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 717: Functional Analysis 1
  Prerequisites: (CASMA 711) or equivalent. - Theory of Banach and Hilbert spaces, and Hahn-Banach and separation theorems. Dual spaces. Banach contraction mapping theorem. Reflexivity and Krein-Milman theorem. Operator theory. Brouwer-Schauder fixed-point theorems. Applications to probability, dynamical systems, and applied mathematics.
• CAS MA 721: Differential Topology 1
  Prerequisites: (CASMA 511) AND (CASMA512) or equivalent. - Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles.
• CAS MA 722: Differential Topology 2
  Prerequisites: (CASMA 721) - Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius' theorem.
• CAS MA 725: Differential Geometry 1
  Prerequisites: (CASMA 721) or consent of instructor. - Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature
• CAS MA 726: Differential Geometry 2
  Prerequisites: (CASMA 725) - Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes.
• CAS MA 727: Algebraic Topology I
  Prerequisites: (CASMA 564) or equivalent. - Covers singular and simplical homology theory. Cohomology and cup products. Duality on manifolds. Lefschetz and fixed-point formula.
• CAS MA 728: Algebraic Topology 2
  Prerequisites: (CASMA 727) - Topics include homotopy theory, theory of characteristic classes and covering spaces, and cobordism theory.
• 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.