TITLE: Applications of Math: Theory vs. Practice
QUESTION [9 upvotes]: I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics are used today (rather than, say, in the pre-computer age).
So if there are non-academics out there reading MO, I would very much like to hear from them about how their use of mathematical tools may or may not differ from the academic training they received. I don't expect we'll get a very representative sample of math-users on MO, but that's quite all right, anecdotal evidence is all I'm after.
Precision: I want to read about the tools being used rather than, say, an underlying mathematical motivation (which is another legitimate role of mathematics, but not really part of my question). So for instance, you may say that Google's PageRank algorithm was motivated by the theory of Markov chains, but (from what I can tell), I would not say it uses Markov chains.

REPLY [6 votes]: My work draws on various bodies of mathematics. Here's a brief description (by no means exhaustive) of how I use math in my work:

Mathematical programming/optimization: Optimization is used for anything from reconciling actual data to a model's (regression), estimating unknown parameters in a system, to finding the best inputs that will extremize some functional in a dynamic system. The applications are endless. When people think mathematical programming, they think Linear Programming. But convex nonlinear programming is actually very well-established. In fact, large problems in nonconvex optimization are routinely solved (although modeling a nonconvex system can be quite an art). 
Real/functional analysis: useful for understanding optimization algorithms. An understanding of convex functions and sets is crucial -- they lead to global solutions (with guarantees) without solving an NP-hard problem, so we exploit convexity properties whenever possible. (Lipschitz) continuity is another important idea, subgradients etc. are important concepts in nonsmooth optimization. Real analysis is not applied directly, but a good understanding of it is required for reading convergence proofs or descriptions of optimization algorithms.
Computational Geometry: ideas like convex hulls, Voronoi diagrams, etc. are useful in optimization. I use them to partition a problem space into convex regions, or to parametrize a space. The region bounded by convex polytopes can be represented by a set of inequality constraints that can be enforced in an optimization problem. Discrete optimization is used to optimally switch between these regions. 
ODE/DAE theory: used for modeling dynamic systems. In particular, understanding the notion of index in DAEs can help one develop models that are amenable to reliable numerical solution. 
Calculus. Differential calculus is used everywhere (e.g. model sensitivity analysis, automatic differentiation, postoptimality analysis)
Statistics: projection methods like the Karhunen-Loewe transform (related to SVD) are used to reduce the dimensionality of large models constructed from data. They're also the only way to handled correlated/collinear data (in practice, most large datasets in the real world are correlated. The assumption of factor independence built into standard regression techniques often does not hold, so methods like multiple linear regression often have to be modified for instance into principal components regression in order for them to be usable on real world large datasets). Also, tools like time-series analysis are used to construct time series models from data. 
Linear algebra: used almost everywhere. They're the basic building blocks for working with nonlinear systems. In particular, efficient numerical solution of sparse structured matrices is crucial to the efficiency of large-scale nonlinear optimization algorithms (the bottleneck is often in the linear algebra solvers, not in the optimization algorithm itself). Tools like SVD are frequently used. 
Numerical methods: used everywhere. Understanding concepts like numerical conditioning is crucial; when modeling, one wants to end up with a system with a Jacobian that is well-conditioned with respect to inversion.
Misc: Diophantine equations are used to derive certain control laws. Laplace transforms are used for modeling linear-time-invariant systems because they allow differential equations to be manipulated as algebraic ones. Algebraic Riccati equations are solved in the derivation of the Kalman gain. Fixed-point iteration is used to converge decomposed models.