In the end, a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant. Typical examples are Black-Scholes-Merton option pricing formula, which has changed the trading practice on Wall street, and general equilibrium under perfect competition, which estabilishes the superiority of market economy.
Belief is not equal to truth. To some extent, theory is some kind of belief. No matter you admit or not, in most cases, we don't care whether something is true or not, we just concern we believe it or not.
Something of beauty and elegance (metaphysics):
MATH:
1.Analysis: real analysis,functional analysis,partial differential equation, calculus of varionation, point set topology,complex analysis
2. Geometry: manifold, algebraic topology
3. Algebra: abstract algebra, group representation
PROBABILITY and STOCHASTIC PROCESS: measure theory, stochastic process(martingale, diffusion, Levy process), stochastic differential equation(on R or Banach space, driven by jump process, or Brownian motion, or Levy process), connection with functional analysis,
weak convergence, Malliavin calculus(differentiable structures on abstract probability space)
CONTROL:
stochastic optimal control(including optimal stopping, implusive control(quasi varational inequality)), markov decision process(discrete time, continous time), robust control(stability, disturbance attenuation), adaptive control, nonlinear control(geometric method), mean field game, filter and prediction.
ECONOMICS:
(1. economic enviroment; 2.behavior assumptions; 3. economic institutional arrangement; 4. equilibrium; 5. evaluation of the outcomes): micro, macro, money and banking, information economics (principal-agent, mechanism design),
SUPPLY CHAIN Management:
inventory, transportation, outsourcing, demand prediction, pricing and revenue management, risk pooling, channel coordination(contract design)
QUANTITATIVE FINANCE(risk allocation and pricing):
derivative pricing (stochastic volatility, interest rate derivative), investment and portfolio theory, financial economics, real option, risk management
What education provide us essentially are the points of view to perceive the nature and social society. It has nothing to do with practical skills.