surfaces. One of the most fascinating theorems in Differential Geometry is Gauss’s Theorema Egregium, which literally translates to ‘Remarkable Theorem’. In this course, we explore the notions of vector spaces, curves in 3D space, quantities that Gauss defined to further understand the structure of surfaces in 3 dimensional space, as well as prove and appreciate the consequences of the Theorema Egregium.
Prerequisites: High school calculus, a high school level familiarity with matrices

Knots and Knot Polynomials
Deeparaj Bhat, MIT
We've all seen knots in our daily lives, but how does one go about distinguishing different knots? A first step towards this was made in 1923 by assigning a polynomial called the Alexander polynomial to each knot. Much later (1960s) it was realised that the Alexander polynomial satisfies a recurrence relation. The significance of this was recognised around the 1980s with the definition of some more `knot polynomials' which were better at distinguishing knots.
In this course, we will go over how to define these various knot polynomials and see some basic properties about them. We then go on to survey some recent developments and mention some examples which show that these polynomials are still a far cry away from giving a classification of knots.
Prerequisites: Familiarity with inductive proofs and recurrence relations.

SymmeTrees
Chaitanya Tappu, Cornell University
This course is about symmetries of trees. We will talk about what symmetry means, and then, via a profusion of examples, describe the abstract mathematical structure of groups which used to keep track of them. Then we will look at symmetries of trees and characterise the absence of fixed points. This leads to a proof of the celebrated NielsenSchreier theorem.
Prerequisites: Familiarity with the language of functions and sets. The main setting of the course is in trees, so graphs and trees will be defined in class; nonetheless, previous exposure will help. The settings for other examples will be sets, the euclidean plane, vector spaces and matrices, the real number line and metric spaces. Familiarity with these will help you understand these examples, but it is okay to not understand all of them.

Invariants and Impossibility: from Geometric Constructions to Solving Polynomial Equations
Toghrul Karimov, MPISWS
Given an angle, is it possible to trisect it into three equal angles using a compass and a straightedge? This is a problem that baffled ancient Greeks for hundreds of years, and we now know that for certain angles such trisection is impossible.
Similarly, we know that there exist polynomials in one variable whose roots cannot be expressed using arithmetic and radical (i.e. square roots, cube roots etc.) operations. We will discuss the idea of invariants, Galois theory and how one obtains such impossibility results in general.
Prerequisites: Knowledge of what is a vector space over a field and what is a group

Pi using Number theory
Hargun Preet Singh Bhatia and Ajay Prajapati
You may have seen this beautiful infinite series 11/3+1/51/7+... = π/4, known as the Leibniz formula for π. In this course, we will give a numbertheoretic proof of this result, which will reveal the circle (hence π) in this formula. Along the way, we will introduce and establish connections between concepts and objects that have now become a fundamental part of modern number theory, like unique factorisation, zeta function(s) and much more! By the end of this course, you will (hopefully!) get a glimpse of what lies at the intersection of two broad subfields of number theory, i.e., algebraic and analytic number theory.
Prerequisites: High school calculus, complex numbers and modular arithmetic

Plane Curves
Swaraj Pande, University of Michigan
This course is about relating the algebraic properties of polynomials with the geometric properties of their graphs. Building on the familiar case of conic sections, we’ll explore more interesting examples of plane curves and their geometry. This leads us to thinking about complex numbers, the fundamental theorem of algebra, projective geometry etc. All these concepts will be introduced briefly. Here are two interesting questions that we will learn the answer to:
We know how to parametrize the circle by the functions (cos(t), sin(t)) i.e. for all values of t, the point (cos(t), sin(t)) will lie on the unit circle. But, can we replace the functions cos and sin by rational functions (quotients of polynomial functions) to parametrize the circle?
Can we find a parametrization by rational functions of the curve defined by the equation y^2 = x^3 + x?
Prerequisites: Some familiarity with plane coordinate geometry and complex numbers.

Complex Dynamics
Malavika Mukundan, University of Michigan
The word ‘dynamics’ suggests movement. In mathematics, dynamics is a subfield that studies what happens to numbers in some domain when we apply the same function over and over again.
This course is an introduction to the study of iterations of polynomials with complex number inputs and outputs. We will explore how this leads to the classification of the complex plane into two sets the Fatou and Julia sets, and generate a lot of really cool looking pictures in the process. Julia sets are a standard source of generating fractals which you may have heard of.
We will also take a look at one of the most famous objects in mathematics the Mandelbrot set, and explore some of its properties.
Prerequisites: High school introduction to complex numbers, set theory, basic calculus. A little bit of modular arithmetic will be introduced.
Introduction to Formal Proofs in Lean
Ashvni Narayanan, LSGNT and Koundinya Vajjha, University of Pittsburgh
A formal proof is a mathematical proof in which every logical inference is checked all the way down to the fundamental axioms of mathematics. No appeal to intuition is made, even if the translation from intuition to logic is routine. This course will introduce the basics of computerassisted formal proofs using the Lean Theorem Prover.
We shall briefly introduce Lean and how to work with it, including lots of exercises in propositional logic, proofs by induction and the natural number game. By the end of the course, students should be comfortable using Lean to verify their own proofs and should be on track to contributing to mathlib, Lean’s standard library for mathematics.
Prerequisites: A fully installed Lean+VS Code/Emacs set up (Installation guide forthcoming).
The following are courses from Camp 2020, some of which were reprised for 2021.

Graph Theory and the Five Color Theorem:
Drake Thomas, University of Minnesota, Twin Cities
I'll start from the basics of graph theory, try to discover and prove Euler's formula for planar graphs, and give a proof of the Five Color Theorem using the tools we've built. Time permitting, we'll think about other interesting topics with ties to graph theory like Ramsey numbers, Eulerian cycles, or de Bruijn sequences.
Prerequisites: Some comfort with proofs and induction would be useful, but isn't essential. If you *have* seen lots of these topics, you might not get as much out of this course.

Extremal graph theory:
Ashwin Sah, Massachusetts Institute of Technology
Turan's theorem, a discussion of ErdosStoneSimonovits and the Zarankiewicz problem, perhaps a discussion of Sidorenko's conjecture for trees.
Prerequisites: Assumes familiarity with graphs, bigO notation.

Introduction to Group Theory using Rubik's Cube:
Praful Shankar, Indian Statistical Institute, Bangalore
First I aim to give a very visual introduction to symmetries and why groups are a useful structure to study. The eventual goal is to be able to grasp the idea of commutator subgroups and how they can be useful in solving the cube, but the central focus will be elementary results about groups and some fundamental theorems.

Curvature in Geometry, Topology and Combinatorics:
Balarka Sen, Indian Statistical Institute, Bangalore
I will introduce a notion of curvature for "smooth spaces", "polyhedral spaces" and "discrete spaces". I will try to convince you that curvature is a fundamental property that can be used to understand various shapes we see everyday from a mathematical perspective.
Prerequisites: Some knowledge of basic differential calculus of one and two variables will be useful, but can be compensated by a good geometric intuition.

Geometry, Symmetry and Hyperbolic Space:
Chinmaya Kausik, Indian Institute of Science
Exposure to a lot of Euclidean geometry may create the impression that higher geometry is the study of generalized distance spaces. This course will try to convince participants that in some cases, a better view of geometry is the interaction between a space and its group of transformations, via material on elementary hyperbolic geometry. We will see basic results in hyperbolic geometry, the hyperbolic GaussBonnet Theorem, the Iwasawa decomposition, a quick version of material on Fuchsian groups and quotienting, and if time permits, the MilnorSvarc lemma.
Prerequisites: High School Calculus and High School Matrices.

An Introduction to Dimension Theory:
Ratul Biswas, University of Minnesota, Twin Cities
In this 3lecture minicourse we are going to see some formal definitions of the dimension of an object, and try to understand why coming up with a definition is not easy. Time permitting, we shall also talk about fractals, which are objects weird in the sense that their dimensions are not integers.
Prerequisites: Basic set theory and the notion of functions.

Analytic number theory:
Ashwin Sah
Chebyshev's bounds for counting primes as well as Bertrand's postulate, and perhaps a discussion of further directions.
Prerequisites: Assumes familiarity with elementary number theory including modular arithmetic, perhaps bigO notation

Fun with padic Numbers:
Suhas Gondi, Indian Statistical Institute, Bangalore
Pure mathematics has a reputation for being highly abstract and exclusive, and there are only few concepts that illustrate this better than padic analysis. In this course, I shall introduce a new set of numbers that were invented (discovered?) in the 1890s to provide number theorists with new techniques in approaching problems. In doing so, I will describe their basic properties by comparing them to the betterknown real numbers. I will then demonstrate a nice application of the padics by showing that a square cannot be dissected into an odd number of triangles of equal area.

Quadratic Reciprocity:
Meghal Gupta, Massachusetts Institute of Technology
Learn how to prove the famous theorem of quadratic reciprocity: determine when a prime is a quadratic residue modulo another.

The Probabilistic Method:
Ritvik Ramanan Radhakrishnan, Indian Statistical Institute, Bangalore
Through examples we will introduce the probabilistic method and illustrate the use of union bound, expectation of random variables, and standard inequalities in problems from combinatorics, graph theory, geometry, and number theory.
Prerequisites: Some familiarity with high school probability will be useful.

Markov Chains and Random Walks
Sidhanth Mohanty, University of California, Berkeley and Parth Karnawat, Indian Statistical Institute, Bangalore
Lec 1: Probability review (SM) Lec 2: Preliminaries of Markov chains along with proposing motivating questions (analyzing random walks on integer lattices) (PK) Lec 3: Probability generating functions and their properties (SM) Lec 4: Recurrence in Z and Z^2, Transience in Z^3 (PK) Lec 5: An electrical networks and resistance approach to proving recurrence in Z and Z^2, transience in Z^3 (SM) Lec 6: Law of Large Numbers (PK)

Linear Logic:
Jason Gross, Massachusetts Institute of Technology
It is a truism of logic that if you have A, then you have A and A. So if the objects you’re studying are dollars, then from a dollar you have a dollar and a dollar, and thus logic proves that from a dollar, you have infinite money. This is absurd! Come learn about linear logic, the formal logic of resources that are neither created nor destroyed.

Löb's Theorem:
Jason Gross
Do you like recursion, selfreference, or being very, very careful about the difference between A, a proof of A, and a proof that A is provable? If you answered "yes" to any of these, this class is for you! We'll be discussing the Santa Claus sentence (a great way to prove that Santa Claus exists by abusing selfreference), Löb's Theorem which is proved by using a version of the Santa Claus sentence that doesn't actually lead to absurdity, and one of Gödel's Incompleteness Theorems (the one that says that any sufficiently powerful system for proving statements that only allows proving true things is necessarily incomplete), which is an easy corollary of Löb's Theorem.

Category Theory:
Jason Gross and Miraya PoddarAgrawal
Category theory is one of the most abstract areas of math (moreorless affectionately named "formal abstract nonsense" by mathematicians). More than just an area of math, category theory is a lens for approaching mathematics in general, and one that draws out the similarities between disparate branches of math, allowing us to see how different highpowered theorems of different fields of math are actually the same theorem. The lens of category theory is that the objects (sets, numbers, points, etc) are not as important as the relations between the objects. Category theory as a field of math is a way to formalize the sorts of reasonings that arise from this lens.
In this class, we'll be taking a look at some very basic category theory, drawing examples and puzzles from basic set theory. We'll assume familiarity with basic set theory, including functions between sets, injections, surjections, bijections, cartesian product, and disjoint union.

To Infinity and Beyond:
Rachana Madhukara, Massachusetts Institute of Technology
Basic introduction to infinite and countable sets, bijections, and The Cantor–Schroder–Bernstein Theorem.

Classical Game Theory:
Miraya PoddarAgrawal, George Mason University
In this class our goal is to use mathematical skills in nonmath places, and observe through examples how we can create useful models. We will rediscover theorems of game theory through puzzles about behaviour.

An Introduction to Minimax:
Robert Cunningham, Massachusetts Institute of Technology
We'll look at an algorithm for finding optimal moves in two player zero sum games, such as Chess and Go. We'll also look at some of its optimizations, both in theory and in practice.

Combinatorial game theory:
Shardul Chiplunkar, Massachusetts Institute of Technology
Combinatorial game theory (CGT) is a field of mathematics that looks at 'mathematical' games: games with two players taking turns, where both players know everything about the state of the game, and there are no random elements. (Many popular board games, including chess, fall under this description!) This class will be a light introduction to the field, looking at some fun games and how to analyze them. We may even end up at an alternative definition of the real numbers and beyond!