Surely the above statement is only true when $$n=1$$, however for $$n=3$$ we have: $$(x+y)^{3}=x^{3}+3x^{2}y+3y^{2}x+y^{3}$$. $$n=5$$ gives: $$(x+y)^{5}=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5}$$. If we ignore multiples of 3 or 5, we obtain precisely the correct result! Try $$n=4$$ yourself! Does it work? What's happening? What can we do with this? The answer is: A lot! In this course we'll discuss in what sense the identity is true, and what the consequences are.

You may have heard the famous saying a topologist can't tell his donut from his coffee cup, because they both have one hole, and if they were made of a very flexible material, you could turn one into the other. In this class, we will discuss things of this sort for example: 1. Why can't I turn a plane into a sphere? 2. Why can't I turn one sphere into two spheres? 3. Why can't I turn a baseball into a donut? In the process, we will talk about notions of homotopy equivalence, homeomorphism, and my favorite, the fundamental group!

Let's suppose you live on some very large planet and you aren't sure if it is shaped like a donut, or like a sphere. How could you tell? It is important to know that this is somehow a "global" problem, since a donut and a sphere both look the same very close up. This is where the largeness of the planet becomes important. Somehow you would need to explore the planet in such a way that would detect the hole. How might you do this? We will explain how to do this, and on the way discover the fundamental group, an incredibly important tool in algebraic topology. The fundamental group is an algebraic object which you associate to a space which keeps track of holes and other obstructions in your space. In this course, we will define the fundamental group and calculate it in lots of interesting cases.

In analysis, we like to prove certain facts about continuous or differentiable functions, and it often becomes tempting to assume certain things that simply aren't true. In this class we will talk about functions with the following properties: 1. Functions that are continuous everywhere but differentiable nowhere. 2. Functions that are continuous at irrationals, and discontinuous at irrationals. 3. A function that rises from 0 to 1 over the unit interval that is continuous everywhere, but has derivative zero almost everywhere. Along the way, we'll encounter interesting concepts such as uniform continuity and measure. It'll be a blast!

In this course, we will examine elliptic curves, which are usually first introduced as equations of the form: $$y^{2}=ax^{3}+bx+c$$. Elliptic curves are interesting because they naturally have a group structure. That is, it makes sense to "add" points on the curve. We will talk about why the points on an elliptic curve form a group, which will bring us into the realm of complex analysis! This will naturally lead us to the theory of elliptic functions and the above equation will jump right out at us! This course will cover a lot in a small amount of time. We will do some explicit calculations, but mostly focus on the abstract concepts.

I'm sure you've all heard of $$\pi$$, and for a very good reason. $$\pi$$ is an extremely important number that shows up everywhere in mathematics. Similarly, you may have heard of a number called $$e$$ which is (arguably) even important than $$\pi$$! Then there's this crazy number called $$i$$ which satisfies $$i^{2}=-1$$. On their own each of these numbers is spectacular, and they are in fact connected in a very important way by the following formula: $$e^{i\pi}=-1$$. If this doesn't shock you, then I don't know what will! So $$e$$, $$i$$, and $$\pi$$ are all very special and interesting numbers. However, there are lots of interesting numbers out there that aren't as "exotic" as $$e$$, $$i$$, $$\pi$$. For example, the number $$163$$ has some very remarkable and mysterious properties related to deep areas of number theory! In this class, we will discuss all sorts of interesting numbers with lots of applications to number theory, and we'll even learn some calculus on the way. It's going to be awesome.

I'm sure you're all familiar with Pythagorean triples like $$3$$, $$4$$, and $$5$$ which satisfy $$3^{2}+4^{2}=5^{2}$$. What if we replace the $$2$$ with a $$3$$? Are there any interesting whole number solutions to $$x^{3}+y^{3}=z^{3}$$? On the other hand, we've just shown $$25$$ is a sum of two squares. What other numbers can be written as a sum of two squares? Can ever number be written in this way? I'm sure you're also familiar with equations such as $$ax+by=c$$, or $$x^{2}+y^{2}=k$$. We know what their graphs look like, but what can we say about points on these graphs? Are there any whole numbered pairs, $$x$$ and $$y$$ on these curves? How many? What if we allow $$x$$ and $$y$$ to be fractions? I'm sure you're ALSO familiar with the idea of a prime number, that is, numbers that are only divisible by $$1$$ and themselves. Whole numbers have the fantastic property that we can factor them uniquely into primes. For example, $$21=3 \cdot 7$$ or $$2012=2^{2} \cdot 503$$. In this way, if we want to understand whole numbers well, it seems like it would suffice to just try and understand these primes. How many primes are there? How are they distributed? Are there patterns to them, or are they sort of randomly distributed? If you can answer this question, you could win $1,000,000! Such is the beauty of mathematics, with every question we answer, there are always more we can ask! In this class, we'll do our best to answer all of these questions and more! The style of the class will be as follows. Each lecture, I'll present a sort of conceptual question about numbers or equations, which will motivate the discussion of important concepts in algebra and number theory. Then, we'll finish off each lecture by looking at an important problem in algebra or number theory, and solving it! (most of the time) Short, fun, and interesting homework problems will be assigned, but they are $$\bf{optional}$$ and solutions will be given out the following class.

Have you ever looked at a map of the Earth and wondered why Greenland isn't a world superpower? I mean, just look at it, it's practically the size of South America! But, is it really? Actually, South America is nine times larger! Unfortunately, in our attempt to map the spherical earth onto a flat piece of paper, we distort areas quite a bit. Imagine trying to flatten out an orange peel onto a table, it doesn't work quite well. Why would we use such a distorted map? Well this particular map, the Mercator projection has the nice property that, while its areas are wrong, its angles are correct, which is extremely important for navigation. Are there any maps that preserve area? Actually, yes, and its very easy to do! Why not use that? Well, as you might be able to predict, that map messes up angles! Is it possible to have a map that accurately displays angles and areas? Or more importantly, one that preserves all distances? The answer is a resounding no, which we shall discuss in this class using some truly amazing theorems of Gauss.

So let's say we have a line in the plane, $$L$$ and a point $$p$$ not on that line. Then I can draw a line through $$p$$ parallel to $$L$$, right? This doesn't seem too difficult to do, but why are we always able to do this? The answer is, it works because we built our geometry in such a way that it had to work! This idea that lines are either parallel or intersect at one point is known as Euclid's Parallel Postulate. So, while it is obvious that Euclid's geometry is immensely useful in applications, it turns out that it isn't the only valid notion of "geometry". That is, we can come up with a new model of geometry that satisfies similar axioms, but fails our parallel postulate. That is, I could find infinitely many (!) lines through $$p$$ parallel to $$L$$. If this isn't weird enough, it turns out that this new geometry lives on a surface that looks like a sphere with radius i. Yes, as in the imaginary number i. Weird! These ideas just scratch the surface of a beautiful and rich field called hyperbolic geometry. Ideas from hyperbolic geometry have applications in number theory (Fermat's Last Theorem), and in Einstein's work on general relativity. It's gonna be awesome!

In this course, we will study polynomial equations in several variables. For example, we will be talking about solving equations like: $$x^{2} + y^{2}= 1$$, or $$y^{2}=x^{3}+1$$, or $$x^{3}+y^{3}=z^{3}$$, where $$x$$ and $$y$$ are whole numbers or fractions. This will lead us to discuss lots of important concepts in algebra and number theory, and I promise it will be awesome. Oh, did I mention that $$2$$ of those equations above are related to two of the hardest problems in modern mathematics, one of which, Fermat's Last Theorem, is solved. The other one, the Birch and Swinnerton-Dyer conjecture, still has a $1,000,000 out for a solution. This class is essentially a condensed version of my HSSP class, Algebra and Number theory, so if you're really interested I have lots of extra materials you can read to learn more.

In this class, we will discuss a very diverse and fundamental class of mathematical objects called rings. Intuitively, rings are things with two operations, usually called addition and multiplication. There are some properties these operations need to satisfy by themselves, and they must also get along with each other. I will give examples and then attempt (and fail) to prove Fermat's Last Theorem. Then, we'll get really crazy and talk about fields, a special class of rings where you can divide. I will of course give more examples. At this point, the class goes in whatever direction you want. I will briefly explain a few amazing applications of field theory and it's up to the class to decide which we pursue. Here are some possibilities: 1. Basic Galois Theory -Why do complex roots come in pairs? -What is the difference between i and -i? -Why is there no quintic formula for polynomials? 2. Geometric Constructions -Why can't I trisect this angle? -Why can't I double this cube? -No seriously... why not? 3. Solving polynomials -I want to solve this polynomial, but my field isn't big enough. What do I do? Note: Due to volume of material, I will most likely not give rigorous proofs. Rather, I intend for this class to give you a broad overview of what ring and field theory and why it is useful and fun!

Now I'm sure you've all heard of differential equations, but I'm willing to bet not all of you are familiar with difference equations. Well, difference equations are simply the finite and under-appreciated analogues of differential equations. I say under-appreciated because difference equations are actually extremely useful in practical applications and in finding patterns. We will start the class off by considering Fibonacci's rabbit problem, arguably one of the most famous difference equations. I will then show you all the wonderful super-secret trick that allows you to solve these equations. Next, I will show you a problem that arose in my own research and how I used difference equations to solve it! Then we'll revisit Fibonacci and give a complete solution. Please, help make a difference. ...get it?

In this course, I will give you a guided tour through a rigorous treatment of the fundamentals of differential and (some) integral calculus. We will start from very modest axioms and I will allow you to figure out how to formulate and prove the theorems you know and love from calculus.

So I'm sure all of you are familiar with the quadratic formula. Well you may also be vaguely familiar with the cubic and quartic formulas. These are very ugly expressions involving square and cube roots and very, very messy polynomial expressions in the coefficients of your original polynomial. Nonetheless, we can write down the solutions. So, what about the quintic? Surely we can solve the quintic with some horrendous expression for the roots. Surprisingly, it is possible to prove that it is impossible for such an expression to exist, and we can prove it. This class will be an introduction to Galois theory, the study of permutations of roots of polynomial equations. We will begin with basic group, ring, and field theory and will quickly move onto computing Galois groups and making connections to fields. Finally, we will finish with a sketch of the proof that the quintic is not solvable.

In this class, we will separate into two groups and face-off in a game of collaborative mathematical jeopardy. Each team will be given 30 seconds to solve a certain question and if they fail to solve it, the other team will have 10 seconds to decide if they would like to try and "steal" the points. However, if you try to "steal" and guess incorrectly, you lose the amount of points the question is worth. The questions range from somewhat elementary to (in my opinion), very difficult. Bring your A-game!

In May 2000, the Clay Mathematics Institute announced a list of seven of the most difficult open problems in mathematics. Here's the fun part: each problem has a $1,000,000 reward for a solution. In this class, we will discuss what each of the problems means, why it is difficult, why it is important, and any work or thoughts about possible solutions. In addition to the problems, we will discuss fun topics and problems related to the fields of each problem. These include: analytic number theory, physics, computability, topology, and more! If you like math, money, or both, this class is a must.

Have you heard about quantum mechanics, but never really knew that much about it? If so, then this class definitely for you! In this class, we will discuss the counterintuitive nature of quantum mechanics and what makes it so awesome!

Do you like math? Me too! In this class I'll show you some of my favorite proofs. At the beginning of class, I'll put a list of proofs on the board, and as a class, we can vote on which ones to do. If you like math, then you'll love this class. Just a few of the things we might do: Proofs by contradiction, proofs by induction, proof of the area of an ellipse (requires calculus knowledge), proofs by infinite descent, proofs without words, and more!

As the title suggests this class is all about primes. The numbers, not the ribs! Prime numbers are an extremely interesting and fundamental subject in mathematics that have prompted many important questions. How many primes are there? What patterns can we discern from the prime numbers? What impact do prime numbers have on our lives? Some of these questions we know a lot about, others very little. Either way, it shows that primes are a very interesting field with surprisingly far-reaching consequences. Come on down to see what the deal is with these amazing numbers!

Number Theory is one of the most beautifully pure areas of mathematics with many beautiful and interesting results. In this ...

In this course I will discuss the research I conducted this summer at the University of Connecticut. My research topic ...