The theory and practice of teaching mathematics at the college level. Basic skills, grading methods, cooperative learning, active learning, use of technology, classroom problems, history of learning theory, reflective practice. Open to graduate students in Mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.

Exploration of some of the major ideas and concepts of the school mathematics curriculum from the advanced perspective of a teacher. Emphasis on mathematical reasoning and deep conceptual understanding. Main focus: Proportional reasoning as it constitutes the backbone structure for higher-level mathematical ideas, and mathematical modeling which provides a solid foundation for learning through meaningful problem solving.

Advanced topics in analysis.

Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields.

Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory.

Topics include, but are not restricted to, Computability Theory, Model Theory, and Set Theory.

Advanced topics in geometry and topology. May be repeated for credit with a change of topic.

Advanced topics in Geometry and Topology.

Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.

Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stielties integral, functions of several variables.

General theory of measure and Lebesgue integration, L^p-spaces.

An introduction to the theory of analytic functions, with emphasis on modern points of view.

Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others.

Normed linear spaces and algebras, the theory of linear operators, spectral analysis.

Normed linear spaces and algebras, the theory of linear operators, spectral analysis.

Foundations of harmonic analysis developed through the study of Fourier series and Fourier transforms.

Harmonic analysis on Abelian and non-Abelian locally compact groups, Pontryagin duality, the Peter-Weyl theorem, various Fourier transforms and connections to unitary representation theory.

Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion.

Contemporary theory of stochastic processes, including stopping times, stochastic integration, stochastic differential equations and Markov processes, Gaussian processes, and empirical and related processes with applications in asymptotic statistics.

Group theory, ring theory and modules, and universal mapping properties.

Linear and multilinear algebra, Galois theory, category theory, and commutative algebra. 

Semi-simple rings, Jacobson radical, density theory, Wedderburn's Theorem, representations and characters of groups, orthogonality relations, Burnside's theorem.

Algebraic integers, ideal class group, ramification, Frobenius elements in Galois groups, Dirichlet's unit theorem, localization, and completion. Further topics (zeta-functions, function fields, non-maximal orders) as time permits.

The LU, QR, symmetric, polar, and singular value matrix decompositions. Schur and Jordan normal forms. Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius theory and graph criteria in the theory of non-negative matrices.

Predicate calculus, completeness, compactness, Lowenheim-Skolem theorems, formal theories with applications to algebra, Godel's incompleteness theorem. Further topics chosen from: axiomatic set theory, model theory, recursion theory, computational complexity, automata theory and formal languages.

Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces.

Smooth manifolds, vector fields, differential forms, de Rham cohomology, homology theory, singular (co)homology, Poincaré duality.

This course is an introduction to algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, algebraic curves, singular points, divisors and line bundles, differentials, intersections.

This course introduces further concepts and methods of modern algebraic geometry, including schemes and cohomology.

This course is an introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. The topics include covariant derivatives and connections, geodesics and exponential map, Riemannian metrics, curvature tensor, Ricci and scalar curvature.

Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.

Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.

Cauchy Kowalewsky Theorem, classification of second-order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in Rn.

The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

The risk-neutral model for pricing and hedging derivative financial instruments within the context of binomial and trinomial models of the stock price process.

The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities.

Long-term insurance products, survival and longevity models, life tables, life insurance, life annuities, premium calculations, reserves.

A continuation of Long-Term Actuarial Mathematics I. Topics include multiple state models, multiple decrements, multiple lives, profit and loss analysis, pension plans and funding, retirement benefits, long-term health and disability.

Data analysis for actuaries, linear models including generalized linear models, time series, principal component analysis, decision trees, cluster analysis, statistical computing with R, actuarial applications.

Models for predictive analytics, model building, selection, estimation, validation and diagnostics, and limitations; actuarial applications, and communication of results.

Loss distribution models for claim frequency and severity, aggregate risk models, coverage modifications, risk measures, construction and selection of parametric models, introduction to simulation.

Credibility theory, pricing for short-term insurance coverages, reinsurance, experience rating, risk classification, introduction to Bayesian statistics.

Techniques for estimating unpaid claims, use of claims triangles, underlying statistical theory behind the techniques, basic adjustments to data and estimation techniques to account for internal and external environments, estimating recoveries, model adequacy and reasonableness.

The continuation of Math 5620, focusing on the mathematics of finance: measurement of financial risk and the opportunity cost of capital, the mathematics of capital budgeting and securities valuation, mathematical analysis of financial decisions and capital structure, and option pricing theory. Provides VEE credit in the Corporate Finance subject area for Society of Actuaries and Casualty Actuarial Society requirements.

An introduction to the standard models of modern financial mathematics including martingales, the binomial asset pricing model, Brownian motion, stochastic integrals, stochastic differential equations, continuous time financial models, completeness of the financial market, the Black-Scholes formula, the fundamental theorem of finance, American options, and term structure models.

The theory and practice of stochastic models to analyze and value interest rate derivatives, and practical issues in the markets where they are traded.

Optimization; linear and non-linear programming; data mining and machine learning in a financial context.

Data structures and algorithms; regression; classification; clustering; recommender systems; anomaly detection; Big Data tools; databases.

Advanced topics in Actuarial Science.

Prerequisites and recommended preparation vary. With a change in content, may be repeated for credit. Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatisfactory).

With a change in content, may be repeated for credit. Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatisfactory).

Prerequisites and recommended preparation vary. With a change in content, may be repeated for credit for a total of 30 credits.

With a change in content, may be repeated for credit.

Students who have well defined mathematical problems worthy of investigation and advanced reading should submit to the department a semester work plan.

Participation in internship and paper describing experiences. May be repeated for a total of six credits.

Participation and presentation of mathematical papers in joint student faculty seminars. Variable topics.