The World’s Best Mathematician 

Terence Tao 陶哲轩 : 

  • Research style: both Collaborative à la  “PolyMath” Project & “Lone Wolf” à la Andrew Wiles (“The Fermat’s Last Theorem”), G. Perelman (“The Poincaré Conjecture”) or Zhang Yitang 张益唐 (“70-million Gap in Twin Primes”)
  • Not attempting the Riemann Hypothese: tools not there yet.
  • His Weakness in Math: Algebraic Topology (拓扑代数) [*]

[*] Algebraic Topology: apply algebra (linear) in Topology: eg. Homology (同调), Co-homology (上同调), Homotopy (同伦) or Homological algebra, etc. A powerful applied mathematics used in Big Data Analytics.

3 Phases of Math Training:

  1. Pre-rigourous: intuition à la Ramanujian 
  2. Rigorous: formal proof
  3. Post-rigourous: 1 + 2 (intuitive bold “cheating” guess, followed by rigorous proof)

Interview With The Smartest Man In The World

Terence Tao 陶哲轩 :
♢ Born in 1975 @ Adelaide. Parents from Hong Kong immigrants to Australia.
♢ Chinese-american
♢ IQ 230-240
♢ Full professor of UCLA at 24
♢ Fields Medalist (2006), the 2nd youngest winner in history (after French J.P. Serre at 27)

The TV interviewer pretending to be an “idiot”, a mentality of most Americans who pride themselves of being poor in Math !

Cayley graphs and the geometry of groups

Terence Tao’s wrote this easy-to-understand Group Theory. It is rare for mathematicians to be also a good writer in explaining difficult math in layman’s terms. Highly recommended!

To be continued (Part 2):

https://terrytao.wordpress.com/2012/05/11/cayley-graphs-and-the-algebra-of-groups/

What's new

In most undergraduate courses, groups are first introduced as a primarily algebraic concept – a set equipped with a number of algebraic operations (group multiplication, multiplicative inverse, and multiplicative identity) and obeying a number of rules of algebra (most notably the associative law). It is only somewhat later that one learns that groups are not solely an algebraic object, but can also be equipped with the structure of a manifold (giving rise to Lie groups) or a topological space (giving rise to topological groups). (See also this post for a number of other ways to think about groups.)

Another important way to enrich the structure of a group $latex {G}&fg=000000$ is to give it some geometry. A fundamental way to provide such a geometric structure is to specify a list of generators $latex {S}&fg=000000$ of the group $latex {G}&fg=000000$. Let us call such a pair $latex {(G,S)}&fg=000000$…

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