Generally speaking, dynamics is the study of how systems change and evolve over time. The theory of dynamical systems is applied in many different fields---physics, biology, economics, and sociology, to name a few---but it is also a captivating study in its own right. Here, we'll look at several examples of mathematical dynamical systems, and in the process we'll develop some language and explore some tools mathematicians use to study and describe this vast and beautiful theory. See you there!

We use numbers all the time: to count and measure things, to solve math problems, and to play games. But what exactly IS a number? Is it a point on a line? A sequence of digits? Or is the answer much more complicated than that? In this class we'll explore some possible answers to these questions, challenging our intuitions and getting a deeper understanding of the building blocks of mathematics in the process.

On the one hand, mathematics is a very old field of study, and so it can seem like many of its questions have been answered. On the other hand, mathematics is still (and always will be!) full of interesting unsolved problems. Many of these problems require a deeper knowledge of higher mathematics to be able to approach, but what's remarkable is that there are many unanswered questions that don't require any deep knowledge to understand! In class, we'll take a look at some of these open problems, and then we'll get our hands dirty and play around with them some. See you there!

Generally speaking, dynamics is the study of how systems change and evolve over time. The theory of dynamical systems is applied in many different fields---physics, biology, economics, and sociology, to name a few---but it is also a captivating study in its own right. Here, we'll look at several examples of mathematical dynamical systems, and in the process we'll develop some language and explore some tools mathematicians use to study and describe this vast and beautiful theory. See you there!

"Mathematics is the art of giving the same name to different things." - Henri Poincaré In mathematics, abstract algebra doesn't have much to do with solving equations for x. Algebra is the study of structure and relation -- how different sets of objects relate to themselves and to each other through various types of transformation. In this course, we'll explore groups, some of the most basic (but by no means simple or boring!) abstract structures. Group theory is essential to many areas of science and mathematics, from quantum physics to relativity to Rubik's cubes. We'll discuss symmetry, isomorphism, and plenty of beautiful and illuminating results.

In mathematics, a "group" is a specific type of structure that encodes information about the symmetry of a system. Not only are groups instrumental in the study of pure math---they also have important and exciting applications in real life: doing cryptography (making and breaking codes), solving a Rubik's cube, and studying quantum physics (quantum groups and supersymmetry). In this class we'll first define a group abstractly and then look at many examples. As time permits, we'll look at both ways to study groups abstractly and ways in which to apply group theory. Also, note: there is a follow up to this course (Group Theory II) if you interested, but it is certainly not required if you just want to take Part I. Feel free to email me (louis.gaudet@yale.edu) with questions. I hope to see you there! :)

This course is a continuation of Group Theory I, another course I am teaching this spring. In Part I, we will have looked at the definition of a group, many examples, a few general properties of groups, and some applications of groups to the real world. Part II will focus more on the purely mathematical side of groups; it will be both algebraic and geometric in flavor. We will ask questions like: How many are groups (of a given size) are there? How can we describe a group's structure? How can we classify groups? Feel free to email me (louis.gaudet@yale.edu) with questions. I hope to see you there! :)

What is a "matrix"? Hey, that's easy! I can tell you right now: a matrix is a "rectangular array of numbers," sort of like a sudoku board! Done! Well, I suppose that's not all...you should be unsatisfied with my answer! Why? It's not incorrect...it's just incomplete! All I've told you is what a matrix looks like written on a piece of paper. I haven't answered the real question about what it IS. That'll take us into new, deeper questions: What does a matrix DO? What sorts of properties can it have? Why do we care about those properties? What does a matrix "look" like, geometrically speaking? These sorts of questions will guide us in our exploration of matrices. I don't want to give too much away just yet, but our questions will take us right into geometry---we'll be looking at shapes and spaces and transformations that manipulate these objects. The types of transformations will give us information about the matrices we study, and vice versa. Then from geometry, we'll look at some of the many places matrices come up---studying networks and how diseases spread, probability, new ways to solve equations, and maybe even "imaginary numbers"---one of my favorite topics. In the end, I hope to be able to show you how matrices are much, much more than algebraic tools---they are rich objects with exciting applications and beautiful geometry! See you in class! P.S. I love to talk about math! If you have any questions at all, about the course or about math in general, please feel free to email me at louis.gaudet@yale.edu!

The remarkable and eccentric 20th century mathematician Paul Erdos (look him up on Wikipedia!) used to speak of "The Book," an "imaginary book in which God had written down the best and most elegant proofs of mathematical theorems" (Wikipedia). In this course, we'll look at some results and proofs that many mathematicians consider particularly "elegant," and we'll talk about a bit about what it means to be elegant. If you have questions, feel free to email me at louis.gaudet@yale.edu. Hopefully I'll see you in class!