Mathematics & Statistics

• 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: Estimation Theory
  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: Hypothesis Testing
  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.
• CAS MA 783: Advanced Stochastic Processes
  Prerequisites: (CASMA 779 or CASMA 780) or consent of instructor. - Proof-based approach to stochastic processes. Brownian motion. Continuous martingales. Stochastic integration. Ito formula. Girsanov's Theorem. Stochastic differential equations. Feynman-Kac formula. Markov Processes. Local times. Levy processes. Semimartingales and the general stochastic integral. Stable processes. Fractional Brownian motion.
• CAS MA 821: TOPICS IN GEOM
  TOPICS IN GEOM
• CAS MA 822: Topics in Geometry and Topology
  rerequisites: (CASMA 725) AND (CASMA 728) AND (CASMA 726 or CASMA 727) - Advanced seminar in topics in differential geometry, topology and mathematical physics of current research interest.
• CAS MA 841: Seminar: Algebra
  ALGEBRA SEMINAR
• CAS MA 842: Seminar: Algebra
  ALGEBRA SEMINAR
• CAS MA 861: Seminar: Applied Mathematics
  APP MA SEM
• CAS MA 876: Seminar: Partial Differential Equations
  PDE SEMINAR