The following courses offered by our Department are on analysis
- MA 515 Analysis I. (Fall semester), Metric spaces: contraction mapping principle, Tietze extension theorem, Ascoli-Arzela lemma, Baire category theorem, Stone-Weierstrass theorem, LP spaces. Banach spaces: linear operators, Hahn-Banach theorem, open mapping and closed graph theorems. Hilbert spaces: projection theorem, Riesz representation theorem, Lax-Milgram theorem, complete orthonormal sets.
- MA 532 Ordinary Differential Equations I. (Fall semester) Existence and uniqueness theorems, systems of linear equations, fundamental matrices, matrix exponential, nonlinear systems, plane autonomous systems, stability theory.
- MA 534 Introduction To Partial Differential Equations. (Fall semester) Linear first order equations, method of characteristics. Classification of second order equations. Solution techniques for the heat equation, wave equation and Laplace’s equation. Maximum principles. Green’s functions and fundamental solutions.
- MA 546 Probability and Stochastic Processes I. (Fall semester) Modern introduction to Probability Theory and Stochastic Processes. The choice of material is motivated by applications to problems such as queueing networks, filtering and financial mathematics. Topics include: review of discrete probability and continuous random variables, random walks, markov chains, martingales, stopping times, erodicity, conditional expectations, continuous-time Markov chains, laws of large numbers, central limit theorem and large deviations.
- MA 715 Analysis II. (Spring semester) Integration: Legesgue measure and integration, Lebesgue-dominated convergence and monotone convergence theorems, Fubini’s theorem, extension of the fundamental theorem of calculus. Banach spaces: Lp spaces, weak convergence, adjoint operators, compact linear operators, Fredholm-Fiesz Schauder theory and spectral theorem.
- MA 716 Advanced Functional Analysis. (Spring semester) Advanced topics in functional analysis such as Spectral theory, Reflexive Banach spaces, Uniformly convex Banach spaces, Differential calculus in Banach spaces, Convex analysis, Nonsmooth analysis, Fixed point theorems, Minimax theory and Monotone operators.
- MA 732 Ordinary Differential Equations II. (Spring semester) Existence-uniqueness theory, periodic solutions, invariant manifolds, bifurcations, Fredholm’s alternative.
- MA 734 Partial Differential Equations. (Spring semester) Linear second order parabolic, elliptic and hyperbolic equations. Initial value problems and boundary value problems. Iterative and variational methods. Existence, uniqueness and regularity. Nonlinear equations and systems.
- MA 746 Introduction To Stochastic Processes. (Spring semester) Markov chains and Markov processes, Poisson process, birth and death processes, queuing theory, renewal theory, stationary processes, Brownian motion.
- MA 797 Convex Analysis. Convex sets, support functions, Generalized interiors and polarity, convex functions. Qualification conditions, Infimal convolution, Infimal post composition, the Legendre transform. Subdifferential calculus, Differential calculus for convex functions, Fermat’s rule and its consequences, Parametric duality, Fenchel-Rockafellar duality, Convexity and nonexpansiveness. Special topics: inequalities in information theory and probability, variational methods in machine learning, PDEs and image processing, convexity methods in mechanics, convex modeling in statistics.