The title says it all (or most of it). This course provides an introduction to proof writing. How do you construct a rigorous proof and when do you know that the proof is complete? What techniques exist and how do we not just make a proof valid but also good?

You will have ample opportunity to construct and critique proofs. This course satisfies a WRIT requirement.

This course is the study of the infinite processes involving numbers. How do you build the real numbers as limits of rational numbers? What does it really mean for a function to be continuous or a space to be connected or a series to converge? Calculus is filled with beautiful ideas but in the calc classes you mostly learn about calculus as a computational tool. This course goes back and revisits the ideas of calculus and examines them for what they are in themselves.

A graph is a collection of points connected by edges. It is a theoretical construct that models many practical situations. You could have a bunch of houses connected by telephone lines, or airports connected by airline flights, or a diagram made from wedding invitees with edges drawn between them if they hate each other. Given such a structure, there are many questions you can ask. For instance, how many tables do you need to seat all the guests at your wedding so that people who hate each other do not sit at the same table? How many telephone calls can be made at the same time? What is the cheapest multi-city flight you can book which reaches every city? Graph theory gives you the tools you to answer questions like these, and many others.

Geometry is the study of shapes: how they fit together, how they can be cut apart, how they sit in space, and how they relate to one another. It is also the study of symmetry and has close connections to algebraic structures like groups. Geometry involves angles, lines, planes, polygons, polyhedra, higher dimensional polytopes, tilings, spheres, curves, and surfaces. It is a subject with a strong visual and tactile component.

This class is the study of the shapes of smooth curves and surfaces in the plane and in space. The idea is to apply tools from calculus and linear algebra to understand what it means for a curve or a surface to curve, and to quantify that idea. A curve in space can both curve and twist, so you study both curvature and another concept called torsion. A surface in space can curve differently in different directions, and so you learn about linear algebra gadgets like quadratic forms which quantify multi-directional curvature.

Imagine you want to send a message to your friend Alice which your friend Bob cannot understand. So you and Alice make an agreement that you will increment each letter by one. You write CPC JT B KFSL. It turns out that Bob is smarter than he looks and is able to decode the message, which reads BOB IS A JERK. So, then you and Alice decide that perhaps it is better to base an encoding system on fancier mathematics that perhaps is related to the ease of multiplying together large primes and the difficulty of factoring a composite number into primes. But then it turns out that Bob is much smarter than he looks and he has access to a supercomputer that can possibly factor large composite numbers using elliptic curves, a mathematical gadget which is both a group and in some sense the surface of a donut. In this class you will learn encoding methods you and Alice can use to keep secrets from Bob and also counterattacks that Bob can use to discover your secrets.

Know the present, know the rules, predict the future: this is the ambition of ordinary differential equations. Given the right input from physics or biology (the rules), one can predict the spread of a pandemic, the motion of planets, the behavior of a spinning top, the rise and disappearance of languages, or how changes in an ecosystem can lead to bifurcations that bring about a sudden increase or drop in animal population. In this course you will solve linear equations relying on linear algebra. You will also study a few common classical nonlinear differential equations and discrete equations (nonlinear pendulum, Newton method...), and develop qualitative tools to study general ODEs, motivating a deeper study of real analysis and topology (not necessary for this course, beyond calculus, but you might pick some along the way). You will also see the limitations of dynamical systems and what they cannot do.

Look around you. Contemplate the waves rolling up on the shore, or the wind whipping through the trees, or a vibrating string, or heat flowing upward along your pot as it sits on the stove. All these processes are described by functions which satisfy certain kinds of partial differential equations -- the wave equation or the heat equation, Navier-Stokes, and so on. In Math 1120 you will learn about these basic equations which control the world around us, how they behave, how to solve them, why they exist.

Complex analysis the study of those functions on the plane which behave like holograms: If you have complete information about the function in any tiny area where it is defined, you can reconstruct the whole function. Polynomials have this property, and the functions in complex analysis are sort of like limits of polynomials. The hologram principle makes possible beautiful formulas, like the Cauchy integral formula, which give recipes for the reconstruction.

Abstract algebra is the study of sets and additional imposed structure which allows you to combine and manipulate the members of a set. For instance, you can add or subtract one integer from another and get yet another integer. The set of symmetries of a cube has the same property: You can "combine'' two symmetries of a cube by doing one then the other, and this gives you yet another symmetry. Both these things are instances of what is called a group, an abstract structure which somehow includes both. When you add in a second combination law, akin to multiplication, you get an object called a ring. When you add in division as well, you get field. In this class you learn all about groups, rings, and fields and their structural properties.

Two thirds of this class are a detailed study of fields, one of the abstract structures studied more briefly studied in Math 1530. The study of fields is closely related to methods of solving polynomial equations, and there is also a hidden underlying group which permutes the roots of a polynomial and is compatible with the associated field. The class builds up to a grand synthesis of polynomials, fields, and symmetry groups called the fundamental theorem of Galois Theory. The last third of the class is usually a further topic in algebra which might or might not be related to Galois Theory. One common topic is elliptic curves, which are simultaneously surfaces called tori and groups.

This course is a deeper foray into real analysis. The first part of the class extends the concepts learned in Math 1010 to more general objects called metric spaces: These are abstract objects in which you can measure distances, just like you can on the real line or in the plane. Metric spaces in general can be much stranger than the real line. You could have an infinite dimensional sphere, for instance. The next part of the class deals with infinite dimensional vector spaces in which each vector represents a function. You will learn about Fourier analysis, which amounts to breaking a sound wave into its component harmonics. This is surprisingly similar to taking a vector and writing it as a linear combination of basis elements. Yet another part of the class deals with the geometry and analysis of partial derivatives of multi-variable functions.

This course is a continuation of M1630. This is the class where you learn about how to deal with really weird sets and weird functions. The first part of the class deals with measure theory, how to assign a notion of size to very complicated sets, such as infinite unions of infinite intersections of infinite unions of infinite intersections of infinite unions of cubes. Relatedly you will learn how to integrate crazy functions, like one which is zero on all points with rational coordinates and nonzero on all points with irrational coordinates. Following this, you will learn about fractal sets, sets whose dimension is not an integer. So, for instance, you'll learn about sets which are somehow halfway between lines and planes. What do they look like? How are they constructed? How to you calculate their dimension? The Hausdorff measure will give you a powerful tool to answer these kinds of questions. The last part of the class goes more deeply into infinite dimensional vector spaces.

Topology is the study of spaces and how they are connected together. From the topological point of view, the surface of a cube and the surface of a ball are the same. They are both "sphere-like''. When you do topology you are a jellyfish swimming around in the ocean and playing with rubber structures. Forgetting about the noise coming from fine scale geometric properties like distances and angles, you can sometimes get a deeper appreciation of shape in a broader sense.

This class is sort of like an n-dimensional version of multivariable calculus. Didn't it seem weird that in multivariable calculus there was a zoo of stuff that was all sort of similar: gradient, curl, divergence, Green's Theorem, and Stokes' Theorem, Gauss's theorem on electric flux. It turns out that all these things fit together into one grand theory called differential forms, and then there is one master theorem called the n-dimensional Stokes' Theorem. The natural setting for Stokes' Theorem is a manifold. A manifold is essentially a higher dimensional surface. The main goal of this class is to bring you to the point where you understand Stokes' Theorem on a manifold.