Courses

The listing of a course description here does not guarantee a course’s being offered in a particular term. Please refer to the published schedule of classes on the MyBU Student Portal for confirmation a class is actually being taught and for specific course meeting dates and times.

• CAS MA 731: Lie Groups and Lie Algebras
  Prerequisites: (CASMA 721) AND (CASMA741) - Classical Lie groups, associated Lie algebras, exponential map, closed subgroups and homogeneous spaces, classification of simple Lie algebras, and elementary representation theory of Lie algebras. Selection of applications to analysis, geometry, or algebra.
• CAS MA 741: Algebra 1
  Prerequisites: (CASMA541) AND (CASMA542) or equivalent, or consent of instructor. – Basic properties of groups, rings, fields, and modules. Specific topics include the Jordan-Holder and Sylow theorems, local rings, theory of localization, modules over PIDs, and Galois theory.
• CAS MA 742: Algebra 2
  Prerequisites: (CASMA 741) or consent of instructor. - Advanced topics in algebra. Linear and multilinear algebra, commutative algebra, and an introduction to category theory and homological algebra. Further topics may include representation of groups, completions, real fields, and elementary algebraic number theory and algebraic geometry.
• CAS MA 743: Algebraic Number Theory 1
  Prerequisites: (CASMA 741) or consent of instructor. - Algebraic integers, completions, ramification and the discriminant, cyclotomic and quadratic fields, ideal class groups, Dirichlet's unit theorem, ideles, and adeles. Further topics are chosen from analytic number theory, class field theory, and the theory of Diophantine equations.
• CAS MA 745: Algebraic Geometry 1
  Prerequisites: (CASMA 741) or consent of instructor. - Affine and projective varieties, morphisms and rational maps, nonsingular varieties, Bezout's theorem, and an introduction to sheaves and schemes. Further topics are chosen from the advanced theory of schemes, algebraic curves, Riemann-Roch theorem, algebraic surfaces, and sheaf cohomology.
• CAS MA 746: Algebraic Geometry 2
  Prerequisites: (CASMA 745) - Continuation of topics in algebraic geometry begun in CASMA 745, including sheaves, schemes, sheaf cohomology, and further study of algebraic curves and surfaces.
• CAS MA 750: Nonparametric and Semiparametric Data Modeling
  Prerequisites: (CASMA 575) AND (CASMA 581) or consent of instructor. - Introduces theory and methods of non- and semiparametric data analysis. Topics include scatterplot smoothers, bias/variance trade-off, selection of smoothing parameter, generalized additive model, smoothing spline, and Bayesian nonparametric models. Applications in various fields are discussed.
• CAS MA 751: Statistical Machine Learning
  Prerequisites: (CASMA 213 or CASMA 681) AND (CASMA 242) AND (CASMA 665) AND (CASMA 666) or consent of instructor. - Provides an overview of statistical time-series modeling for neuroscience applications. Topics include regression and generalized linear modeling, state space modeling, and parametric and nonparametric spectral analysis. Special emphasis on reading and discussing applications in recent literature.
• CAS MA 752: Mathematical Foundations of Deep Learning
  Rigorous introduction to mathematical foundations of deep learning. Universal approximation theory, stochastic gradient descent algorithms and their convergence properties, approximation theory, neural tangent kernel, mean field overparametrized regime, different deep neural network architectures, reinforcement learning, deep learning for dynamical systems.
• CAS MA 765: Time Series Analysis for Neuroscience Research
  Prerequisites: (CASMA 213 or CASMA 681) AND (CASMA 242) AND (CASMA 665) AND (CASMA 666) or consent of instructor. - Provides an overview of statistical time-series modeling for neuroscience applications. Topics include regression and generalized linear modeling, state space modeling, and parametric and nonparametric spectral analysis. Special emphasis on reading and discussing applications in recent literature.
• CAS MA 769: Mathematical Neuroscience
  Fundamental questions, models, and methods in mathematical and theoretical neuroscience. For example: biophysical and reduced single-neuron models, synaptic plasticity and learning, population density and mean field approaches. Mathematical methods as needed, such as applied dynamical systems and stochastic processes.
• CAS MA 770: Mathematical and Statistical Methods of Bioinformatics
  Prerequisites: graduate standing or advanced undergraduate math/statistics major, (CASMA 225) AND (CASMA 242), and previous work in mathematical analysis and probability. - Mathematical and statistical bases of bioinformatics methods and their applications. Hidden Markov models, kernel methods, mathematics of machine learning approaches, probabilistic sequence alignment, Markov chain Monte Carlo and Gibbs sampling, mathematics of phylogenetic trees, and statistical methods in microarray analysis.
• CAS MA 771: Introduction to Dynamical Systems
  Prerequisites: (CASMA 711) or equivalent, or consent of instructor.- Diffeomorphisms and flows; periodic points, nonwandering points, and recurrent points; hyperbolicity, topological conjugacy, and structural stability; stable manifold theorem; symbolic dynamics; Axiom A and chaotic systems; ergodic theory.
• CAS MA 775: Ordinary Differential Equations
  Prerequisites: (CASMA 242 & CASMA 511 & CASMA 512) or equivalent, or consent of instructor.- Stable and center manifolds theorem, linearization of vector fields, variational equations, Floquet theory and Poincare maps for periodic orbits, bifurcation of rest points, averaging theory, topics from singular perturbations, Hamiltonian systems, nonlinear oscillations, normal forms, and applications.
• CAS MA 776: Partial Differential Equations
  Prerequisites: (CASMA 711) equivalent, or consent of instructor. - Hyperbolic, elliptic, and parabolic equations. Characteristics and separation of variables. Eigenvalue problems, Fourier techniques, Sobolev spaces, and potential theory. Introduction to pseudodifferential operators.
• CAS MA 777: Multiscale Methods for Stochastic Processes and Differential Equations
  Prerequisites: (CASMA 581) AND (CASMA 583) or equivalent, AND (CASMA 226 or CAS MA 231) or equivalent. - Methods and models for the analysis of systems that possess many characteristic length and time scales. Asymptotic expansions, coarse-graining of multiscale stochastic models, mathematical analysis and statistical inference. Balance of theory and concepts illustrated via various applications.
• CAS MA 779: Probability Theory 1
  Prerequisites: (CASMA 511) or consent of instructor. - Introduction to probability with measure theoretic foundations. Fundamentals of measure theory. Probability space. Measurable functions and random variables. Expectation and conditional expectation. Zero-one laws and Borel-Cantelli lemmas. Chracteristic functions. Modes of convergence. Uniform integrability. Skorokhod representation theorem. Basic limit theorems.
• CAS MA 780: Probability Theory 2
  Prerequisites: (CASMA 711) - Probability topics important in applications and research. Laws of large numbers. Three series theorem. Central limit theorems for independent and non-identically distributed random variables. Speed of convergence. Large deviations. Laws of the iterated logarithm. Stable and infinitely divisible distributions. Discrete time martingales and applications.
• CAS MA 781: Advanced Mathematical Statistics 1
  Prerequisites: (CASMA 581) AND (CASMA 582) or consent of instructor. - Review of probability, populations, samples, sampling distributions, and delta theorems. Parametric point estimation. Rao-Cramer inequality, sufficient statistics, Rao-Blackwell theorem, maximum likelihood estimation, least squares estimation, and general linear model of full rank. Confidence intervals. Bayesian analysis and decision theory.
• CAS MA 782: Advanced Mathematical Statistics 2
  Prerequisites: (CASMA 781) or consent of instructor. - Parametric hypothesis testing, uniformly and locally the most powerful tests, similar tests, invariant tests, likelihood ratio tests, linear model testing, asymptotic theory of likelihood ratio, and chi-squared test. Logit and log-lin analysis of contingency tables.