Posted on 23 February, 2018 in Students

National Mathematics Summer School

National Mathematics Summer School

The National Mathematics Summer School (NMSS) is a two week program held each year in January at the Australian National University for mathematically gifted and talented senior high school students. Students who have completed Year 11 can apply and this year Darwin High School student Nafi Mazid attended. 

By student Nafi Mazid

From January 7th to January 20th, I had the opportunity to attend the National Mathematics Summer School at the Australian National University. NMSS is a two-week residential school at which mathematically gifted and talented students who have finished Year 11 are able to explore and expand their understanding of a wide range of mathematical topics.

The broadest topic covered at NMSS was Number Theory, which ran for the entirety of the two weeks. The course had a focus on prime numbers, the Euclidean Algorithm, the Division Algorithm, and the Chinese Remainder Theorem. However, the course diverged into a wide variety of subtopics in number theory, such as modulo arithmetic, Gaussian integers, polynomial rings and fields, Diophantine equations, greatest common divisors, Magic Tables, continued fractions and many more. The lecturer of this course was director of the NMSS, Professor Terry Gagan, whose stories about famous and influential mathematicians such as Archimedes, Gauss, Euler, and Ramanujan were a great start to our morning lectures.

The two other courses taught at NMSS were Knot Theory in the first week by lecturer Ben Burton, and Projective Geometry in the second week by lecturer Leanne Rylands. Knot Theory, as the name suggests, is a branch of mathematics that deals with knots. In Knot Theory, we create what are known as ‘invariants’, functions ranging from tricolouring (simply colouring strands of knots) to the Jones polynomial, to try and distinguish whether knots are equivalent, or whether they can be untangled into and unknot (simple loop). In Projective Geometry, we attempt to represent higher dimensions in lower dimensions, such projecting 3-dimensions in 2-dimensions. This is done with the projective plane, which uses homogenous coordinates to add a line of infinity to the real plane, a line at which parallel lines meet.

As you can imagine, there is a lot of diverging threads in the courses for students to consider and work on. In fact, we did not have to necessarily do all the problems in order, but we could choose a few problems from the problem sets that interested us and work on those problems. There was no expectation to solve every problem, as there simply is not enough time in the two weeks. However, it is satisfying solving a difficult problem after persevering after many hours, until a tutor points out a small nagging detail in your proof that needs a bit more thinking. That’s where the focus of the NMSS lies. It’s not about being shown the answer by one of the lecturers of tutors, but about being given the problem-solving skills and processes required to tackle challenging and thought-provoking maths problems. A lot of time is given at NMSS for working on problems individually, during morning and afternoon tutorials and private study in the evening. There is also time to collaborate and share ideas with other students.

On the weekend, we had the chance to explore the many interesting parts of Canberra. There were also organised trips to Black Mountain tower, Mount Stromlo Observatory, and the supercomputer on the ANU campus, which is the largest supercomputer in Australia. We also had the chance to explore other parts of the ANU campus, which you may be interested in especially if you plan to study at ANU. There were guest lectures from an NMSS alumni, a software engineer at Google, and from the ANU vice-chancellor and Nobel Prize Laureate Professor Brian Schmidt. His lecture was on ‘The State of the Universe’.

The NMSS was a wonderful experience and an opportunity I am glad I took. While it may be challenging, I felt like I developed lifelong skills and connections. It is a great way to experience fields of mathematics that you would not be able to explore in the classroom.

“Think deeply of simple things”

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