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.

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. Theorems of Green, Stokes and Gauss. Honors section available. (Gen.Ed. R2)

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

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 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). 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.

Satisfies Junior Year Writing requirement. Develops research and writing skills in mathematics through peer review and revision. Students write on mathematical subject areas, prominent mathematicians, and famous mathematical problems.

Essentially all serious mathematics is written using some variant of the LaTeX software, and developing proficiency with this tool is an important part of the course. We will cover a variety of types of writing related to mathematics, such as: critique of seminar presentations and written articles; (auto)biography of mathematicians; expository writing about mathematical topics; aspects of writing a research article; opinion pieces; precise short communications (e.g. abstract, memo, tweet, blog post, newspaper headline); essays about aspects of mathematical culture (e.g. jokes, songs, anecdotes, sociological aspects).

Satisfies Junior Year Writing requirement.

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 on projects, both individually and in groups, 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.

Approximate Schedule:

• Group axioms. Examples of groups. Numbers and matrices. Transformation groups.
• Multiplication tables and Cayley graphs. Properties of groups.
• Subgroups. Order of an element. Cyclic groups.
• Symmetric group. Properties of permutations. Cayley theorem. Isomorphisms.
• Cosets. Lagrange theorem. Normal subgroups.
• Quotient group. Isomorphism theorems. Homomorphisms.
• Direct product of groups. Classification of finite Abelian groups.
• Group actions. Orbit/stabilizer theorem. Burnside lemma. Conjugacy classes. Cauchy theorem. Sylow theorems (without proof).
• Classification of crystallographic groups.
• Group-theoretic aspects of public-key cryptography.
• Subgroups of the symmetric group. Solving puzzles using groups.
• Classification of groups of small order.

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.

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, and matchings; big-Oh notation, the pigeonhole principle, induction and recursion, generating functions, and discrete probability proofs (time permitting). 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 concept or an application related to discrete mathematics, 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.

Mathematics is usually termed as “the language of nature”. Complex physical phenomenon can be described by mathematical models with sufficient accuracy. In this course, students learn how to formulate and analyze some real-world problems by utilizing concepts, methods and theories from mathematics, thus coming to understand the interplay between mathematical theory and practice. Since mathematical models can be very broad and can appear in every discipline, this course will mainly focus on problems rising from data science. The goal is to discuss how to build and learn mathematical models in data-driven applications. Students will form several groups to investigate a modeling problem and each group will report their findings in a final presentation. Prerequisites of the course include Calculus (Math 131, 132, 233), Linear Algebra (Math 235) and Differential Equations (Math 331). Some familiarity with a programming language is desirable (Matlab, Python, etc.)

The goal of this course is to give a rigorous introduction to elementary number theory. While no prior background in number theory will be assumed, the ability to read and write proofs is essential for this course. The list of topics include (but is not limited to): Euclidean Algorithm, Linear Diophantine Equations, the Fundamental Theorem of Arithmetic, congruence arithmetic, continued fractions, the theory of prime numbers, primitive roots, and quadratic reciprocity, with an emphasis on applications to and connections with cryptography. Homework will consist of both rewritten assignments and computer projects.

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.

Introduction to the fascinating theory of knots, links, and surfaces in 3- and 4-dimensional spaces. This course will combine geometric, algebraic, and combinatorial methods, where the students will learn how to utilize visualization and make rigorous arguments.

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.

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.

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.

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.

This is a second course in Probability, studying stochastic/random process, intended for majors in Applied Math, Statistics and related fields. The prerequisite is STAT 515 or similar upper-division course. If you did not get at least a B in that course then you will find this course very tough. Moreover, students are required to have some knowledge of Python, as we will cover simulation topics, including sampling of probability distributions, Monte Carlo algorithms, etc. A major focus of the course is on solving problems extending the scope of the lectures, developing analytical skills and probabilistic intuition. The course will cover the following topics in the core of the theory of Random and Stochastic Processes.

1. Review of probability theory : probability space, random variables, expectations, independence, conditional expectations.
2. Random walks and finite state Markov chains: Transition matrix, transience and recurrence, limiting distributions, convergence of Markov Chains.
3. Poisson processes: definition, inter-arrival and waiting time.
4. Continuous Markov chains: Strong Markov properties, Chapman Kolmogorov equations, irreducible and recurrence. Long time behaviour.
5. Brownian Motions: Definitions, scaled random walk, Brownian motion.

In addition, we also add a parallel part to the theoretical lectures - the stochastic simulations.

1.Sampling of basic probability distributions, generation of pseudorandom numbers,
2. Monte Carlo integration Simulation of random samples from discrete distributions and continuous distributions
3. Discrete event simulation for stochastic models of queueing systems
4. MCMC

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.

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.

This course will introduce computing tools needed for statistical analysis including data acquisition from database, data exploration and analysis, numerical analysis and result presentation. Advanced topics include parallel computing, simulation and optimization, and package creation. The class will be taught in a modern statistical computing language.

This fast-paced course (and its continuation - Math 612) will introduce modern algebra concepts with an emphasis on topics required for the qualifying exam in algebra. Syllabus of Math 611 - Math 612:

I. Group Theory and Representation Theory.

Group actions. Counting with groups. P-groups and Sylow theorems. Composition series. Jordan-Holder theorem. Solvable groups. Automorphisms. Semi-direct products. Finitely generated Abelian groups. Complex representations of finite groups. Schur's Lemma. Maschke's Theorem. Representations of Abelian groups. Characters. Schur's orthogonality relations. The number of irreducible representations is equal to the number of conjugacy classes. The sum of squares of dimensions of irreducible representation is equal to the size of the group. The dimension of any irreducible representation divides the size of the group.

II. Linear Algebra and Commutative Algebra.

Euclidean domain is a PID. PID is a UFD. Gauss Lemma. Eisenstein's Criterion. Exact sequences. First and second isomorphism theorems for R-modules. Free R-modules. Hom. Tensor product of vector spaces, Abelian groups, and R-modules. Right-exactness of the tensor product. Restriction and extension of scalars. Complexification. Bilinear forms. Symmetric and alternating forms. Symmetric and exterior algebras. Structure Theorem for finitely generated modules of a PID. Rational canonical form. Jordan canonical form. Chain conditions. Noetherian rings. Hilbert's Basis Theorem. Prime and maximal ideals. Field of fractions. Localization of rings and modules. Exactness of localization. Local rings. Nakayama's Lemma. Integral extensions. Noether's Normalization Lemma. Integral closure. Nullstellensatz. Closed affine algebraic sets.

III. Field Theory and Galois Theory

Algebraic extensions. Finite extensions. Degree. Minimal polynomial. Adjoining roots of polynomials. Splitting field. Algebraic closure. Separable extensions. Theorem of the primitive element. Galois extensions. Fundamental Theorem of Galois Theory. Finite fields and their Galois groups. Frobenius endomorphism. Cyclotomic polynomial. Cyclotomic fields and their Galois groups. Cyclic extensions. Lagrange resolvents. Solvable extensions. Solving polynomial equations in radicals. Norm and trace. Transcendence degree.

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Linear systems and exponential matrix solutions. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant set and manifolds. Stability theory: linearization about an equilibrium, Lyapunov functions. Autonomous two-dimensional systems and other special systems.

This fast-paced course (and its sequel, Math 672) is an introduction to topology, from point-set to geometric and algebraic topology.
Part I: Basic point-set topology, constructions of topological spaces, connectedness, compactness, countability and separation axioms, topological manifolds.
Part II: Introduction to algebraic topology, cell complexes, homotopy, fundamental group, covering spaces.
Grade will be based on regularly assigned homework, as well as exams.

This course is the group project that is required for the MS program in Applied Mathematics. Each academic year we undertake an in-depth study of select applied science problems, combining modeling, theory, and computation to understand it. The main goal of the course is to emulate the process of teamwork in problem solving, such as is the norm in industrial applied mathematics.

Topics to be covered: smooth manifolds, smooth maps, tangent vectors, vector fields, vector bundles (in particular, tangent and cotangent bundles), submersions,immersions and embeddings, sub-manifolds, Lie groups and Lie group actions, Whitney's theorems and transversality, tensors and tensor fields, differential forms, orientations and integration on manifolds, The De Rham Cohomology, integral curves and flows, Lie derivatives, The Frobenius Theorem.

We first review some basic probability and useful tools, including random walk, Law of large numbers and central limit theorem. Conditional expectation and martingales. The topics of the course include the theory of stochastic differential equations oriented towards topics useful in applications, such as Brownian motion, stochastic integrals, and diffusion as solutions of stochastic differential equations. Then we study about diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus, as well as Girsanov's theorem, Feynman-Kac formula, Martingale representation theorem. We will also include some applications to mathematical finance as time permits.

This is a class on the mathematical foundations and the applications of distances and divergences between probability measures. Among the topics to be treated: Entropies, Kullback-Leibler and general f-divergences; Optimal transport and Wasserstein metrics; Maximum mean discrepancy and reproducing kernel Hilbert spaces, Stein discrepancies, and integral probability metrics. We will provide the analytical and probabilistic foundations for these objects with an emphasis on variational representations, We will discuss inequalities in information theory, and also present algorithms for the computations or statistical estimation of divergences or metrics, including generative adversarial networks and gradient flows. We illustrate the theory with examples from applied mathematics and data science, for example mutual information, generative adversarial networks, and so on.

The first part of a two-semester graduate level sequence in probability and statistics, this course develops probability theory at an intermediate level (i.e., non measure-theoretic - Stat 605 is a course in measure-theoretic probability) and introduces the basic concepts of statistics.
Topics include: general probability concepts; discrete probability; random variables (including special discrete and continuous distributions) and random vectors; independence; laws of large numbers; central limit theorem; statistical models and sampling distributions; and a brief introduction to statistical inference. Statistical inference will be developed more fully in Stat 608.
This course is also suitable for graduate students in a wide variety of disciplines and will give strong preparation for further courses in statistics, econometrics, and stochastic processes, time series, decision theory, operations research, etc.
You will be expected to read sections of the text book in parallel with topics covered in lectures, since important part of graduate study is to learn how to study independently.

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.

In this course, we will apply several novel machine learning algorithms, including normalization methods, classification and regression analysis on cancer patient data sets to arrive at personalized cancer treatments. We will develop several algorithms for analyzing cancer data sets, including gene expression data sets. We will review, develop, and evaluate some computational biology methods. We will implement most of these methods in Python. Although programming skills, machine learning, or computational biology background are preferred, they are not required for this course. Importantly, this is a research based course; it is an introduction on how to do research in computational biology. We all work as a team to learn cutting-edge methods in computational biology and hopefully find ways to improve them. We will read some recently published papers and implement methods that have been introduced in these papers. Except the first few lectures, a team of students will present the papers and their implementation of methods. Students should be interested in Python programming, computational biology, and doing research as a team member. There is no exam. Students will be evaluated based on their participation, presentations, and works, including their codes and HWs.

This class focuses on skills statisticians need to bring their classroom knowledge to practical problems. Half of the class will meet concurrently with the Statistical Consulting Practicum, and be focused around real consulting problems brought in by clients. The other half will explore in greater depths methods, techniques, and approaches useful in practical problems but not usually covered in the standard curriculum. Specific topics covered may vary slightly based on the mix of clients this semester, but may include methods like power analysis, missing data, practical mixed models, and practical model selection and interpretation, as well as soft skills like maintaining professional relationships, asking questions, listening, and communicating with non-statisticians. Wherever possible, content will be motivated by specific current or past consulting projects.