Course Descriptions

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.

One-semester review of manipulative algebra, introduction to functions, some topics in analytic geometry, and that portion of trigonometry needed for calculus.

Various topics that might enrich an elementary school mathematics program, including probability and statistics, the integers, rational and real numbers, clock arithmetic, diophantine equations, geometry and transformations, the metric system, relations and functions. For Pre-Early Childhood and Pre-Elementary Education majors only.

Basic calculus with applications to problems in the life and social sciences. Functions and graphs, the derivative, techniques of differentiation, curve sketching, maximum-minimum problems, exponential and logarithmic functions, exponential growth and decay, and introduction to integration.

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

Techniques of calculus in two and three dimensions. Vectors, partial derivatives, multiple integrals, line integrals. Theorems of Green, Stokes and Gauss. Honors section available. (Gen.Ed. R2)

Introduction to ordinary differential equations. First and second order linear differential equations, systems of linear differential equations, Laplace transform, numerical methods, applications. (This course is considered upper division with respect to the requirements for the major and minor in mathematics.)

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.

Math 300 is an introduction to rigorous, abstract mathematics. In lower-level courses like calculus, the emphasis is on applying theorems and formulas to solve specific, often numerical, problems. More advanced math classes are concerned with developing the theorems and formulas and solving general classes of problems. In particular, it's important to know why those theorems and formulas are true. In mathematics, the way we know a statement is true is by giving a proof of it, and this course is about learning what proof is, how to read, create, and present proofs, and how to tell a correct proof from an incorrect one. In many ways, this is like learning a language. We need to learn the grammar (logical deduction) and vocabulary (sets, functions, and other basic structures), but it also helps to have something to say. So we will also study some important and beautiful mathematics along the way. Starting with explicit axioms and precisely stated definitions, we will systematically develop basic propositions about integers and modular arithmetic, induction and recursion, equivalence classes and rational numbers, and such other topics as time allows.

This course will introduce students to writing in mathematics, both technical and otherwise. Writing assignments will include proofs, instructional handouts, resumes, cover letters, presentations, and a final paper. All assignments will be completed using LaTeX using offline and online editors. By the end of the semester, students should be able to clearly convey mathematical ideas through their writing, and to tailor that writing for a particular audience. Students will also learn about LaTeX offline and online editors, file sharing and version control (GitHub).

While the mathematicians of the pre-internet age often spread their mathematical ideas within the community via written letters prior to publication, modern mathematical correspondence and exposition is rapidly facilitated by a variety of digital tools. Of great importance to the publishing process in mathematical sciences is the LaTeX markup language, used to typeset virtually all modern mathematical publications, even at the pre-print stage. In this course we will develop facility with LaTeX, and develop a variety of writing practices important to participation in the mathematical community. There will be regular written assignments completed in LaTeX, as well as collaborative writing assignments, owing to the importance of collaborative writing in mathematical research. Writing topics may include proofs, assignment creation, pre-professional writing (resumes/cover letters, research and teaching statements), expository writing for a general audience, recreational mathematics, and the history of mathematics. Short writing assignments on such topics will be assigned in response to regular assigned readings from a variety of accessible/provided sources. Towards the end of the semester groups will complete a research paper of an expository nature and craft a seminar style presentation. This course meets the junior year writing requirement.

This course is an introduction to group theory, which is one of the oldest branches of modern algebra. It was invented in 1830 by a 19-year-old, Evariste Galois, with a goal of proving that there is no algebraic formula expressing the roots of every equation of degree 5 in terms of its coefficients. Since then, group theory has become the crucial tool in uncovering hidden symmetries of the world. The emphasis of this class will be on using concrete examples to develop problem-solving and proof-writing skills as we explore the abstract theory of groups. We will discuss permutations, cyclic and Abelian groups, cosets and Lagrange's theorem, quotient groups, group actions, and counting with groups.

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 Laurent series. Classification of isolated singularities. The Argument Principle and Rouche's Theorem. Evaluation of Improper integrals via residues. Conformal mappings.

Calculus of several variables, Jacobians, implicit functions, inverse functions; multiple integrals, line and surface integrals, divergence theorem, Stokes' theorem.

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.

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. In the beginning, we'll focus on differential equation based models. For the second half, we will study a number of topics from games and gambling, economics, social sciences, for which we will use elementary tools from probability, game theory, information theory, and optimization. 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 is an introduction to the history of mathematics from ancient civilizations to present day. Students will study major mathematical discoveries in their cultural, historical, and scientific contexts. This course explores how the study of mathematics evolved through time, and the ways of thinking of mathematicians of different eras - their breakthroughs and failures. Students will have an opportunity to integrate their knowledge of mathematical theories with material covered in General Education courses. Forms of evaluation will include presentations, class discussions, and a final paper. Satisfies the Integrative Experience requirement for BA-MATH and BS-MATH majors.

Topology of the euclidean space and functions of several variables (implicit function theorem), introduction to Fourier analysis, metric spaces and normed spaces. Applications to differential equations, calculus of variations, and others.

This course is based on the first examination of the Society of Actuaries. Its content is largely dependent on that examination. Presently, it covers: calculus of a single variable (integration, differentiation, infinite series, Taylor's series etc.); calculus of several variables (Jacobians, Lagrange multipliers, double and triple integrals, etc.); probability Theory (discrete and continuous distributions, conditional probability and expectations, Bayes' rule, joint distributions, moment generating functions, the central limit theorem, etc.) The problems are drawn from old SOA examinations and most will have an insurance industry emphasis. Much of the material is a review of several courses, but this review is extensive and probably exceeds most interested students' backgrounds.

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.

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.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis.

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; global and variational aspects of geometry will be a central theme of the course.

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).

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.

An applied statistics course on planning, statistical analysis and interpretation of experiments of various types. Coverage includes factorial designs, randomized blocks, incomplete block designs, nested and crossover designs. Computer analysis of data using the statistical package SAS (no prior SAS experience assumed).

Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

This course covers classical methods in applied mathematics and math modeling, including dimensional analysis, asymptotics, regular and singular perturbation theory for ordinary differential equations, random walks and the diffusion limit, and classical solution techniques for PDE. The techniques will be applied to models arising throughout the natural sciences.

This fast-paced course is an introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, rings, vector spaces, and homomorphism between them). Emphasis will be placed on being able to compute these invariants. Topics include: Cell complexes, homotopy, fundamental group, Van-Kampen's theorem (all reviewed from Math 671), covering spaces, simplicial complexes, singular and cellular homology, exact sequences, Mayer-Vietoris, cohomology, cup products, universal coefficients theorem, Künneth formulas, Poincaré and Lefschetz dualities.

Grade will be based on regularly assigned homework, an in-class exam, and a final presentation.

This is the second semester of a 2-semester, year long, class on probabilistic artificial intelligence. The main emphasis of the course is on building and understanding the mathematical tools and conceptual foundations of AI, especially from a probabilistic point of view. Furthermore, group projects of the students will serve to explore applications and implementations.

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.

Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with a strong emphasis on examples.

In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. Passing from local to global data is delicate (as in complex analysis) and is either accomplished by working in projective space (corresponding to a graded polynomial ring) or by using sheaves and their cohomology.

Topics will include projective varieties, singularities, differential forms, line bundles, and sheaf cohomology, including the Riemann--Roch theorem and Serre duality for algebraic curves. Examples will include projective space, the Grassmannian, the group law on an elliptic curve, blow-ups and resolutions of singularities, algebraic curves of low genus, and hypersurfaces in projective 3-space.

Prerequisites: Commutative algebra (rings and modules) as covered in 611-612. Some prior experience of manifolds would be useful (but not essential).

This is the second part of a two semester sequence on probability and mathematical statistics. Stat 607 covered probability, discrete/continuous distributions, basic convergence theory and basic statistical modelling. In Stat 608 we cover an introduction to the basic methods of statistical inference, 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 Resampling methods. The theory is utilized in addressing problems in parametric/nonparametric methods, two and multi-sample problems, and regression. As with Stat 607, this is primarily a theory course emphasizing fundamental concepts and techniques.

This course provides an introduction to the statistical techniques that are most applicable to data science. Topics include regression, classification, resampling, linear model selection and regularization, tree-based methods, support vector machines and unsupervised learning. The course includes a computing component using statistical software. Students must have prior experience with a statistical programming language such as R, Python or MATLAB.