Systematic approach to solving non-standard and competition level math problems on inequalities, infinite sums and products, combinatorics, number theory and games.

Probability spaces. Main families of discrete and continuous distributions and relations among them. Moments, characteristic functions, sequences of random variables, laws of large numbers, Central Limit Theorem.

Principles for testing hypotheses and estimation, confidence intervals, methods of moments, maximum likelihood, information inequality, likelihood ratio tests, goodness of fit and nonparametric methods.

The real number system, metric spaces, limits, continuity, derivatives and integrals, infinite series.

Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and the argument principle.

Methods for static, dynamic, unconstrained, constrained optimization. Gradient, conjugate gradient, penalty methods. Lagrange multipliers, least squares, linear, nonlinear dynamic programming. Application to control and estimation.

Topics in Linear Algebra. Polynomial rings, vector spaces, linear transformations, canonical forms, inner product spaces.

Linear equations and matrices, Gauss elimination, error estimates, iteration techniques; contractive mappings, Newton's method; matrix eigenvalue problems; least-squares approximation, Newton-Cotes and Gaussian quadratures; finite difference methods.

Markov processes in discrete or continuous time; renewal processes; martingales; Brownian motion and diffusion theory; random walks, inventory models, population growth, queuing models, shot noise.

Probability spaces; distributions and characteristic functions; laws of large numbers, central limit problems; stable and infinitely divisible laws; conditional distributions.

Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini's theorem, convergence theorems, differentiation.

Parametric families of distributions, sufficiency. Estimation: methods of moments, maximum likelihood, unbiased estimation. Comparison of estimators, optimality, information inequality, asymptotic efficiency. EM algorithm, jackknife and bootstrap.

Stochastic system models, Dynamic programming, Linear quadratic control, Kalman filtering and estimation, System identification, approximate dynamic programming methods, adaptive control, reinforcement and online learning.

Statistical methods for building intelligent and adaptive systems that improve performance from experiences; focus on theoretical understanding of these methods and their computational implications.

Reasoning under uncertainty, statistical directed and undirected graphical models, temporal modeling, inference in graphical models, parameter learning, decisions under uncertainty.

Modern stochastic control theory, recent developments in reinforcement learning theory, game theory, mean-field analysis and reinforcement learning in practice.

Rational decision making; value and utility functions; distribution theory; invariance relations; copulas; value of information and control; multiattribute utility; decision and game theory; behavioral research.

Consumer/firm behavior; uncertainty; market structure; efficiency; externalities; policy analysis.

National income, employment, inflation; economic cycles; fiscal/monetary policy; financial markets; public debt.

Corporate finance. Valuation; capital budgeting; risk/return analysis; CAPM; cost of capital; corporate policy.

Financial & managerial accounting foundations for users of accounting information. Balance sheets, income statements, cash flows.

Fundraising; startup valuation methods; deal structuring; VC securities; exits & IPOs; fund dynamics.

Applied Finance in Fixed Income. Bond pricing; yield curves; hedging tools (duration/convexity); portfolio analysis; derivatives.