Course Descriptions

Second semester of the two-semester sequence MATH 101-102. Detailed treatment of analytic geometry, including conic sections and exponential and logarithmic functions. Same trigonometry as in 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.

Continuation of MATH 127. Elementary techniques of integration, introduction to differential equations, applications to several mathematical models in the life and social sciences, partial derivatives, and some additional topics.

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.

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

The field of Data Science encompasses methods, processes, and systems that enable the extraction of useful knowledge from data. Foundations of Data Science introduces core data science concepts including computational and inferential thinking, along with core data science skills including computer programming and statistical methods. The course presents these topics in the context of hands-on analysis of real-world data sets, including economic data, document collections, geographical data, and social networks. The course also explores social issues surrounding data analysis such as privacy and design.

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.

This course is about how to write and use computer code to explore and solve problems in pure and applied mathematics. The first part of the course will be an introduction to programming in Python. The remainder of the course (and its goal) is to help students develop the skills to translate mathematical problems and solution techniques into algorithms and code. Students will work together on group projects with a variety applications throughout the curriculum.

The focus of the course will be on learning group theory. A group is a central concept of mathematics which is used to describe algebraic operations and symmetries of every possible kind, from modular arithmetic to symmetries of geometric objects.

Learning objectives: The emphasis will be on development of careful mathematical reasoning. Theoretical constructions and applications will be tested on many examples, both by hand and using computer algebra systems, specifically Wolfram Mathematica.

The following topics will be covered: Group axioms. Examples of groups. Numbers and matrices. Transformation groups. Multiplication tables. The group of units modulo n. Properties of groups. Generators and Cayley graphs. Subgroups. Symmetric, Dihedral and Cyclic groups. Cyclic subgroups of a group. Order of an element. Properties of permutations. Cayley theorem. Isomorphisms.  Rubik's cube and other permutation puzzles. Subgroups of permutations. Sign of a permutation. Cosets. Lagrange Theorem. Normal Subgroups. Quotient group. Homomorphisms. Isomorphism theorems.  Direct product of groups. Classification of finite Abelian groups.  Group actions. Orbit/stabilizer theorem.  Burnside lemma.  Crystallographic groups. Conjugacy classes. Cauchy theorem. Sylow theorems (w/o proof). Matrix groups over Fp. Group-theoretic aspects of public-key cryptography. Simple groups. Classification of groups of small order.

This course is a continuation of Math 411. We will study properties of rings and fields. A ring is an algebraic system with two operations (addition and multiplication) satisfying various axioms. Main examples are the ring of integers and the ring of polynomials in one variable. Later in the course we will apply some of the results of ring theory to construct and study fields. At the end we will outline the main results of Galois theory, which relates properties of algebraic equations to properties of certain finite groups called Galois groups. For example, we will see that a general equation of degree 5 can not be solved in radicals.

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 mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Small student groups will be formed to investigate a modeling problem independently, and each group will report its findings to the class in a final presentation. Satisfies the Integrative Experience for BS-Math and BA-Math majors.

We learn how to build, use, and critique mathematical models. In modeling we translate scientific questions into mathematical language, and thereby we aim to explain the scientific phenomena under investigation. Models can be simple or very complex, easy to understand or extremely difficult to analyze. We introduce some classic models from different branches of science that serve as prototypes for all models. Student groups will be formed to investigate a modeling problem themselves and each group will report its findings to the class in a final presentation. The choice of modeling topics will be largely determined by the interests and background of the enrolled students. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

This course is a proof-based introduction to elementary number theory. We will quickly review basic properties of the integers including modular arithmetic and linear Diophantine equations covered in Math 300 or CS250. We will proceed to study primitive roots, quadratic reciprocity, Gaussian integers, and some non-linear Diophantine equations. Some important applications to cryptography will be discussed.

This course is an introduction to mathematical analysis. A rigorous treatment of the topics covered in calculus will be presented with a particular emphasis on proofs. Topics include: properties of real numbers, sequences and series, continuity, Riemann integral, differentiability, sequences of functions and uniform convergence.

The course will start with the study of transport equations, and an introduction to and classification of second-order partial differential equations and their applications. Subsequently, we will proceed to examine in more detail the wave equation, heat equation and Laplace equation. For instance, we will touch upon the D'Alembert solution to the wave equation, the solution of the heat equation, the maximum principle, energy methods, separation of variables, Fourier series and Fourier transform methods, as well as operator eigenvalue problems, starting with one spatial dimension and then considering generalizations of some of these ideas to higher dimensions.

Time-permitting, we will briefly examine numerical methods for partial differential equations and relevant implementation thereof, e.g., in Matlab, as well as some select examples of nonlinear partial differential equations and the traveling or standing wave solutions possible therein.

The final grade will be determined on the basis of attendance/in class participation, homework, an in-class midterm and a final exam.

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 course in Advanced Linear Algebra and Applications. We will cover LU decomposition, Vector and Inner Product Spaces, Orthogonality and Least Squares, Determinants and Eigenvalues, Jordan form, Spectral theorem, symmetric positive definite matrices. Other decompositions such as SVD and QR, will be covered as well. If time permitting. basic numerical linear algebra will be included. There will be elements of proof and computation in the course. No coding will be taught in the class, but the students will have the option to do a final project instead of the exam. Homework will be assigned every one or two weeks. Late homework will NOT be accepted. The final grade will be based on both exams and homework assignments.

Basic concepts (over real or complex numbers): vector spaces, basis, dimension, linear transformations and matrices, change of basis, similarity. Study of a single linear operator: minimal and characteristic polynomial, eigenvalues, invariant subspaces, triangular form, Cayley-Hamilton theorem. Inner product spaces and special types of linear operators (over real or complex fields): orthogonal, unitary, self-adjoint, hermitian. Diagonalization of symmetric matrices, applications. Homeworks include programming projects.

A practical overview of computational methods used in science, statistics, industry, and machine learning. Topics will include: an introduction to python programming and software for scientific computing such as NumPy and LAPACK, numerical linear algebra, optimization and root-finding, approximation of functions by splines and trigonometric polynomials, and the Fast Fourier Transform. Applications may include regression problems in statistics, audio and image processing, and the calculation of properties of molecules. Homework will be assigned frequently. Each assignment will involve both mathematical theory and python programming. There will be no exams. Instead, each student will pursue an open-ended project related to a topic discussed in class.

Introduction to the application of computational methods to models arising in science and engineering, concentrating mainly the numerical solution of ordinary, partial differential equations, and stochastic simulations of particle systems. Topics include finite differences, finite elements, spectral methods, boundary value problems, ODE integrators, and fast Fourier transforms. If time permits we will discuss nonlinear optimisation methods with applications to data science.

Applications and examples will be selected from biomedical engineering, cell biology, and population dynamics.

This course is an introduction to differential geometry, where we apply theory and computational techniques from linear algebra, multivariable calculus and differential equations to study the geometry of curves, surfaces and (as time permits) higher dimensional objects.

This course is an introduction to the fundamental principles of statistical science. It does not rely on detailed derivations of mathematical concepts, but does require mathematical sophistication and reasoning. It is an introduction to statistical thinking/reasoning, data management, statistical analysis, and statistical computation. Concepts in this course will be developed in greater mathematical rigor later in the statistical curriculum, including in STAT 515, 516, 525, and 535. It is intended to be the first course in statistics taken by math majors interested in statistics. Concepts covered include point estimation, interval estimation, prediction, testing, and regression, with focus on sampling distributions and the properties of statistical procedures. The course will be taught in a hands-on manner, introducing powerful statistical software used in practical settings and including methods for descriptive statistics, visualization, and data management.

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, probability inequalities, the central limit theorem, the Poisson approximation, 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, probability inequalities, the central limit theorem, the Poisson approximation, 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, probability inequalities, the central limit theorem, the Poisson approximation, 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, probability inequalities, the central limit theorem, the Poisson approximation, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

Continuation of Stat 515. The overall objective of the course is the development of basic theory and methods for statistical inference from a mathematical and probabilistic perspective. Topics include: sampling distributions; point estimators and their properties; method of moments; maximum likelihood estimation; Rao-Blackwell Theorem; confidence intervals, hypothesis testing; contingency tables; and non-parametric methods (time permitting).

Regression analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, 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; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed).  Requires prior coursework in Statistics, preferably  ST516, and basic matrix algebra.  Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

Regression analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, 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; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed). Requires prior coursework in Statistics, preferably ST516, and basic matrix algebra. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

This course provides an introduction to fundamental computer science concepts relevant to the statistical analysis of large-scale data sets. Students will collaborate in a team to design and implement analyses of real-world data sets, and communicate their results using mathematical, verbal and visual means. Students will learn how to analyze computational complexity and how to choose an appropriate data structure for an analysis procedure. Students will learn and use the python language to implement and study data structure and statistical algorithms.

This is the second part of a 2-semester introduction to Real Analysis (namely Math 623 in the Fall, and Math 624 in the Spring) which covers parts of Vol. III and of Vol. IV of Stein & Shakarchi texts.

Math 624 is a continuation of Math 623. We start where Math 623 left off in the Fall and cover in particular the following topics: signed measures; Hilbert spaces and L2 theory; compact operators; the Fourier transform; Banach spaces; elementary operator theory and linear functionals; Lp spaces theory: duality, interpolation, fundamental inequalities and approximation theorems. Time permitting we will discuss some applications in harmonic analysis and some distribution theory.

The prerequisites for this class are Math 623 covering most of chapters 1, 2, 3 (part) of Stein-Shakarchi’s Real Analysis book (Vol. III) (namely having a working knowledge of Measure theory: Lebesgue measure and Integrable functions; Integration theory: Lebesgue integral, convergence theorems and Fubini theorem; Differentiation theorems, covering lemmas, approximation to the identity, functions of bounded variation, etc as well as a working knowledge of undergraduate Analysis (as for example taught in classes like M523H and M524H).

Presentation of the classical finite difference methods for the solution of the prototype linear partial differential equations of elliptic, hyperbolic, and parabolic type in one and two dimensions. Finite element methods developed for two dimensional elliptic equations. Major topics include consistency, convergence and stability, error bounds, and efficiency of algorithm. Prerequisites: Math 651, familiarity with partial differential equations.

Data science and machine learning (deep learning in particular) have become a burgeoning domain with a great number of successes in science and technology. Most of the recently developed deep learning techniques are still at the “engineering” level based on trial and error. A complete theory of deep learning is still under development.

The purpose of this course is to introduce the theoretical foundation of data science with an emphasis on the mathematical understanding of machine learning. The course is divided into two
semesters:

• In the first semester, we will introduce the basic set up of statistical learning, optimization, classical learning methods such as support vector machines, kernel methods, dimensionality reduction, as well as advanced learning theories, providing a mathematical foundation for the study of neural networks in the following semester.

• The second semester will contain two parts. The first half of the course introduces some useful fundamental tools from probability and statistics, and more extensively the theory of neural networks including their approximation power and generalization properties. The second half is a seminar part of the course, covering selected advanced topics on optimization and generative modeling.

The expected outcome of this course is to prepare students with solid mathematical background of modern machine learning, and to get students engaged with new research topics in this area.

This course complements some earlier courses on machine learning and data sciences, such as MATH 697PA: ST-Math Foundtns/ProbabilistAI and STAT 697ML: ST- Stat Machine Learning.

In this course, we will consider maximization and minimization problems in graphs and networks. We will cover a broad range of topics such as matchings in bipartite graphs and in general graphs, assignment problem, polyhedral combinatorics, total unimodularity, matroids, matroid intersection, min arborescence, max flow;min cut, max cut, traveling salesman problem, stable sets and perfect graphs. One of our main tools will be integer programming, and we will also sometimes rely on semidefinite programming. Many of these problems come from real-world applications, so we will also sometimes discuss the algorithms necessary to solve them. This is a rigorous mathematical introduction to combinatorial optimization with proofs.

This course aims to give an introduction to the fundamental topics in modern differential geometry, as organized in the following five units.

1. Theory of fiber bundles and connections.
2. Characteristics classes via Chern-Weil theory.
3. Riemannian geometry.
4. Hermitian and Kahler geometry.
5. Hodge theory.

We will present the basic concepts and theorems in each unit listed above, illustrated with interesting examples and detailed proofs of some selected results to demonstrate the various basic techniques in these subjects. We also hope to enhance the learning experience with homework assignments/projects, which form the basis of the course grade. Lecture notes will be provided.

The study of randomness in partial differential equations (PDEs) goes back more than seventy years and
include examples such as the modeling of random vibrations of strings, or the scattering of waves by objects
that are imbedded in random media.
Nonlinear dispersive PDEs naturally appear as models describing wave phenomena in quantum mechanics,
nonlinear optics, plasma physics, water waves, and atmospheric sciences. Due to their ubiquitousness
they have been at the center of profound research both from the applied community as well as from the
theoretical one. One way in which randomness enters the field of nonlinear dispersive PDE is via the
random data Cauchy initial value problem for (deterministic) equations, such as the nonlinear Schr¨odinger
(NLS) and the nonlinear wave equations (NLW). The interest comes from two fundamental problems:
(1) invariance of measures such as Gibbs measures which are physical equilibria for these systems;
arising naturally in statistical mechanics and closely related also to QFT models such as the  4, and
(2) the study of generic behavior of solutions in the probabilistic sense, and how they are expected to be
better than worst case (exceptional) scenarios.
The study of this subject in the context of dispersive PDEs can be traced back to Lebowitz–Rose–Speer
(1988, 1989) and Bourgain (1994, 1996) concerning the Gibbs measure for NLS. Since then there have been
substantial developments of their ideas by many different researchers, extending them in different directions
(geometric, infinite volume, other dispersive relations). In recent years, especially since 2018, this field
has seen significant progress and many new ideas and methods have been introduced that go beyond the
original ideas of Bourgain, including hyperbolic versions of paracontrolled calculus (Gubinelli–Koch–Oh
and Bringmann), the method of random averaging operators and the theory of random tensor (both by
Deng–Nahmod–Yue). These new methods have led to the resolution of several important open questions in
this field, and are expected to play more important roles in future developments. The aim of this course is to
provide the foundations upon which these recent developments have built upon, and in particular have a
working knowledge of Bourgain’s seminal works in the subject.

This is the second part of a two semester sequence on probability and mathematical statistics. Stat 607 covered probability, basic statistical modelling, and an introduction to the basic methods of statistical inference with application to mainly one sample problem. In Stat 608 we pick up some additional probability topics as needed and examine further issues in methods of inference including more on likelihood based methods, optimal methods of inference, more large sample methods, Bayesian inference and decision theoretic approaches. The theory is utilized in addressing problems in nonparametric methods, two and multi-sample problems, and categorial, regression and survival models. As with Stat 607 this is primarily a theory course emphasizing fundamental concepts and techniques.

This course will introduce students to Bayesian data analysis, including modeling and computation. We will begin with a description of the components of a Bayesian model and analysis (including the likelihood, prior, posterior, conjugacy and credible intervals). We will then develop Bayesian approaches to models such as regression models, hierarchical models and ANOVA. Computing topics include Markov chain Monte Carlo methods. The course will have students carry out analyses using statistical programming languages and software packages.

This course provides an introduction to the more commonly-used multivariate statistical methods. Topics include principal component analysis, factor analysis, clustering, discrimination and classification, multivariate analysis of variance (MANOVA), and repeated measures analysis. The course includes a computing component.