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June 25, 2010

Coolest Mathematical Ideas Learnt in University

Filed under: mathematics — ckrao @ 12:53 pm

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
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