MATHEMATICS CAMP 1 - REAL ANALYSIS AND LINEAR ALGEBRA

Instructor: Jubayer Ibn Hamid

Real analysis and linear algebra are not just fascinating in and of themselves; they are extremely useful. More modern sciences, like machine learning and optimization, employ these fields of mathematics almost every step of their way.

In real analysis, we will first take a close look at what defines integers, rationals and real numbers - their various axioms and consequences. After that, we will study sequences of real numbers - their limits, properties of subsequences and special kinds of sequences like monotone, bounded, etc. Then, we will naturally move on to series and functions - their limits, continuity, etc. We will close by taking a brief look at metric spaces - open, closed sets and continuous functions on metric spaces. After doing so, we will gain a much better understanding of reals in general. In linear algebra, we will take a brief look at fields and then focus on vector spaces defined on them - in particular, finite-dimensional ones. We will then study linear maps which will motivate the study of matrices, null spaces, etc. followed by the study of dual spaces. We will close by taking a look at eigenspaces and inner product spaces.

These techniques will become very useful in any field of science you go to. In optimization, one would frequently face problems like optimizing resources under certain constraints or scheduling usage of batteries to conserve electricity or optimizing a portfolio at an investment bank; you would often find yourself formulating these problems through things like “linear programming”, “second order cone programming”, etc which, more often than not, are in the language of linear algebra. In machine learning, you would often find yourself trying to analyze local properties of a loss surface using real analysis. The most rigorous study of probability and statistics is founded on real analysis.

Prerequisites: we hope that you know the mathematics that is taught up until 10th grade. We will also learn everything through proving them. Do not worry if you do not know proof-writing yet - we will teach an introduction to proof-writing as well. While the course content may seem daunting, we promise we will go at a pace that you can understand and, more importantly, enjoy.

MATHEMATICS CAMP 2 - ABSTRACT ALGEBRA AND DISCRETE MATH

Instructors: Jubayer Ibn Hamid, Ramisha Sharika

Algebra is one of the most beautiful fields of mathematics. This is where you will see the power of abstraction. Groups and rings are very simple, abstract umbrellas under which many things we see around the world fall - from numbers to matrices to polynomials to the symmetries of the universe (take a look at Noether’s Theorem for example). It is truly startling how much information these abstract objects contain - rings and ideals have been used to describe complex relations and geometries and to get very powerful results. For a basic example, take a look at Hilber’s Nullstellensatz or how physicists constantly discuss symmetries whose natural language is group theory.

We will introduce groups and rings - how to construct them, various contexts in which they show up, their defining characteristics, their subgroups/subrings and generators. We will then study homomorphism, the very useful isomorphism theorems and actions on groups. Watch this video to get a sense of what this field is all about!

It is hard to overstate the impact of algebra on the modern world. Today it is no longer a separate field of mathematics; algebraic number theory, algebraic geometry, etc. are all very hot topics of mathematics today.

Mathematics forms the foundations of many diverse fields, a key one being Computer Science. A course in discrete structures bridges the gap between pure mathematics and theoretical computer science. It allows us to apply tools such as logic, probability, and proofwriting to formalize our understanding of algorithms and better grasp the theoretical underpinnings of advanced topics in computer science, such as cryptography and machine learning. In this course, we will start by doing a review of different proof techniques with a strong emphasis on how to write inductive proofs. We will then continue to practice formal proofwriting while learning about set theory, graph theory, and number theory. The focus of this course will be on learning the basics of graph theory and number theory.

PHYSICS CAMP 1 - ADVANCED CLASSICAL MECHANICS: FROM LAGRANGIANS AND HAMILTONIANS, CHAOS THEORY TO COMPUTATIONAL PHYSICS AND AN INTRODUCTION TO PARTICLE PHYSICS

Instructor: Tausif Hossain

We were always taught in school that energy is conserved in a closed system. But have you wondered why energy is conserved? This simple question we will learn, has a surprisingly beautiful answer: energy is conserved when there is time reversal symmetry. That is, if physics remains the same forwards and backwards in time, we can say energy is conserved! There’s numerous such symmetries in our universe, such as translation symmetry leading to momentum conservation etc. These symmetries are the bedrock of modern physics and all of this starts in our well known classical mechanics!

Using symmetry as our basis, in this course we will delve deep into classical physics and actually derive Newton’s Laws instead of taking them as a given. We will solve famous problems like the Brachistochrone problem, find simpler and beautiful methods to solve difficult mechanics problems, and in general look at the usefulness of Lagrangians and Hamiltonians in a wide variety of fields. We will also show how simple symmetries can form the basis of all of Particle Physics, and we will do a quick introduction to many fascinating results and ideas in this field.

We will also focus some time on learning computational physics skills like how we can actually simulate complicated physical systems. We will use Python to learn together how we can not only study but also animate a double pendulum or other chaotic systems such as a 3-Body Problem! Thus, by the end of the course you will be able to make many novel connections on how interconnected physics is. You will hopefully also have the computational tools to be able to do your own simulations and research projects, which I think will be a great foundation to have for a future in any science or data analysis field.

Prerequisites: We hope you’ve learnt calculus at an introductory level, as taught till grade 10. We will work through physics from first principles so no advanced knowledge is necessary. We will also start the computational physics code from the basics so no prior coding knowledge is needed!

COMPUTER SCIENCE CAMP 1 - PYTHON, BASIC DATA STRUCTURES AND ALGORITHMS

Instructor: Farhan Tajwar Romit

Computers seem like the closest thing we have to magic. But if we peek behind the curtain just a little bit, we can see for ourselves the glorious mess that a computer really is. Everything, starting from the contents of our hard drives, to the pixels on our screen, are composed of just some 1s lost in a big sea of 0s. Surely, you'll think, this does not make sense to even the computers themselves.

And you would be right. In reality, the CPU inside our phone, laptop, or PC is actually a very simple machine that just does exactly what it is told. As we will see, human ingenuity is actually the glue that holds a computer and its seemingly arbitrary actions together. With our decades of experience in theorizing about, and working with, computers, we have been able to formulate certain techniques of storing data on our computers (data structures), and certain techniques of using and manipulating that data (algorithms), which form the bedrock of all the awesome things our computers can do. In this course, we will study the fundamentals of these techniques.

We will begin by taking a look at the motivations behind some common data structures like arrays and linked lists, and then study some commonly-used algorithms for those data structures (such as insertion and sorting algorithms). Later on, we will encounter some more specialized data structures, such as trees (for storing hierarchical data), graphs (for networked data), and heaps (for ordered data).

The language for this course will be Python. Don't worry - as we go along, we will cover all the necessary Python skills you will need for completing this course!

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