As usual, we’ll have a range of classes from A*ccessible *** (A)** classes that anyone at camp can enjoy to

*Intermediate*

**classes that will be a bit more formal and where some mathematical maturity helps and also some**

*(I)**Prerequisite*

**courses that can get quite advanced and take us well beyond calculus. If a course is marked with more than one tag, that means that it serves as both: an**

*(P)***(A-I)**course will be accessible, but offer extra challenge for the students who are ready.

**Introduction to Computation (A)**

**Rolfe Schmidt**

Do you know what a computer is? Could you build one? Do you know what computers can do? What they can’t do? In this class we will organize transistors into gates, organize gates into register machines, and see how a simple register machine can be programmed to do most anything we can think of. We will play around with a few programs, we will experience the frustration of slow algorithms and the joy of fast ones. Then we’ll wrap up the course by exploring the limits of computers seeing that some questions simply cannot be answered by a computer. This class touches on material deep enough to challenge all students at the camp, but we will approach it with hands-on group activities and examples that make it accessible to all students too.

**Art and Math (A)**

**Martin Strauss**

In this course we’ll explore art history/criticism, math concepts, and art execution over topics that may include: symmetry, perspective and other projection, curvature, dimension (including fractals and dimension greater than three) and self-reference.

**Continued Fraction Expansion (A)**

**Yo’av Rieck**

We all know how to write down integers and rational numbers, the quotient of two integers. But how do we represent real numbers in general? Even rational numbers can be hard to represent if the integers involved are very big: it would make sense to write 2/3 instead of 2,000,001/3,000,001, getting a much simpler quotient at a very low cost. So how do we represent “complicated” rational numbers, or even irrational numbers (such as the square root of 2 or 5) efficiently?

In this course we will discuss what “good approximation” means and study a method, called *Continued Fraction Expansion, *that allows calculating *best* approximations.

*Open to students of all backgrounds and ages.*

*This course pairs well with the course **SL**(2,**Z**). Students taking both may discover they are two sides of the same coin!*

**Indo-European Linguistics (A)**

**Todd Krause**

Did you know English is related to German? Look at some common words to check: *Vater* ‘father’, *Mutter* ‘mother’, *zwei* ‘two’, *Hand* ‘hand’. But it’s also related to Italian, Greek, Russian, and even Hindi. How does that work? What does it mean for languages to be “related”?

We will study how Historical Linguistics establishes these relationships. We will mine English for clues as to how languages evolve over time. We will compare Old English to other ancient languages looking for similar patterns in both grammar and vocabulary: maybe Latin? Ancient Greek? Gothic? (I thought that’s just a way of dressing…) Sanskrit? Old Church Slavonic? Tocharian? (OK, now you’re just making names up…) Get ready to climb the family tree and learn some new — old — languages!

*Open to students of all backgrounds and ages.*

**Mayan Hieroglyphs (A)**

**Todd Krause**

Completely independent writing systems have only arisen a handful of times across the globe. One such development of writing occurred in Mesoamerica — a region including parts of modern Mexico, Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica — where a rich system of hieroglyphs arose. This system perhaps reached its zenith among the ancient Maya, who adorned their temples and pottery with ornate art that also served to record their language. In this class you will learn how to read and write these hieroglyphs. We will also learn about the unique Mayan calendar, the contexts of hieroglyphic inscriptions, and some basics of how the Classic Mayan language worked. Let’s dive into the ancient jungles of the Americas and read the temple inscriptions!

*Open to students of all backgrounds and ages.*

**Ancient Chinese Language & Writing (A)**

**Todd Krause**

The Chinese writing system, or 漢字 *hànzì*, is a unique creation stemming back to 1200 BCE, if not farther. But how does this writing system work? In this class we will study the origin and evolution of the system, how characters are classified, and even how to write them ourselves! Of course, the writing system makes a bit more sense if you know something about the language they represent. So we’ll also devote some time to learning how the language works: How are words formed? How are sentences structured? How do you speak a language with *tones*… and where do they come from? We’ll provide some answers to these questions and more. And we will learn enough to read some of the ancient sayings of the great philosopher Confucius, or 孔子 *Kǒngzǐ*. Come learn about a language and culture both ancient and modern at the same time!

*No particular linguistic knowledge assumed. But you’ll be helped by a willingness to get out of your comfort zone and try new things!*

**How Language Works (A)**

**Todd Krause**

This class will provide an introduction to the basics of modern linguistic analysis. Have you ever wondered what it means for a word to be a noun? If you think you know what nouns are in English, you might wonder: do all languages have nouns? Or verbs: do verbs work the same way in all languages? In what ways might they differ? In this class we will learn to analyze the fundamental components of a language: the different types of words that a language has. We will also explore how we identify the rules by which the elements fit together. How do components and structures combine to create meaning? The real challenge facing you as a budding linguist: how can you figure all this out for a language you don’t even know yet?!? You might be surprised how some of the thinking you use in mathematics will keep you in good stead when analyzing languages. So come explore one of the great mysteries of the human mind: how it creates language!

*No particular linguistic knowledge assumed. But you’ll want to be comfortable with logical thinking and banging your head against concepts and data that might not immediately seem to make sense. Some familiarity with the basic elements of English grammar — nouns, verbs, conjunctions … participles?… clauses? — will be helpful, though not required.*

**Symmetry and Topology (A)**

**Chaim Goodman-Strauss**

Why settle for anything less! The orbifold theory is the modern and complete approach of planar symmetry, using tools from topology to describe and classify geometric patterns. Conway’s Magic Theorem counts out the possibilities in the sphere, plane and hyperbolic plane: each part of each orbifold symbol has a cost– the Euclidean wall-paper groups cost exactly $2, the same 2 as in the Euler Map Theorem!

With paper scissors and tape, we’ll go through many brand new exercises preparing for a new edition of the Symmetries of Things. The course is suitable for anyone who can handle scissors and is fun for families.

*There are no prerequisites beyond interest in geometrical thinking. This course is family oriented with an invitation extended to parent(s) and younger sibling(s) — accompanied by parent — to sit in and participate. *

**Low Dimensional Topology (A)**

**Chaim Goodman-Strauss**

Our own universe is a topological space — what are the possibilities? What would it be like to live in other spaces? We will explore the topology and geometry of 3-manifolds (the hypersphere!, the three-torus!, knot-complements! the Poincaré dodecahedral space! Solve! Nil!), getting our hands on them directly with models and imagination.

This course is suitable for anyone who enjoys geometrical thinking.

*There are no prerequisites beyond interest in geometrical thinking.*

**The Mathematics of Fairness (A) **

**Wendy K. Tam Cho**

Social science is not often seen as mathematical in nature. Voting, however, involves the “adding up” of votes. Representation is about “dividing” a small set of representatives among a larger group of constituents. Both of these “mathematical exercises” attempt to achieve fairness in some sense. In this course, we will explore the mathematics behind notions of fairness and discover that this math is simultaneously simple, as well as, as complex as any you will encounter.

**Statistical Simulation (I) **

**Wendy K. Tam Cho**

Very few real-world problems have tidy, closed-form solutions. With numerical methods and approximation theorems in hand, you won’t need to let this slow you down. In this class we will learn about simulations and how to write R code to model systems and analyze their statistical properties, which will, in turn, inform best practices for decision-making in real-world situations.

*Prerequisites: Some prior exposure to programming (e.g. ‘if statements’ and ‘for loops’). *

**Interpolation (P)**

**Martin Strauss**

Two points determine a line; three points determine a quadratic. That basic principle is the heart of: the Fast Fourier Transform, to multiply (ax+b)(cx+d) with less than four multiplications of coefficients; error-correcting codes for communicating reliably over noisy

channels; secret sharing, so that any two of Google cloud, dropbox, and iCloud can keep your secret, but no one service learns anything; and the uncertainty principle, that says we can’t know both the time and frequency of any sound.

*Prerequisites (may be waived, depending on interest): linear algebra and modular arithmetic.*

**Introduction to Analytic Number Theory (P)**

**Rolfe Schmidt**

How many primes are there? You may be familiar with Euclid’s proof that there are infinitely many, but can you say how many primes there are less than one trillion? Could you tell me the probability that a random 10-digit number is prime? In this class you’ll learn a set of tools that are key to answering these sorts of questions (and more!) accurately and easily. We’ll focus our attention on the Riemann zeta function, but meet many more characters along the way. This is a deep subject full of hard and unsolved problems. You will not master it in this class, but hopefully you’ll come away convinced that you want to try.

*Prerequisite: Integral Calculus, some number theory will be helpful*

**A number out of this world! (P)**

**Yo’av Rieck**

We all know that some numbers, for example the square root of 2, are not rational. This can be described by saying that these numbers are not solutions to equations of the form ax+b, with a and b integers. A *transcendental number* is a number that is not the root of *any* polynomial with integer coefficients. Transcendental numbers have been discussed for hundreds of years, but their existence was only established by Liouville in 1844.

In this course we will discuss three strategies for showing that transcendental numbers exist: the first is extremely easy but does not allow us to conclude that any *particular* number is transcendental (surprisingly, this easy strategy was discovered last). The second is Liouville’s proof, which allows constructing specific transcendental numbers (for example, the sum of 10^-n!). But these are not numbers that we usually encounter.

The main part of the course will be a proof for Hermite’s theorem: *e* is transcendental. We will then discuss Lindemann’s Theorem, which is more involved: π is transcendental.

*Prerequisites: Calculus and mathematical maturity. *

**Knot Theory (P)**

**Jeremy Van Horn-Morris**

Over the past 150 years, mathematicians developed tools to understand the shape of not only our universe but any possible universe. Surprisingly, a significant part of that story requires understanding the classification of knotted loops in our standard 3-dimensions. We call this study Knot Theory and some of the notions we’ll look at are isotopy, Reidemeister moves, the knot invariants of colorability and the Alexander polynomial, and Seifert surfaces.

*Prerequisites: Mathematical maturity*

**SL(2,Z) (P)**

**Jeremy Van Horn-Morris**

SL(2,Z) is a magical mathematical object that is many different things to many different people and shows up all over mathematics. We’ll spend some time exploring what it is and its connections to linear algebra, group theory, combinatorics, dynamics, number theory, and topology, all by starting with the basics of matrix algebra.

*Prerequisites: Mathematical maturity*