Camp 2021 course list will be released after camp.
The following are courses from Camp 2020 along with prerequisites for each course. If no prerequisites were mentioned, the course was meant to be accessible to all students at the camp. We also plan to increase the number of low-prerequisite classes this year. It is completely alright if you are not familiar with any of the content or terms here - that’s what the camp is for!
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 Erdos-Stone-Simonovits and the Zarankiewicz problem, perhaps a discussion of Sidorenko's conjecture for trees.
Prerequisites: Assumes familiarity with graphs, big-O 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 Gauss-Bonnet Theorem, the Iwasawa decomposition, a quick version of material on Fuchsian groups and quotienting, and if time permits, the Milnor-Svarc lemma.
Prerequisites: High School Calculus and High School Matrices.
An Introduction to Dimension Theory:
Ratul Biswas, University of Minnesota, Twin Cities
In this 3-lecture mini-course 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:
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 big-O notation
Fun with p-adic 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 p-adic 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 better-known real numbers. I will then demonstrate a nice application of the p-adics by showing that a square cannot be dissected into an odd number of triangles of equal area.
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)
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.
Do you like recursion, self-reference, 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 self-reference), 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.
Jason Gross and Rajashree Agrawal
Category theory is one of the most abstract areas of math (more-or-less 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 high-powered 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:
Rajashree Agrawal, George Mason University
In this class our goal is to use mathematical skills in non-math 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!