Introduction to Cambridge IA Analysis I 2014

Prof Timothy Gowers is the Cambridge Professor who won the 1998 Fields Medal. Surprisingly he teaches such “low level” undergrad course (Analysis I), but he takes a higher-level approach to tackle the subject with much deeper and broader view.

A master can teach the same subject with a ‘helicopter’ view than an ordinary prof who only confuses the students with over-detailed views of ‘trees’ without letting them see the ‘forest’. (见树不见林).

For example, Prof Gowers brilliantly points out that Analysis is all about Real Number Structure with ONLY 1 AXIOM : Least Upper Bound.

I really enjoy Prof Gowers’s blog. He is the man whom David Hilbert was looking for : “The Pied Piper”, able to bring complicated Math down approachable to the ordinary men on the streets.

Abel also advised us, “Read Direct from the Masters“. Prof Gowers is the Master. 行家一出手, 便知有没有.

Note: I have a ‘lay-man’ analogy of the “Least Upper Bound” of Real Number (or the scary name: Complete Ordered Field). Example: The Bible said God gave the descendants of Noah (i.e. include all mankind now) after The Flood a life span of maximum 120 years. This week the Hong Kong Movie tycoon, billionaire and philanthropist, Sir Run Run Shaw, died at 107 years-old. His Life Span = (0,120), but his extraordinary Least Upper Bound is 107 🙂 Prof Gowers, do you agree with my analogy for non-mathematicians ?

Gowers's Weblog

This term I shall be giving Cambridge’s course Analysis I, a standard first course in analysis, covering convergence, infinite sums, continuity, differentiation and integration. This post is aimed at people attending that course. I plan to write a few posts as I go along, in which I will attempt to provide further explanations of the new concepts that will be covered, as well as giving advice about how to solve routine problems in the area. (This advice will be heavily influenced by my experience in attempting to teach a computer, about which I have reported elsewhere on this blog.)

I cannot promise to follow the amazing example of Vicky Neale, my predecessor on this course, who posted after every single lecture. However, her posts are still available online, so in some ways you are better off than the people who took Analysis I last year, since you will have…

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Senior Wrangler: Singapore Prime Minister

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek Wanless
1973 Lee Hsien Loong (Singapore 3rd Prime Minister)
1977 Glyn Moody
1982 Christopher Budd
1985 Nick Mee
1990 Kevin Buzzard
1992 Ruth Hendry
1993 Ian Dowker
1996 David W. Essex
1997 Alexander Barnar
1998 Ben Joseph Green
1999 Paul Russell
2000 Toby Gee
2001 Mohan Ganesalingam
2002 Jeremy Young
2003 Thomas Barnet-Lamb
2004 David Loeffler
2005 Timothy Austin
2007 Paul Jefferys
2008 Le Hung Viet Bao
2009 Thomas Beck
2010 Zihan Hans Liu
2011 Sean Eberhard
2012 Sean Moss
2013 Arran Fernandez (@18 years old 0 month)

Note * :
Hua Luogeng, like Ramanujian, was a self-paced study mathematician, with only secondary education. Hardy took Hua as research student, let Hua choose either spend 3 years in Cambridge to get a PhD working on one Math topic, or learn everything from Hardy but no PhD degree. Hua chose the “no PhD degree” option but got to become the greatest Mathematician in modern China. Later in 1980s Hua and his students reformed the Chinese Secondary School Math education, endorsed by Deng Xiaoping, propelled China to the World’s Championship of Math Olympiad for nearly 2 decades, ahead of Russia, USA, UK and Europe.