Courses

The modern world is dominated by chance; we walk through life speaking of hopes, fears, failures, and more in the language of statistics. In this course, we will explore fundamental concepts in probability with an emphasis on computational examples, from apartment hunting to computer-generated art. We’ll look at the human history of chance—especially over the 19th and 20th centuries—to think about how we got here. We’ll see that, in a life of finite spacetime but infinitely complex problems, sometimes embracing randomness can be just what we need. But what is chance, how do we understand it, and why?

An introduction to the central ideas of calculus with review and practice of those skills needed for the continued study of calculus. Problem solving strategies will be emphasized. In addition to regular MWF class time, students will be expected to attend two problem-solving sessions each week, one on Monday or Tuesday, and one on Wednesday or Thursday.  Details will be provided on the first day of class.

Inverse functions, integration by parts, improper integrals, modeling with differential equations, vectors, calculus of functions of two independent variables including directional derivatives and double integrals, Lagrange multipliers.

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.

Linear algebra centers on the study of highly structured functions called linear transformations. Given the abundance of nonlinear functions in mathematics, it may come as a surprise that restricting to linear ones opens the door to a rich and powerful theory that finds applications throughout mathematics, statistics, computer science, and the natural and social sciences. Linear transformations are everywhere, once we know what to look for. They appear in calculus as the functions that are used to define lines and planes in Euclidean space. In fact, differentiation is also a linear transformation that takes one function to another. The course focuses on developing geometric intuition as well as computational matrix methods. Topics include kernel and image of a linear transformation, vector spaces, determinants, eigenvectors and eigenvalues. 

(Formerly Mathematics 265) Introduction to probability and its applications. Topics include discrete probability, random variables, independence, joint and conditional distributions, expectation, limit laws and properties of common probability distributions.

An exploration of the behavior of non-linear dynamical systems. Topics include one and two-dimensional dynamics, Sarkovskii’s Theorem, chaos, symbolic dynamics,and the Hénon Map.

An introduction to mathematical ideas from numerical approximation, scientific computing, and/or data analysis. Topics will be selected from numerical linear algebra, numerical analysis, and optimization. Theory, implementation, and application of computational methods will be emphasized.

A first course in number 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.

In a democratic society we are confronted with problems of implementing fairness. How can we build a representative government, measure society’s preferences, or fairly divide power? Many of these problems are amenable to mathematical analysis and, in many cases, there exist deep theories and rich historical narratives of attempts at solution. We will study three such problems all of current political and mathematical interest: apportionment of representation, voting, and gerrymandering. We will approach these problems by considering what abstract properties a “fair” solution should have then attempting to construct and analyze procedures that maximize our measures of fairness.

An introduction to mathematical methods for studying planetary climate. The focus will be on low-dimensional models, whose simplicity allows insight into fundamental mechanisms of climate change. We will use tools from algebra, geometry, and calculus to study topics including energy balance, greenhouse gas forcing, and ice-albedo feedback. This course will count towards the Applied Math area of the math major.

An introduction to partial differential equations with emphasis on the heat equation, wave equation, and Laplace’s equation. Topics include the method of characteristics, separation of variables, Fourier series, Fourier transforms and existence/uniqueness of solutions.

An introduction to the study of topological spaces. We develop concepts from point-set and algebraic topology in order to distinguish between different topological spaces up to homeomorphism. Topics include methods of construction of topological spaces; continuity, connectedness, compactness, Hausdorff condition; fundamental group, homotopy of maps.

The theoretical foundations for the calculus of functions of a complex variable.

Geometric group theory is the study of (infinite) groups using geometric tools. The underlying principle of geometric group theory is that if a group G acts “nicely” on a space, then information about that space tells us information about the group. This class will introduce tools from topology, graph theory, and geometry and use them to study groups. Topics will include groups acting on trees and (more generally) hyperbolic groups. This course counts toward the Algebra area of the math major.

An introduction to the techniques and principles of analytic number theory. Topics covered include arithmetical functions, Dirichlet multiplication, averages of arithmetical functions, elementary theorems on the distribution of the primes, and Dirichlet’s theorem on primes in arithmetic progressions. 

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

(Formerly MATH 215) Introduction to statistics and data analysis. Practical aspects of statistics, including extensive use of statistical software, interpretation and communication of results, will be emphasized. Topics include: exploratory data analysis, correlation and linear regression, design of experiments, basic probability, the normal distribution, randomization approach to inference, sampling distributions, estimation, hypothesis testing, and two-way tables. Students who have taken Mathematics 211 are encouraged to consider the more advanced Mathematics 240/Statistics 250 (formerly Mathematics 265 and 275) Probability/Statistical Inference sequence.

(Formerly Mathematics 285) 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, supervised and unsupervised classification methods, and understanding and visualizing spatial data. We will use the statistics software R in this course.

(Formerly Mathematics 245) 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 to analyze real-life data.

(Formerly Mathematics 275) Introduction to modern mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: resampling methods (permutation tests, bootstrap intervals), classical methods (parametric hypothesis tests and confidence intervals), parameter estimation, goodness-of-fit tests, regression, and Bayesian methods. The statistical package R will be used to analyze data sets.

(Formerly MATH 255) Covers sampling design issues beyond the basic simple random sample: stratification, clustering, domains, and complex designs like two-phase and multistage designs. Inference and estimation techniques for most of these designs will be covered and the idea of sampling weights for a survey will be introduced. We may also cover topics like graphing complex survey data and exploring relationships in complex survey data using regression and chi-square tests.

(Formerly MATH 280) Students will apply their statistical knowledge by analyzing data problems solicited from the Northfield community. Students will also learn basic consulting skills, including communication and ethics.

An independent study course intended for students who have completed an external activity related to the statistics major (for example, an internship or an externship) to communicate (both in written and oral forms) and assess their statistical learning from that activity.

Spatial data is becoming increasingly available in a wide range of disciplines, including social sciences such as political science and criminology, as well as natural sciences such as geosciences and ecology. This course will introduce methods for exploring and analyzing spatial data. Methods will be covered to describe and analyze three main types of spatial data: areal, point process, and point-referenced (geostatistical) data. The course will also extensively cover tools for working with spatial data in R. The goals are that by the end of the course, students will be able to read, explore, plot, and describe spatial data in R, determine appropriate methods for analyzing a given spatial dataset, and work with their own spatial dataset(s) in R and derive conclusions about an application through statistical inference.

(Formerly MATH 315) 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).

(Formerly MATH 315) 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, generalized additive models or models for spatial data.

Formerly MATH 315) An introduction to statistical inference and modeling in the Bayesian paradigm. Topics include Bayes’ Theorem, common prior and posterior distributions, hierarchical models, Markov chain Monte Carlo methods (e.g., the Metropolis-Hastings algorithm and Gibbs sampler) and model adequacy and posterior predictive checks. The course uses R extensively for simulations.

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