The Immerse 2019 Mathematics summer school programme comprises a number of key topics which are explored in depth, with the guidance of our expert tutors. The course aims to provide participants with a thorough grounding in topics necessary for success as an undergraduate. That being said, the programme is designed to provide much more than understanding of key topics. At Immerse, our aim is to develop a student’s passion for their chosen subject, by providing them with an opportunity to put theory into practice, and discover the vast range of applications this intricate subject offers. Over the course of the programme, participants will explore a range of complex topics.
Constructive mathematics is the mathematics of iterative methods, which arise everywhere in the subject. For example, Niels Abel famously proved that we cannot in general write down the solution to polynomial equations of degree 5 or higher; but of course there will be cases where we need to know the answer! The first place to turn to is an iterative method – a process in which we continually apply the same “action” to a given input, in the (mathematically justified) hope that this process eventually outputs something close to the true answer that we seek.
However, in the context of real world applications, imagine that the solution of this problem is an essential parameter in the design of a satellite or a heavy load carrying bridge, for example. In this case, one needs to be sure that the approximate solution is sufficiently close to the true solution – this requires careful, rigorous, and rewarding mathematical analysis.
Furthermore, the famous “Monty Hall problem” will tell you that the theory of probability is not always as simple as you may think; what does it really mean to say that two events occur independently (successively flipping a coin for example)? What is your probability of success in a game of dice? However, there are numerous complications in probability where the events are not independent, for example: two golfers (A and B) are at a driving range, their golf balls are both different, but they have ended up in the same bucket. After picking out 5 balls at random, what is the probability that all of the balls belong to “golfer A”? Of course, we need more information to determine the answer in this case, but it is immediately clear that this “conditional probability” makes things more complex.
These examples outline only a couple of examples of the challenges that participants will tackle during the Immerse Mathematics summer course.
Below is a list of topics that participants will additionally cover during the maths summer programme: