Detailed Information
Awe-Sums: The Majesty of Maths
In this course, I will deliver an expository treatment of some of the landmark theorems in mathematics. Their history will be presented and their importance in the modern world illustrated by applications in engineering, technology, medicine, finance, geophysics and the arts.
We all love music and enjoy it without being musicians. We all appreciate beautiful paintings but may have little artistic talent. We all delight in great literature without being accomplished writers. It is the same with mathematics: we can enjoy the elegance of brilliant logical arguments and appreciate the beauty of mathematical structures and symmetries without being skilled creators of new theorems.
Mathematics pervades the modern world. It underlies technological advances and governs the social networks and systems that are crucial in our world. Our future wellbeing depends on the application of mathematics. While we may never acquire advanced technical skills in mathematics, it is very valuable and empowering to have a broad appreciation of the subject.
To have a concrete goal, I will aim to give students a general understanding of one of the greatest unsolved problems in mathematics, the Riemann hypothesis. This will mean introducing ideas from several branches of mathematics: set theory, algebra, geometry and analysis.
The emphasis will be on giving a broad qualitative exposition of the key results without becoming embroiled in technicalities. Proofs will be presented only where they illustrate crucial concepts like induction and where they are uncomplicated and short.
Peter Lynch is Emeritus Professor in the UCD School of Mathematical Sciences. Formerly Deputy Director of Met Eireann, he was Professor of Meteorology from 2004 to 2013. He writes a regular mathematical column in The Irish Times and maintains a mathematical blog, thatsmaths.com. He has a passionate interest in all things mathematical.
Whether you loved or hated maths at school, you will find this course fascinating and fulfilling: it is intended is for everyone with a general interest in mathematics. It will also appeal to those with interests in the history of science and maths. No previous knowledge beyond basic school mathematics (pass level) will be assumed.
All topics beyond elementary school maths will be motivated and introduced as required. Ample historical context will be provided. Some key topics are:
- Early History: Archimedes formulas for the Circle and Sphere.
- Elementary Set Theory. Functions. Graphs.
- Prime Numbers (Euclid, Fermat, Gauss, Riemann).
- Index Notation. Powers and Roots. Logarithms
- The Real Number Line. Complex Numbers and the Complex Plane.
- The Idea of Infinity. Infinite Series. (Euler and the Basel Problem).
- The key ideas of Riemann's Hypothesis. Present status.
John Darbyshire (2004): Prime Obsession. Plume Publishing. This tells the story of the Bernhard Riemann and the greatest unsolved problem in mathematics. It is an excellent account of the background leading to Riemann's Hypothesis and subsequent efforts to prove it.
William Dunham (1991): Journey through Genius. Penguin Books. Each chapter of this splendid book is devoted to a “Great Theorem”. Mathematical details are interspersed with interesting historical context.
Marcus Du Sautoy (2004): The Music of the Primes. Harper Perennial. This book traces the history of mathematics leading to the Riemann Hypothesis. It is excellently written and enjoyable and informative to read.
Peter Lynch (2016): That's Maths. Gill Books [To be published in October 2016]. A collection of readable articles on all areas of mathematics and its applications. Articles are self-contained, so that they can be read in any order and the contents assimilated in small doses.