GRAD > MATH

## Mathematics

#### MATH 545 Probability & Statistics I for Secondary Teachers +

**Description:**

This course presents the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. Topics also include basic ideas and techniques of statistical analysis. More Info

**Offered in:**#### MATH 570 History of Mathematics for Secondary Teachers +

**Description:**

This course traces the development of mathematics from ancient times up to and including 17th century developments in the calculus. Emphasis is on the development of mathematical ideas and methods of problems solving. Attention will also be paid to the relevance of history to mathematics teaching as well as investigation into the origins of non-Euclidean geometry even though this comes well after Newton and Leibniz, because of its relatively elementary character and fascinating nature. More Info

**Offered in:**#### MATH 597 Special Topics +

**Description:**

An advanced course offering intensive study of selected topics in mathematics. More Info

**Offered in:**#### MATH 620 Combinatorial Analysis +

**Description:**

This course is an introduction to combinatorics: a branch of mathematics that studies the existence, enumeration, analysis, and optimization of discrete structures that satisfy certain properties. Topics include counting distributions and colorings, sieve methods, generating functions, permutation spaces, partially ordered sets, Ramsey theory, and matching theory, with applications to computational problems. More Info

**Offered in:**#### MATH 625 Numerical Analysis I +

**Description:**

This course introduces the essential ideas and computational techniques that modern scientists or engineers will need in order to carry out their work. In most scientific modeling projects, investigators have to deal with very large systems of linear equations, understanding of which requires powerful computers, and a firm understanding of the vast number of existing pertinent algorithms. The main goal of the course is to provide an introduction to algorithmic and mathematical foundations of high-performance matrix computations. Topics include linear algebraic systems, matrix decompositions, least squares problems, eigenvalue problems, sparse linear systems and linear dynamical systems. The course will emphasize mathematical and software engineering methods that will allow students to fully participate at all levels of algorithm design and implementation. More Info

**Offered in:**#### MATH 626 Numerical Analysis II +

**Description:**

This course introduces the essential ideas and computational techniques that modern scientists or engineers will need in order to carry out their work. In most scientific modeling projects, investigators have to deal with very large systems of linear equations, understanding of which requires powerful computers, and a firm understanding of the vast number of existing pertinent algorithms. The main goal of this course is to provide an introduction to numerical techniques for approximating solutions of nonlinear problems. Topics include Polynomial Interpolation and Approximation, Numerical Methods for Root finding, Numerical Integration, Numerical Differentiation, and Numerical Solutions of Differential Equations. The course will emphasize mathematical and software engineering methods that will allow students to fully participate at all levels of algorithm design and implementation. More Info

**Offered in:**#### MATH 640 Computational Algebraic Topology +

**Description:**

This course covers foundational aspects of combinatorial algebraic topology with a view towards applications to computational data analysis. It will cover basic geometric-combinatorial constructions, and it will concentrate on the study of invariants associated to topological spaces, such as homology, Euler characteristic, Betti numbers, etc. The mathematical formalism will be as basic as possible and the course will focus on examples. The concept of cubical homology will be discussed and its applications to images. Some other invariants to understand the underlying topology of data sets will be discussed, such as persistent homology as well as other homology theories associated to data sets "approximating" a space. We will give an introduction to computational environments such as JavaPlex and CHomP, to obtain Betti numbers and bar-codes. Some examples to be discussed can include the invariants associated to conformation spaces of proteins, the space of natural images and other higher dimensional examples. More Info

**Offered in:**#### MATH 642 Probabilistic Simulation +

**Description:**

Simulation is a powerful tool in dealing with systems that are too complex to solve analytically. Probabilistic simulations using Monte Carlo techniques provide a way to emulate the behavior of the system and generate random samples for each output variable of interest that can then be analyzed by statistical methods. This course provides a practical introduction to Monte Carlo simulations and statistical methods for analyzing random samples generated by such simulations. The following topics will be discussed: random number generators, generating continuous and discrete random variables, generating multivariate random variables, statistical analysis of the output data and methods for fitting probability distributions to the data. The course will emphasize the practical implementations of these techniques using the R statistical program language. More Info

**Offered in:**#### MATH 647 Probability Models +

**Description:**

This is an introductory course on probability models with a strong emphasis on stochastic processes. The aim is to enable students to approach real-world phenomena probabilistically and build effective models. The course emphasizes models and their applications over the rigorous theoretical framework behind them, yet critical theory that is important for understanding the material is also covered.Topics include: discrete Markov chains, continuous-time Markov chains, Poisson processes, renewal theory, Brownian motion, and martingales. Optional topics: queuing theory, reliability theory, and random sampling techniques. Applications to biology, physics, computer science, economics, and engineering will be presented. More Info

**Offered in:**#### MATH 648 Statistical Learning +

**Description:**

This course will provide an introduction to methods in statistical learning that are commonly used to extract important patterns and information from data. Topics include: linear methods for regression and classification, regularization, kernel smoothing methods, statistical model assessment and selection, and support vector machines. Unsupervised learning techniques such as principal component analysis and generalized principal component analysis will also be discussed. The topics and their applications will be illustrated using the statistical programing language R. More Info

**Offered in:**#### MATH 673 Structure and Dynamics of Complex Networks I: Structural Properties +

**Description:**

This course on complex networks is intended for graduate students in mathematics, physics, biology, computer science and engineering who wish to learn about the major ideas and techniques developed in--and the results recently discovered in--one of the most important interdisciplinary research fields. The main concepts and results are structured so as to be accessible to those with only a good knowledge of basic calculus and probability. The ideas and methods of network theory covered form a foundation for the study of the structure of complex networks.The course is devoted to the introduction of essential network concepts, the development of new network models, and the characterization of the structural properties of real world networks. It will contain topics from graph theory, social networks analysis, statistical physics, systems biology, ecology, and computer science. The course will combine lectures, readings, and discussions of the recent literature. Throughout the course theoretical ideas and methods will be presented in concert with numerous applications. During the course computational methods will be emphasized and appropriate software for network analysis will be used. More Info

**Offered in:**#### MATH 674 Structure and Dynamics of Complex Networks II: Dynamical Processes +

**Description:**

This course on complex networks is intended for graduate students in mathematics, physics, biology, computer science and engineering who wish to learn about the major ideas and techniques developed in - and the results recently discovered in - one of the most important interdisciplinary research fields. The students will find the main concepts an results presented in a way that is accessible to those with only a good knowledge of basic calculus and probability. They will learn the ideas and methods of network theory that will allow them to study the structure of complex networks. The course is devoted to the introduction of essential network concepts, the development of new network models, and the characterization of the structural properties of real world networks. It will contain topics from graph theory, social networks analysis, statistical physics, systems biology, ecology, and computer science. The course will combine lectures, readings, and discussions of the recent research literature. Throughout the course theoretical ideas and methods will be presented in concert with numerous applications. During the course computational methods will be emphasized and appropriate software for network analysis will be used. More Info

**Offered in:**#### MATH 677 Symbolic Computation +

**Description:**

The course will cover computational arithmetic and algorithms in a number of contexts: floating point, multiple precision, large integer, rational, polynomial and power series, with an emphasis on exact symbolic calculations. Additional topics, including sparse matrix and polynomial operations will be included. The course will be evaluated via programming course projects in a high level language such as C++. The class will not only give theoretical understanding, but will also provide "hands-on" experience in writing mathematical software. By the end of the course, students will be expected to have all necessary practical skills to write and test a mathematical library in a high level language. More Info

**Offered in:**#### MATH 680 Introduction to Computational Algebraic Geometry +

**Description:**

This course provides a strong foundation for the study of computational algebraic geometry and its applications, both within and outside mathematics. It has two foci. The first is the algebra-geometry dictionary, going back to the ideas of Descartes, by which one can translate geometric ideas into algebraic ones, and vice versa. The second is Buchberger's algorithm, which extends the familiar Gauss-Jordan elimination procedure to systems of polynomial equation. By means of this algorithm one can compute almost everything worth knowing about affine algebraic varieties. Computer algebra systems will be used for computation and visualization of this algorithm and its ramifications. Applied areas of exploration may include robotics, computer aided design, automatic theorem proving, invariant theory, projective geometry, and computer vision. In addition, highly motivated students will be prepared to participate meaningfully in current research in invariant theory and the geometry of nilpotent orbits. More Info

**Offered in:**#### MATH 696 Independent Study +