Coolest Mathematical Ideas Learnt in University

After studying mathematics at the undergrad level and electrical engineering at the post-graduate level, here is a list of what I think are the most interesting mathematical ideas I have learnt. It’s good to reflect on one’s studies after a while and see what ideas dominate and gave the greatest sense of satisfaction to learn. Sometimes an idea takes years to fester in the mind before one gets an ‘aha’ moment, as happened with some of those below. I may modify the list over time if I see fit.

• Linear Algebra and Multivariable calculus: a beautiful blend of ideas from algebra and geometry to go from the scalar to the multi-dimensional world:
  • the derivative generalises to a linear map in higher dimensions
  • integration by substitution (change of variable) generalises to the Jacobian determinant
  • the generalisation of the fundamental theorem of calculus (to find line, surface, volume integrals, flux etc.) using the unifying language of differential forms
  • how to solve differential equations such as those of Maxwell and Schrödinger, with particular boundary conditions
  • the spectral theorem – many linear maps (e.g. normal matrices, self-adjoint operators) are orthogonally diagonalisable (convertible to scalar multiplication)
  • the integral transform is simply a change of basis (analogous to a rotation)
  • orthogonal projections can be applied to function spaces for linear least squares estimation/prediction
  • Quantum Mechanics can be formulated using the language of Hilbert spaces
  • interchanging the order of limits (e.g. differentiation, integration) can simplify calculations
  • quadratic optimisation problems can be solved by completing the square in the matrix case
• Signals and Systems
  • many real-world signals can be viewed as functions, and many input-output relations can be described by linear systems
  • for an LTI (linear time-invariant) system the impulse response determines its response to any input
  • concepts of transfer function and the frequency domain, via the Fourier transform
  • compression involves finding an appropriate signal representation (basis), where insignificant terms can be ignored
• The ubiquity of complex numbers and the exponential function – found to be indispensable tools in so many areas:
  • describing growth and decay, circular and oscillatory motion
  • the Gaussian distribution
  • the exponential distribution (to model fading or interarrival times of a Poisson process)
  • as eigenfunctions of LTI systems, so that Laplace/Fourier/z- transforms can be used to convert to frequency domain or convert differential/difference equations to algebraic ones
  • Statistical Mechanics (e.g. Fermi-Dirac) to describe the distribution of particles
  • the characteristic function in probability theory
  • more applications in the fields of control, communications, signal processing
• Probability through measure theory
  • we can view summation as a special case of integration (with respect to the counting measure), so discrete and continuous ideas can be combined
  • the probability of an event is a special case of expectation (integral of an indicator random variable)
  • information theory as a significant application of probability theory (the notion of capacity is based on conditional probability)
• Miscellaneous
  • the use of generating functions in combinatorics
  • point-set topology to generalise ideas such as limits and continuity
  • simplifying spaces or algebraic objects by factoring out equivalence relations to form a quotient space/group/ring (e.g. modular arithmetic works this way)
  • the classification of spaces by assigning topological invariants to them (e.g. Euler characteristic, homology group)
  • quadratic reciprocity – a delightful, unexpected result enabling one to solve quadratic congruences
  • the method of characteristics to solve partial differential equations
  • algorithms and data structures applied to computer science

1. Complex Analysis and the Exponential Function
2. Signals and Systems – DSP, Communications, Control, Information Theory

Best winning streaks in sport

After Roger Federer’s amazing semi-final streak in grand slams (tennis) ended at 23, I read about some of the great winning streaks in sport… here are some of the most impressive ones that I sampled from the internet. Later I will include a list of consistency streaks (e.g. consecutive games played).

• Squash player Jahangir Khan is believed to have the longest winning streak in any professional sport. He was undefeated between 1981 and 1986 winning 555 consecutive matches! Astonishing.
• Dutch wheelchair tennis star Esther Vergeer had a 383 396 429 470-match winning streak, from January 2003 to her last match in 2012. At one point she won 250 sets in a row, with only one going to a tie-breaker!
• Freestyle wrestler Osamu Watanabe was the only modern Olympic wrestler to go undefeated in his entire career – 186 straight wins over two years up to his retirement in 1964.
• Romanian Iolanda Balaş won 140-150 consecutive women’s high jump competitions between Dec 1956 and Jun 1967, improving the world record from 1.74m to 1.91m.
• Track and Field star Edwin Moses won every 400m hurdles race he competed in for 9 years, 9 months and 9 days (Sep 1977 – Jun 1987) – 122 races (including 107 finals) in total!
• American Beach volleyballers Misty May-Treanor and Kerri Walsh won 112 straight matches as a duo from Aug 2007 to Aug 2008.
• US College basketball team UCLA won 88 consecutive NCAA games between 1971 and 1974.
• Julio César Chávez has the longest winning streak in professional boxing – 87 consecutive wins from his first bout in Feb 1980 to Sep 1993.
• Wang Meng has won 83 consecutive 500m women’s short track speed skating races at World Cup, World Championship and Olympic level.
• Carl Lewis won 65 consecutive long jump meets over 10 years until this.
• In Greco-Roman wresting, Russian Alexander Karelin was undefeated over 13 years, winning <does anyone know how many?> consecutive matches.
• Chris Evert won 125 straight tennis matches on clay, Rafael Nadal 81.
• Martina Navratilova won 74 consecutive matches (all surfaces) in 1984, Guillermo Vilas 46.
• Martina Navratilova and Pam Shriver had 109 consecutive wins as a doubles pair between April 1983 and Wimbledon 1985 in which they only lost 14 sets!
• Puerto Rican horse Camarero won 56 straight races between Apr 1953 and Aug 1955.
• The United States won the first 25 America’s Cups, spanning 132 years.
• Byron Nelson won 11 consecutive PGA Tour golf tournaments in 1945.

More info here:
http://www.sportingo.com/more-sports/a10456_longest-winning-streaks-professional-sport

http://en.wikipedia.org/wiki/Winning_streak_%28sports%29

Esther Vergeer

Welcome!

I intend to use this place to write about areas of interest to me that might appeal to others. I will start with maths and sports-related themes.