Applications

Some people focus on the beauty and elegance of Mathematics.
Others focus on using mathematics to solve real world problems.
This page is for the latter.

I remember hearing that studying mathematics could prepare me to work in many different fields.
This was a little deceptive because the mathematics turned out to be only a fraction of the preparation needed to work in those fields.

For example:

• Pandemics create a demand for measuring viruses in blood.
• Biochemistry provides the method of Polymerase Chain Reaction (PCR) to amplify a DNA sample (e.g. from blood) to a measureable concentration.
  In each cycle, the target DNA fluoresces and the light is measured.
  The amount of light is a function of the amount of DNA.
• Chemistry provides models of reaction rates as functions of reagent concentrations.
• Mathematics provides an algorithm for finding the model parameters (reagent concentrations) which best match observed measurements.
• Computer science is used to implement the algorithm.
• Statistics is used to evaluate the reliability of the implementation.
  Perhaps the reliability is too low due to noisy data.
  Understanding the basic science underlying the optics and electronic sensors used to make the measurements allows you to
  • Safely identify and remove expected data artifacts using software
  • Improve the hardware design.
    Note: You probably don't need to be able to create new lense coatings or design a new sensor chip from scratch.
    But it would be handy to be able to write the specifications for the components you'd want.

Key Idea: Mathematics is often necessary, but not sufficient.
Solving real world problems often requires tools from multiple academic disciplines.

Many useful tools are found in departments like Statistics, Computer Science, Operations Research or Economics:

• Probability and Statistics
• Linear Programming
• Constrained Optimization
• Portfolio Optimization
• Monte Carlo Simulation

Although some of the mathematical tools might be basic, whole fields have been developed for their application.
For example, linear programming can be seen as applied linear algebra (taken by college freshmen or sophomores), but an in-depth understanding of this method could require years of graduate school.

Many problems are solvable with known techniques on known hardware.
In these cases, using an off-the-shelf library can be sufficient.

But if you have problems at the edge of solvability, memory and computation limits may come up:

• Memory
  
  • How much overhead is being used for function pointers?
  • How are sparse matrices represented?
  • Does memory layout affect algorithm implementation?
  • Data compression: Lossless or Lossy?
• Computation
  • How do numeric gradients compare to recursive analytic chain rule derivatives?
  • How much communication is needed in multi-threaded implementations?
  • Is custom code needed for manipulating sparse matrices?
  • Is there a good enough approximation that runs faster?
  • Would a probabilistic algorithm be adequate?

An of course, any time mathematics is implemented in computer code, there are process questions.

• How is the code stored and shared?
• How does one make the code easy to understand and modify?
• What is the right balance between high performance and maintainability?
• How is the data acquired and processed?

This short outline of related disciplines and potential issues should suggest that those looking to use mathematics to solve real world problems will probably want tools from outside of pure mathematics.