The summer is a great time to try something new! In this case, we will be putting a bit of a new spin on an “old idea”—counting. We all know how to do it (hopefully), but what may be more interesting is that many “simple-sounding” counting problems have surprisingly complex solutions. For example, suppose eight students take a test and hand it in. The next day, a the teacher is out sick and the substitute teacher attempts to hand back the tests without knowing the students’ names. How many ways can she hand back the tests so that no single student gets back his or her original test? It is relatively easy to understand the question—but its solution is a bit involved, as we will see when we study derangements. That is essentially what this summer class will be about—solving problems that involve a “how many ways” circumstance with different methods and techniques which we will acquire! Class-time will be used for mini-“lectures”, group activities, individual problem-solving, and presentations. There will be short, but meaningful assignments for this class. On average, it should take you between an hour and an hour and a half for each weekly assignment. If this is a time commitment you cannot or do not want to make, you may want to consider registering for a different class. You will not be expected to get everything correct, but you will be expected to attempt every problem.Be prepared to participate actively and be fully engaged in the discussion at hand! This is not a lecture-based class like you may be used to—so don’t be afraid to try something new! Topics will include Derangements, the inclusion-exclusion principle, the Fibonacci Sequence, its motivation, the golden ratio, recursion relations, and elementary Ramsey theory.

So, why do we care so much about these stupid numbers that are partially 'imaginary?' Seriously?! Why would we bother with numbers that aren't even real? Well, as a matter of fact, complex numbers creep their way into electrical engineering! Woah! In this class, we will cover the algebra of complex numbers, functions of a complex variable, some basic calculus with complex numbers, and applications to engineering and real life. Be prepared for rapid neuron growth in this class!

Calculus BC can be offered by schools that are able to complete all the prerequisites before the course. Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics. Both courses represent college-level mathematics for which most colleges grant advanced placement and credit. The content of Calculus BC is designed to qualify the student for placement and credit in a course that is one course beyond that granted for Calculus AB. (from collegeboard.com)

Mainly for students who are enrolled in the multivariable concentration course. This will cover basics of differential and integral calculus, and will serve as a refresher of single-variable calculus. We will spend time both rebuilding calculus intuition and refreshing formulas. There will be many examples given and worked out, plus supplemental review sheets and helpful formula sheets. You may find this helpful

In this seminar, we cover the basics of functions involving complex numbers! Topics include Euler's formula, Taylor series in complex variables, and applications to fluid flow and electrostatics.

A stochastic process is a fancy way of saying a random process. Supposed a rat is trapped in a maze with three doors, one of which is an exit that takes two minutes to get out of, and the other two of which are three and four minute detours, respectively, which lead back to where the rat started. How long can we expect the rat to be in the maze before he exits if he immediately selects a door upon returning to the maze and selects each door with the same probability? You’ll be able to answer this question on the first day of the class!

Suppose you planned a vacation in Italy, starting at the base of the boot--each week, you travel 150 miles up the boot until you reach the top. Because you have strong value for culture and language, you try to learn as much of the native language as you can when you're at any given place. By the end of your vacation you will be speaking a vastly different language than you were when you started--but science has shown that you are very unlikely to be aware of this change. Why does that happen? Language is a science. That's right. We will look at particular parameters of language including sound, sentence structure, word structure, word meaning, and pragmatics.

Have you ever wondered if there was a more general fundamental theorem of calculus? As a matter of fact…there is. It’s called Stokes’ Theorem, and it applies to functions that take in several variables, spit out several variables, and sit in several dimensional space. Woah! Don’t try to visualize it. But come to this class to learn the general fundamental theorem of calculus, and see the craziest math you’ve ever seen before in your life!

Stochastic Processes are, in short, just random process. In this two-hour seminar, we discuss some of the following topics: +Conditional expectation and Wald's Lemma; +Gambler's Ruin and Game Strategy +Random Walks +Brownian Motion Chances are, if you have enough interest in these topics, you will have sufficient motivation to master the material. This is the heart of applications of probability theory.

So, we know how to differentiate a function of a single variable. The derivative of a function at a point gives you the slope of the tangent line, of course! But suppose you're standing on a surface in three dimensions, and you want to know what the instantaneous rate of change would be should you choose to move ANY direction whatsoever in three-space...is there a way to figure this out? Of course there is! In this seminar we introduce partial derivatives of functions of several variables, the gradient vector, and the directional derivative, which will answer questions that will suddenly make math seem even cooler than it already is!

So you're told that there are these things called "complex numbers" that are of the form a+bi, where i is the square root of -1; but what does that actually mean? How did people come up with this garbage? We will discuss functions of a complex variable, the motivation for them, isomorphism (what? come and find out!) between the complex plane and two dimensional space, geometry of functions of a complex variable. Did I mention that e^(pi*i) = -1? Well anyway, that is true. Come and find out why! This seminar will be quick-paced, activity-based, and a lot of fun. Three hours goes by very, very quickly on this topic, so it's an investment of your time you probably want to make, and an opportunity you don't want to miss!

So, matrices are arrays of numbers, and vectors are arrows with magnitude and direction that are defined component-wise. But, what do they actually do? In this class, we will discuss some of the cool things that matrices and vectors can do, and we will try to classify some linear transformations and endomorphisms. Possible topics also include eigenvalues and eigenvectors.

The summer isn't fun without getting your feet wet with something new! This summer, we're going to take a new spin on some old topics in mathematics, with an emphasis on enumerative combinatorics. Woah, what?! Enumerative combinatorics is the study of counting; of course you all know how to count, but in this course we will be able to answer questions like "How many ways can I rearrange 5 blue books and 3 red books so that no two red books are adjacent." What sometimes these questions seem trivial, we will see that they are in fact the root of a lot of problems in mathematics. Expect to think in new ways; expect to work in groups; expect to have fun! Don't be afraid to try something new.

Have you ever wondered if there was a way to generate Pythagorean Triples in a simple way? What is the difference between a coffee cup and a doughnut? How can we construct the real numbers from the rational numbers? Why does e^($$\pi$$i) = -1 ? Let's face the facts...there are so many cool things in math that go unexplored. That is why instead of covering one big topic, each week will be devoted to one particular field in mathematics. If the above questions are interesting to you, ALL will be answered in this course, plus many more. The topics have been chosen very carefully from a wide variety of mathematical fields, including Topology, Analysis, Number Theory, and Abstract Algebra. Classes will include hands-on activities, group work, and exciting demonstrations.

How do you find the area under a parabola? How can you prove the formula for the area of a triangle? What about a circle? These are questions that can be answered with integral calculus, the study of continuous sums. We will briefly review limits, continuity, and differentiability. Then, we will go on to define the integral as a limit of Riemann sums, and prove the Fundamental Theorem of Calculus. Techniques for integration will be introduced as time permits.

There are a lot of things that high school mathematics teachers often don't have time to fit into their curriculum these days. This course will seek to teach you interesting little things about mathematics that may have been overlooked. Some topics may include: mappings and functions, reading complex roots off of a real graph, elementary number theory, and derivation of common formulas that you may take for granted.

We will first look at the concept of a limit, and why it is so important in calculus. Then we will derive a formula for the slope of a tangent line, called the derivative. As time permits, we will introduce techniques of differentiation, the derivative as a function, and applications. The attempt is to introduce students to the theoretical and conceptual aspects of an introductory calculus course. It could also serve as a refresher for people who have already taken it.

A full year's calculus class, covering differential and integral calculus. The prerequisites for this class were: Algebra I and II, ...

You learn about the complex numbers in high school, numbers of the form a+bi. But what does that actually mean? ...

Have you ever wondered what the difference is between a coffee cup and a doughnut? Now that I asked the ...

In Calculus I, they show you rules for taking derivatives. Once you know all the rules, you know how to ...

Kind of scared of AP Calculus, or just calculus in general? This is a perfect time to try it out ...

Ever wonder what a mathematician actually does? How are things proven? How do mathematicians think? Why should we care? More ...

Real analysis is the study of functions of a real variable. We are all "told" about real numbers in high ...

Number theory is such a great subject because it is accessible at many levels. Really, it is as simplistic or ...