First semester of the two-semester sequence MATH 101-102. Detailed, in-depth review of manipulative algebra; introduction to functions and graphs, including linear, quadratic, and rational functions.

The trigonometry topics of MATH 104.

Fundamental and relevant mathematics for prospective elementary school teachers, including whole numbers and place value operations with whole numbers, number theory, fractions, ratio and proportion, decimals, and percents. For Pre-Early Childhood and Pre-Elementary Education majors only.

Linear equations and inequalities, matrices, linear programming with applications to business, probability and discrete random variables.

Honors section of Math 127.

Continuity, limits, and the derivative for algebraic, trigonometric, logarithmic, exponential, and inverse functions. Applications to physics, chemistry, and engineering.

The definite integral, techniques of integration, and applications to physics, chemistry, and engineering. Sequences, series, and power series. Taylor and MacLaurin series. Students expected to have and use a Texas Instruments 86 graphics, programmable calculator.

Techniques of calculus in two and three dimensions. Vectors, partial derivatives, multiple integrals, line integrals.

Basic concepts of linear algebra. Matrices, determinants, systems of linear equations, vector spaces, linear transformations, and eigenvalues.

Descriptive statistics, elements of probability theory, and basic ideas of statistical inference. Topics include frequency distributions, measures of central tendency and dispersion, commonly occurring distributions (binomial, normal, etc.), estimation, and testing of hypotheses.

The goal of this course is to help students learn the language of rigorous mathematics. Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers), and number theory (divisibility, Euclidean algorithm, congruences). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

Students will develop skills in writing, oral presentation, and teamwork. The first part of the course will focus on pre-professional skills, such as writing a resume, cover letter, and graduate school essay and preparing for interviews. Subsequent topics will include presenting mathematics to a general audience, the role of mathematics in society, mathematics education, and clear communication of mathematical content. The end of the term will be dedicated to the research process in mathematics and will include grant writing, research paper, and professional presentation.

Introduction to abstract algebra, focusing primarily on the theory of groups. There will be regular problem assignments and quizzes, as well as midterm and final examinations.

An introduction to functions of a complex variable. Topics include: Complex numbers, functions of a complex variable and their derivatives (Cauchy-Riemann equations). Harmonic functions. Contour integration and Cauchy's integral formula. Liouville's theorem, Maximum modulus theorem, and the Fundamental Theorem of Algebra. Taylor and Laurant series. Classification of isolated singularities. Evaluation of Improper integrals via residues.
Conformal mappings.

This course covers the basics of differential and integral calculus in many variables: differentiability, directional and partial derivatives and gradient of functions; critical points without or with constraints (Lagrange-multipliers/tangential-gradient) and the Hessian; vector fields and differential forms; divergence, curl and exterior derivative; line- and surface-integrals; the fundamental theorem of calculus (Gauss/Green/Stokes/Thomson). If time and taste permit, topics from physics (fluids and electromagnetism) and differential geometry (curves and surfaces in space) may also be explored.

This is a rigorous introduction to some topics in mathematics that underlie areas in computer science and computer engineering, including graphs and trees, spanning trees, colorings and matchings; the pigeonhole principle, induction and recursion, generating functions, and (if time permits) combinatorial geometry. The course integrates learning mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Student groups will be formed to investigate a modeling problem and each group will report its findings to the class in a final presentation. This course satisfies the university's Integrative Experience (IE) requirement for math majors.

We will explore several approaches to geometry: constructions with straight-edge and compass, axiomatic approach of Euclid and Hilbert, analytic geometry via linear algebra, and Klein's approach using symmetries and transformations. This will open the doors to many non-Euclidean flavors of geometry. Projective and spherical geometry will be studied in some detail.

The William Lowell Putnam Mathematics Competition is the most prestigious annual contest for college students. While the problems employ topics from a standard undergraduate curriculum, the ability to solve them requires a great deal of ingenuity, which can be developed through systematic and specific training. This class aims to assist the interested students in their preparation for the Putnam exam, and also, more generally, to treat some topics in undergraduate mathematics through the use of competition problems.

This course presents a rigorous development of the real number system and fundamental results of calculus. We will study the real numbers and their topology, convergence of sequences, integration and differentiation, and sequences and series of functions. Emphasis will be placed on rigorous proofs.

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing financial instruments, or "derivatives." The central topic will be options, culminating in the Black-Scholes formula. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis.

This is a second course in Linear Algebra building upon the concepts and techniques introduced in Math 235. We will study the decomposition of matrices, particularly the LU, QR, and singular value decompositions. We also study vector spaces and linear transformations, inner product spaces, orthogonality, and spectral theory. We will emphasize applications of these techniques to various problems including, as time permits: solutions of linear systems, least-square fitting, search engine algorithms, error-correcting codes, fast Fourier transform, dynamical systems.

The coursework will be a mix of proof and computation. For the latter students may choose Matlab or one of the open source programs: Scilab or Octave (both similar to Matlab).

The course will introduce basic numerical methods used for solving problems that arise in different scientific fields. Properties such as accuracy of methods, their stability and efficiency will be studied. Students will gain practical programming experience in implementing the methods. We will cover the following topics (not necessarily in the order listed): Finite Precision Arithmetic and Error Propagation, Linear Systems of Equations, Root Finding, Interpolation, least squares, Numerical Integration, Numerical Solution of Ordinary Differential Equations.

This course provides a calculus based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

This course provides a calculus based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

This course provides a calculus based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

Regression analysis answers questions about the dependence of a response variable on one or more predictors, including prediction of future values of a response, discovering which predictors are important, and estimating the impact of changing a predictor or a treatment on the value of the response. This course focuses on linear regression, which is the basis for many modern, advanced regression techniques, including those used by statisticians, machine learners, and data scientists.

In addition to the usual topics in linear regression, this course also emphasizes (1) graphical methods, because it’s important to visualize your data and not just rely on numerical output from computer packages, and (2) diagnostics, because it’s important to check that any regression analysis accurately represents your data. The course will teach and use the R statistical language.

State space models in general, and dynamic linear models in particular, are useful for many types of data and have proven especially popular for time series. After a general introduction to state space models, this course focuses on dynamic linear models, emphasizing their Bayesian analysis. When possible, we show how to calculate estimates and forecasts in closed form; but for more complex models, we use simulation and the dlm package in R. The course includes many detailed examples based on real data sets. No prior knowledge of Bayesian statistics or time series analysis is required, although familiarity with basic statistics and R is assumed.

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Topics to be chosen from: Linear systems and exponential matrix solutions; Well-posedness for nonlinear systems; Floquet theory for linear periodic systems. Qualitative theory: limit sets, invariant sets and manifolds. Stability theory: linearization about equilibria and periodic orbits, Lyapunov functions. Numerical simulations will be used to illustrate the behavior of solutions and to motivate the theoretical discussion.

The first part of the course concerns general topology, often called point-set topology. At the most basic level this is just the study of continuous functions between spaces and properties that are preserved by them. The language of point-set topology underlies modern formulations of many areas of geometry and analysis. In the second part of the course we will do some elementary algebraic topology, including defining the fundamental group, one of the most basic and useful topological invariants of a space.

In this course we will cover some of fundamental ideas from analysis, statistics, and optimization that are relevant to methods in machine learning and statistical inference. The class will cover not only the most well known linear methods, but also the more recently developed nonlinear methods that use the intuition from classical topics in PDE and the calculus of variations, such as the theory minimal surfaces, optimal transport, and gradient flows. The class will be more theoretical than an ordering machine learning class, however, there will still be plenty of coding as we put the theory into practice.

The target audience includes but is not limited to:
(1) Mathematics graduate students interested in areas such as: convex analysis, statistics, probability, optimization, elliptic PDE, calculus of variations, graph theory, and even Riemannian geometry.
(2) Computer science and electrical engineering graduate students interested in image/signal processing, statistical inference, analysis on networks, linear programming, nonlinear optimization.

An introduction to algebraic number theory. Topics include rings of integers, Dedekind domains, ideal theory in algebraic number fields, class groups, units, factorization in Galois extensions, and an introduction to L-functions.

We discuss first what are rare events, why are they important, how are they modeled and why their simulation is a significant high performance computing challenge. We will discuss some of the existing methods and will apply them in prototype systems in applications selected from (among others) queueing theory, molecular dynamics, biochemical reaction networks and finance. Furthermore, we will discuss mathematical theories for rare events and in particular elements of the theory of large deviations and its insights in providing better sampling algorithms for rare events. Finally, will also discuss recent progress on the uncertainty quantification and sensitivity analysis of rare events, that allow us to assess their impact on the predictions of statistical models.

Math 605: In this course we will build up the tools and foundations of probability theory needed for understanding and using statistical theories, stochastic simulation, and stochastic processes. Topics covered in this class are measure and integration (construction of probability spaces), distribution functions, random variables and their simulation, convergence of random variables, laws of large numbers and Monte-Carlo methods, concentration inequalities, central limit theorem, information theory, random walks.

A class in real analysis (integration theory, Math 623) is not required for this class as we will review, as needed, the basic constructions of probability spaces. It can be beneficial to take these two classes concurrently for the student who wants and in-depth understanding of the mathematical foundations of probability.

The primary goal of the class is to understand and master the basic probability tools and concepts needed in modern applications such as in stochastic simulation, data science and machine learning, engineering, and economics. Motivated students with various backgrounds are welcome.

The class is the first part of the sequence Math605/606 and Math 606 will cover the theory of stochastic processes and stochastic simulation: Martingale theory, Poisson processes, Markov chains in discrete and continuous time, Markov chain Monte-carlo, Brownian motion.

Regression is the most widely used statistical technique. In addition to learning about regression methods this course will also reinforce basic statistical concepts and introduce students to "statistical thinking" in a broader context. This is primarily an applied statistics course. While models and methods are written out carefully with some basic derivations, the primary focus of the course is on the understanding and presentation of regression models and associated methods, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection, regression models and methods in matrix form. With time permitting, further topics include an introduction to weighted least squares, regression with correlated errors and nonlinear (including binary) regression.

This course is intended to provide a foundation in statistical machine learning with emphasis on statistical methodology as it applies to large-scale data applications using the graphical model framework. At the end of this course, students will be able to build and test a latent variable statistical model with companion inference algorithm to solve real problems in a domain of their interest including genomics, recommendation systems, topic modeling, and other high-dimensional data problems. Course topics include: exponential families, sufficiency and conjugacy, graphical model framework and approximate inference methods such as expectation-maximization, variational inference, and sampling-based methods. Additional topics may include: nonparametric Bayesian methods, cross-validation, bootstrap, empirical Bayes, and deep learning networks.