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Course Criteria
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3.00 Credits
Review of Riemann integration. Introduction to Lebesgue measure and integration. Holomorphic functions, residue theory, and conformal mappings. PREREQ: MATH600. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Piecewise polynomial and global interpolation, adaptive, Gaussian, and multidimensional quadrature, Runge-Kutta and multistep methods for initial value problems, finite differences for boundary value problems, method of lines for partial differential equations. PREREQ: Multivariate calculus and ordinary differential equations. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
LU and QR factorizations, singular value and eigenvalue decompositions, matrix conditioning, solution of linear systems and linear least-squares problems, iterative methods in linear algebra, descent and quasi-Newton methods of optimization, globalizing convergence, constrained optimization, applications. PREREQ: Elementary linear algebra and programming. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Introduction to modeling and analytical techniques used in solving problems arising in a variety of physical settings. Biological modeling. Derivation of the equations of mathematical physics. Solution behavior of nonlinear systems of ODE's. Use of software to explore solutions to physical systems. PREREQ: One semester of advanced calculus. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Introduction to techniques used in solving problems arising in a variety of physical settings. Sturm-Liouville problems and Green's functions. Methods of solution for the wave, heat and Laplace equations. Variational principles. PREREQ: One semester of advanced calculus. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Terminology of mathematical finance. Asset pricing and interest rate models. Discrete and continuous models for option pricing and fixed-income products. Discussion of various payoff structures, including path-dependent options. May include use of technology to simulate trading and pricing research. PREREQ: Undergraduate-level knowledge of linear algebra, probability, and ordinary differential equations, equivalents of MATH349, MATH302, and MATH350 or STAT470. RESTRICTIONS: Students who received credit in MATH460 are not eligible to take this course without permission.
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3.00 Credits
Introduction to probability theory as background for further work in statistics or stochastic processes. Sample spaces and axioms of probability; conditional probability and independence; random variables and describing their distributions; classical discrete and continuous random variables; mathematical expectation and moments of a distribution; the distribution of a function of a random variable; Chebyshev's inequality; infinite sequences of independent random variables; the weak and strong laws of large numbers; central limit theorems. PREREQ: One semester of advanced calculus. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Classical stochastic processes emphasizing Martingales and Markov chains in discrete and continuous time with examples from random walk, birth and death processes, branching processes and Markov chain Monte Carlo. Possible additional topics include renewal and Markov renewal processes; queues; basic notions of Brownian motion and second-order processes; Markov random fields; point processes. PREREQ: MATH630 or equivalent. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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3.00 Credits
Linear methods for regression (subset Selection, ridge, lasso), Logistic regression. Analysis of the convergence and complexity of common algorithms. Linear discriminant analysis, Principal component analysis, Additive Models, Kernel Smoothing. Cross-validation, Bootstrap, Support Vector Machines, Cluster analysis (K-means, spectral clustering), Undirected graphical models, Expectation maximization algorithm, Introduction to deep learning, Introduction to Bayesian methods. PREREQ: Probability theory and basic statistics (e.g. MATH 350), Multivariable calculus (e.g. MATH 243), Linear Algebra (e.g. MATH 349), Optimization background (e.g. MATH 529) desirable but not necessary, basic computing skills.
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3.00 Credits
Properties of integers, commutative rings, finite fields, elementary group theory, and classification of finite abelian groups. PREREQ: MATH672 or permission of instructor. RESTRICTIONS: Undergraduate students must have a B or better in six credits of MATH at the 400 level or 500 level, exclusive of MATH518 and MATH540.
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