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.

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 technical writing, research paper, and professional presentation.

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

This course is an introduction to group theory, which is one of the oldest branches of modern algebra. It was invented by a 19-year-old Evariste Galois with a goal of proving that there is no algebraic formula expressing the roots of an 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. We will cover permutations, cyclic and Abelian groups, cosets and Lagrange's theorem, quotient groups, group actions and Burnside's theorem.

This course is an introduction to group theory, which is one of the oldest branches of modern algebra. It was invented by a 19-year-old Evariste Galois with a goal of proving that there is no algebraic formula expressing the roots of an 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. We will cover permutations, cyclic and Abelian groups, cosets and Lagrange's theorem, quotient groups, and group actions.

This 3 credit hours course serves as a preparation for SOA's second actuarial exam in financial mathematics, known as Exam FM or Exam 2. The course provides an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. The main topics include time value of money, annuities, loans, bonds, general cash flows and portfolios, immunization, interest rate swaps and determinants of interest rates etc. Many questions from old exam FM will be practiced in the course.

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. Prerequisites: Calculus (Math 131, 132, 233), required; Linear Algebra (Math 235) and Differential Equations (Math 331), or permission of instructor required.

There are three types of surfaces which look the same at every point and in every direction: the plane, the sphere, and the hyperbolic plane. The hyperbolic plane is a remarkable surface in which the circumference of a circle grows exponentially as the radius increases; it was only discovered in the 18th century. We will begin by studying the geometry of the plane and the sphere and their symmetries. Then we will describe and study the hyperbolic plane. The emphasis will be on developing our geometric intuition in each case.

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 class is designed to help students review and prepare for the GRE Mathematics subject exam, which is a required exam for entrance into many PhD programs in mathematics. Students should have completed the three courses in calculus, a course in linear algebra, and have some familiarity with differential equations. The focus will be on solving problems based on the core material covered in the exam. Students are expected to do practice problems before each meeting and discuss the solutions in class.

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.

Basic concepts (mainly over the field of 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, normal. Diagonalization of real symmetric matrices, generalizations (like singular value decomposition) and applications (for instances, to the geometry of "propellors" – a current research topic).

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. Prerequisites: MATH 233 and 235, or consent of instructor, and knowledge of a scientific programming language.

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.

For graduate and upper-level undergraduate students, with focus on practical aspects of statistical methods.Topics include: data description and display, probability, random variables, random sampling, estimation and hypothesis testing, one and two sample problems, analysis of variance, simple and multiple linear regression, contingency tables. Includes data analysis using a computer package (R).

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.

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.

Seeking answers to questions of causality is a fundamental part of the scientific process and the advancement of human knowledge. Answers to causal questions are imperative to supporting decision-making in areas such as healthcare and public policy. This course provides an introduction to causal inference in statistics. We will introduce the potential outcomes framework to causal modeling, and use it to study core causal models including randomized experiments, backdoor adjustment, instrumental variables, difference-in-difference, regression discontinuity, and mediation analysis. Directed Acyclic Graphs will be introduced as an alternative approach to causal reasoning and as a tool for assessing conditional independence assumptions.  In each model, we will study the causal and observed data structures, the causal estimand(s) of interest, the causal identification conditions and the associated graphical model, the identification result, statistical inference for the identified parameter, and R code implementing statistical inference. We will use case studies from the real scientific literature to illustrate each model. In the final project, students will apply the techniques learned in class to conduct causal analysis for a scientific question of their choosing and write up their results.

Introduction to groups, rings, and fields. Direct sums and products of groups, cosets, Lagrange's theorem, normal subgroups, quotient groups. Polynomial rings, UFDs and PIDs, division rings. Fields of fractions, GCD and LCM, irreducibility criteria for polynomials. Prime field, characteristic, field extension, finite fields. Some topics from Math 612 included.

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.

In this course, 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 novel methods in computational biology and hopefully find ways to improve them. We will read some recently published papers, which include four cutting-edge papers on computational oncology, implement the methods that have been introduced in these papers, and reproduce their results. 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. Students will be evaluated based on their participation, presentations, and works, including their codes and HWs.

This course explores mathematical vignettes motivated by the biophysics of cell shape, cell motility, cell signalling, and pattern formation in cells. We consider how cells interact with each other, and how they coordinate to form tissues. The two primary goals are: (1) learn to use the tools and techniques to formulate models of biological systems, and (2) learn to use mathematical analysis techniques to provide insights into the underlying biological mechanisms.

The two primary goals are: (1) learn to formulate models of biological systems, and (2) learn mathematical analysis tools and techniques to provide insights into the underlying biological mechanisms. Here we concentrate on ordinary and partial differential equation models (i.e.\ deterministic models), and the Cellular-Potts model to describe multiscale tissues. This course will include a tutorial and hands-on exploration of the open source software, Morpheus, which can simulate computational models of cells and tissues.

From a mathematical perspective, we will showcase several important models (with application beyond cellular biology) where analysis provide insights and helps to understand the underlying mechanisms. Many of these models are useful in many other areas of science (e.g.\ ecology, population biology, social aggregation). Hence this course will also serve as a survey of partial differential equations that are important to STEM.

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.

This course introduces Riemann surfaces from the points of view of 1-dimensional complex manifolds and also 2-dimensional real oriented conformal manifolds. Topics covered included the structure of holomorphic maps between Riemann Surfaces (Riemann-Hurwitz Theorem), holomorphic line;vector bundles, Chern classes, the Picard group of holomorphic line bundles, the Abel-Jacobi map, and the basic theorems of Riemann Surface theory: Mittag-Leffler, Riemann-Roch, Serre duality, Kodaira embedding and Serre's GAGA principle.

This class is a second course in algebraic topology, taking the next steps after (co)homology and homotopy.  The central objects of study are vector bundles, which are families of vector spaces parametrized by a topological space.  Associated to a vector bundle are cohomology classes called characteristic classes, which provide a measure of how "twisted" the bundle is.  The first part of the course will focus on motivating examples, the axiomatic approach, and culminate with the notion of a classifying space, which provides a concrete source for characteristic classes.
The second part of the course will develop the natural context for studying vector bundles: K-theory, as a generalized cohomology theory.  We will also develop special features of K-theory, notably Bott periodicity and the Chern character, which relates K-theory to characteristic classes.  Additional topics such as cobordism, the Atiyah-Singer index theorem, or Adams operations and the Hopf invariant one problem may be discussed subject to time and student interest.

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.

This course presents the fundamental principles of time series analysis including
mathematical modeling of time series data and methods for statistical inference.
Topics covered will include: modeling and inference for linear autoregressive time
series models, autoregressive (AR) and autoregressive moving average (ARMA)
models, (nonseasonal/seasonal) autoregressive integrated moving average (ARIMA)
models, unit root and differencing, spectral analysis,
(generalized) autoregressive conditionally heteroscedastic models,
Kalman filtering and smoothing, and state-space models.

This course is designed for students to complete the master's project requirement in statistics, with guidance from faculty. The course will begin with determining student topics and groups. Each student will complete a group project. Each group will work together for one semester and be responsible for its own schedule, work plan, and final report. Regular class meetings will involve student presentations on progress of projects, with input from the instructor. Students will learn about the statistical methods employed by each group. Students in the course will learn new statistical methods, how to work collaboratively, how to use R and other software packages, and how to present oral and written reports.