Mathematics

What does it mean for a mathematical problem to be unsolvable? The very concept does not seem to have been much considered, until, in 1824, a young Norwegian named Niels Henrik Abel published a small pamphlet on an old problem. The pamphlet was one of the first markers of a sea change in mathematics, and by the time Abel died, six years later at the age of twenty-six,

This course covers the breadth of university calculus: differentiation, integration, infinite series, and ordinary differential equations. It focuses on concepts and interconnections. In order to cover this much material, computational techniques are de-emphasized. The approach is historically based and classical, following original texts where possible. Further techniques

This class focuses on what is most intellectually interesting about calculus: the problems it was invented to solve, the fundamental ideas, and the interconnection between the ideas. The class approaches integration, infinite series, differentials, and differential equations, in a unified way. It focuses on concepts and interconnections. In the process, the class builds

Discrete mathematics studies problems that can be broken up into distinct pieces. Some examples of these sorts of systems are letters or numbers in a password, pixels on a computer screen, the connections between friends on Facebook, and driving directions (along established roads) between two cities. In this course we will develop the tools needed to solve relevant, real

In this experimental class, we will create space for you to pursue work within Mathematics. This course is intended for students at a variety of levels of experience, with a solid interest in following questions and curiosity, to lead to a deeper understanding. You will lead your work, in collaboration, and with the support of the class. 

Topics explored in this

Throughout history, people have played games — games of chance and games of skill. Many of us grew up playing all kinds of different games, and most of those are infused with the core tenets of statistical reasoning and understanding: probability, risk assessment, expected value, and game theory. This course will look at statistics and probability through this lens. We will

This foundational class covers modes of reasoning used in quantitative sciences and mathematics. While learning the art of mathematical modeling, i.e. translating the physical systems/real-life situations into mathematics, we will apply problem solving and practice effective communication of mathematics. This process involves isolating the essential variables and

This foundational class covers modes of reasoning used in quantitative sciences and mathematics. While learning the art of mathematical modeling, i.e. translating the physical systems/real-life situations into mathematics, we will apply problem solving and practice effective communication of mathematics. This process involves isolating the essential variables and

Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." The thesis of this class is that this is still true today. Leonhard Euler's (1707–1783) collected works run to 81 volumes and over 35,000 pages, the publication only having been (mostly) completed in 2022. Most of

Together with calculus, linear algebra is one of the foundations of higher-level mathematics and its applications. This is NOT just the algebra you know from high school. There are several perspectives one can take on linear algebra: it is a method for handling large systems of linear equations, it is a theory of linear geometry (including in dimensions larger than three),

Together with calculus, linear algebra is one of the foundations of higher-level mathematics and its applications. This is NOT just the algebra you know from high school. There are several perspectives one can take on linear algebra: it is a method for handling large systems of linear equations, it is a theory of linear geometry (including in dimensions larger than three),

Everything is geometry! This class is about two things: first, about how mathematicians have extended the concept of "geometry" beyond triangles and circles, into higher-dimensional spaces, curved spaces, spaces of functions, discrete spaces, and more. Second, about how this extension of "geometry" can allow us to apply our powerful geometric intuition to a wide range of

Multivariable calculus is one of the core parts of an undergraduate mathematics curriculum. Introductory calculus mostly concentrates on situations where there is one input and one output variable; multivariable extends differentiation, integration, and differential equations to cases where there are multiple input and output variables. In this way, multivariable calculus

I would (and will) argue that Newton's Principia is the most important book yet written. It is certainly the most important book that a vanishingly small number of people have actually read.

Written about 150 CE, Ptolemy's Almagest collected and systematized the knowledge of astronomers of the time to give a system which roughly predicted the

Differential equations are a powerful and pervasive mathematical tool in the sciences and are fundamental in pure mathematics as well. Almost every system whose components interact continuously over time can be modeled by a differential equation, and differential equation models and analyses of these systems are common in the literature in many fields including physics,

In the eighteenth and nineteenth centuries, mathematics underwent a vast expansion, into new, exciting, and increasingly counter-intuitive realms. The subject risked mystification and mutual incomprehensibility between experts in different sub-fields. In the first part of the twentieth century, a group of French mathematicians, under the pseudonym Bourbaki, undertook an

This intermediate/advanced linear algebra class will focus mostly on two main topics: spectra, and tensors. The theory of matrix spectra, (eigenvalues and eigenvectors), is arguably one of the most useful ideas in all of mathematics. We will discuss some major applications, such as input-output models in economics, the Page Rank algorithm, JPEG compression, the Fast Fourier

In this course, we will focus on developing the statistical skills needed to answer questions by collecting data, designing experimental studies, and analyzing large publicly available datasets. The skills learned will also help students to be critical consumers of statistical results. We will use a variety of datasets to develop skills in data management, analysis, and

In this course, we will focus on developing the statistical skills needed to answer questions by collecting data, designing experimental studies, and analyzing large publicly available datasets. The skills learned will also help students to be critical consumers of statistical results. We will use a variety of datasets to develop skills in data management, analysis, and

In this experimental class, we will create space for you to pursue work within or near mathematics, statistics, data science, etc. This course is intended for students at a variety of levels of experience with a solid interest in following questions and curiosity to lead to a deeper understanding. You will lead your work, in collaboration and with the support of the class.