Courses

The tour consists of a series of eight presentations given by a variety of Mathematics and Statistics department faculty. The course is intended for first- or second-year students considering a Mathematics or Statistics major or minor. The emphasis of these talks will be on presenting engaging ideas and research in various areas of mathematics and statistics, rather than on developing extensive knowledge or techniques in any particular subject area.

Linear algebra centers on the geometry, algebra, and applications of linear equations.  It is pivotal to many areas of mathematics, natural sciences, computer science, and engineering. To study linear equations, we will develop concepts including matrix algebra, linear independence, determinants, eigenvectors, and orthogonality.  Students will use these tools to model real world problems and solve these problems using computational software. 

Vectors, curves, calculus of functions of three independent variables, including directional derivatives and triple integrals, cylindrical and spherical coordinates, line integrals, Green's theorem, sequences and series, power series, Taylor series. This course cannot be substituted for MATH 211.

Ordinary differential equations are a fundamental language used by mathematicians, scientists, and engineers to describe processes involving continuous change. In this course we develop ordinary differential equations as models of real world phenomena and explore the mathematical ideas that arise within these models. Topics include separation of variables; phase portraits; equilibria and their stability; non-dimensionalization; bifurcation analysis; and modeling of physical, biological, chemical, and social processes.

Dynamics is the branch of mathematics that deals with the study of change. In this course we will focus on simple discrete non-linear dynamical systems that produce astoundingly rich and unpredictable behavior — something that is colloquially referred to as "chaos". Topics will include one dimensional dynamics (including fixed points and their classifications), Sharkovsky's Theorem, a careful formulation/definition of "chaos", symbolic dynamics, complex dynamics (including Julia and Mandelbrot sets), iterated function systems, fractals and more. 

Optimization is all about selecting the "best" thing. Finding the most likely strategy to win a game, the route that gets you there the fastest, or the curve that most closely fits given data are all examples of optimization problems. In this course we study linear optimization (also known as linear programming), the simplex method, and duality from both a theoretical and a computational perspective. Applications will be selected from statistics, economics, computer science, and more. Additional topics in nonlinear and convex optimization will be covered as time permits.

A first course in number theory, covering properties of the integers. Topics include the Euclidean algorithm, prime factorization, Diophantine equations, congruences, divisibility, Euler’s phi function and other multiplicative functions, primitive roots, and quadratic reciprocity. Along the way we will encounter and explore several famous unsolved problems in number theory. If time permits, we may discuss further topics, including integers as sums of squares, continued fractions, distribution of primes, Mersenne primes, the RSA cryptosystem.

Students work on a research project related to a faculty member's research interests, and directed by that faculty member. Student activities vary according to the field and stage of the project. The long-run goal of these projects normally includes dissemination to a scholarly community beyond Carleton. The faculty member will meet regularly with the student and actively direct the work of the student, who will submit an end-of-term product, typically a paper or presentation.

This course explores a particularly visual sort of mathematical pattern: a tessellation. A tessellation is a way of covering the plane with shapes (called “tiles”) that don’t overlap. This class will explore questions like: Is it possible to make a tessellation out of a given set of tiles? How many different tessellations can I create from this set of tiles? We’ll cover both classical results (it is impossible to tile the plane with heptagons!), and the 2023 construction of the “Einstein tile”: the first known polygon that tiles the plane but never periodically. Links will be made with graph theory, topology, and geometry.

Repeatable: This course is repeatable provided the topics are different.

A systematic study of single-variable functions on the real numbers. This course develops the mathematical concepts and tools needed to understand why calculus really works: the topology of the real numbers, limits, differentiation, integration, convergence of sequences, and series of functions.

Differential geometry is the study of shapes (like curves and surfaces) using tools from linear algebra and calculus. In this course we focus on the differential geometry of curves and surfaces and the concepts of curvature, geodesics, and first and second fundamental forms. These concepts will lead us to remarkable results like the Theorem Egregium and the Gauss-Bonnet Theorem, which relate the ways that curvature and shape interact.

Students work on a research project related to a faculty member's research interests, and directed by that faculty member. Student activities vary according to the field and stage of the project. The long-run goal of these projects normally includes dissemination to a scholarly community beyond Carleton. The faculty member will meet regularly with the student and actively direct the work of the student, who will submit an end-of-term product, typically a paper or presentation.

As part of their senior capstone experience, majors will work together in teams to develop advanced knowledge in a faculty-specified area or application of mathematics, and to design and implement the first stage of a project completed the following term.

Either a supervised group project or an individual, independent project. Required of all senior majors.

Introduction to statistics and data analysis. Practical aspects of statistics will be emphasized, including extensive use of programming in the statistical software R, interpretation and communication of results. Topics include: exploratory data analysis, correlation and linear regression, design of experiments, the normal distribution, randomization approach to inference, sampling distributions, estimation, and hypothesis testing. Students who have taken Mathematics 211 are encouraged to consider the more advanced Mathematics 240/Statistics 250 Probability/Statistical Inference sequence.

This course will cover the computational side of data analysis, including data acquisition, management, and visualization tools. Topics may include: data scraping, data wrangling, data visualization using packages such as ggplots, interactive graphics using tools such as Shiny, an introduction to classification methods, and understanding and visualizing spatial data. We will use the statistics software R in this course.

A second course in statistics covering simple linear regression, multiple regression and ANOVA, and logistic regression. Exploratory graphical methods, model building and model checking techniques will be emphasized with extensive use of statistical software R to analyze real-life data.

Statistical learning (sometimes called statistical machine learning) centers on the discovery of structural patterns and making predictions using complex data sets. This course explores supervised and unsupervised statistical learning methods, and the ethical considerations of their use. Topics may include nonparametric regression, classification, cross validation, linear model selection techniques and regularization, and clustering. Students will implement these concepts using open-source computational tools, such as the R language.

Students work on a research project related to a faculty member's research interests, and directed by that faculty member. Student activities vary according to the field and stage of the project. The long-run goal of these projects normally includes dissemination to a scholarly community beyond Carleton. The faculty member will meet regularly with the student and actively direct the work of the student, who will submit an end-of-term product, typically a paper or presentation.

Models and methods for characterizing dependence in data that are ordered in time. Emphasis on univariate, quantitative data observed over evenly spaced intervals. Topics include perspectives from both the time domain (e.g., autoregressive and moving average models, and their extensions) and the frequency domain (e.g., periodogram smoothing and parametric models for the spectral density). Exposure to matrix algebra may be helpful but is not required.

Topics include linear mixed effects models for repeated measures, longitudinal or hierarchical data and generalized linear models (of which logistic and Poisson regression are special cases) including zero-inflated Poisson models. Depending on time, additional topics could include survival analysis or generalized additive models. 

The Bayesian approach to statistics provides a powerful framework for incorporating prior knowledge into statistical analyses, updating this knowledge with data, and quantifying uncertainty in results. This course serves as a comprehensive introduction to Bayesian statistical inference and modeling, an alternative to the frequentist approach to statistics covered in previous classes. Topics include: Bayes’ Theorem; prior and posterior distributions; Bayesian regression; hierarchical models; and model adequacy and posterior predictive checks. Computational techniques will also be covered, including Markov Chain Monte Carlo methods, and modern Bayesian modeling packages in R.

Students work on a research project related to a faculty member's research interests, and directed by that faculty member. Student activities vary according to the field and stage of the project. The long-run goal of these projects normally includes dissemination to a scholarly community beyond Carleton. The faculty member will meet regularly with the student and actively direct the work of the student, who will submit an end-of-term product, typically a paper or presentation.

As part of their senior capstone experience, majors will work together in teams to develop advanced knowledge in a faculty-specified area or application of statistics, and to design and implement the first stage of a project completed the following term.

A supervised group project. Required of all senior majors.