MathPath Breakout Catalog 2021 Final Version, 7/12/2021

Week 1, Morning

Elementary Graph Theory, Dr. Jane (Jane Butterfield)

Most people think of a “graph” as a visual representation of data - a function, assorted

information, etc. In the study of graph theory, we define a “graph” to be a set of points, called

“vertices,” and a set of lines connecting two of those “vertices,” called “edges.” Graphs of this

sort can be used to diagram, understand, and solve many mathematical problems; some of which

may surprise you! In this breakout, we will be investigating the fundamentals of graph theory,

and discovering problems that can be solved by being diagrammed as graphs. 1 star

The Amazing Algorithmic Race, Ben (Ben Chaplin)

Welcome to the world of algorithms, where speed is everything! Suppose I'm tasked with sorting

my breakout students into alphabetical order, however it's the first day and I don't yet know their

names. Every step I take has a cost—I could go one by one, searching for the student with the

next-in-order name and placing them in a line, but how long would it take? Can I do better? In

this breakout, we'll discuss the theory of algorithms, and work on building algorithms for two

functions: searching and sorting. No programming experience is necessary, but we'll play with

some basic Python code in order to run our algorithms and race them! There are no other

prerequisites for the course, just come ready to think about simple problems in a new way! 2 stars

Exploding Dots, Prof S (Daniel Showalter)

Mathematicians know the value of looking at a problem from several different perspectives. This

course will introduce a perspective known as the “exploding dots machine,” which can be helpful

in approaching various problems in algebra, calculus, and number theory. Consider a row of

adjacent boxes with a nonnegative integer number of dots in each box. In a “3 to 1” exploding

dots machine, for example, dots follow these two rules: Any group of 3 dots in a box can explode

into a single dot in the box to the left, and any single dot can explode into 3 dots when moving to

the right. Using just these simple rules, exploding dots can be used to pose and approach complex

mathematical problems in a simple visual format. This course will begin with an introduction to

exploding dots and move through polynomial manipulations to advanced mathematical questions.

1 star

Mathematical Card Tricks, Prof Su (Francis Su)

In this breakout, we'll learn about the mathematics of card-shuffling, and examine some card

tricks that have mathematical ideas behind them. Mathematical card tricks are a great way to

experience how math helps us understand patterns and use them in clever ways. Plus, there's no

sleight of hand required! Bring a deck of cards with you. 1 star

Neutral Geometry, Dr. T (George Thomas)

The geometry generated by the first four axioms of Euclid is called Neutral geometry. It is called

‘neutral’ because although it allows through a point not on a given line one or more lines parallel

to the given line it neither affirms nor denies that there is exactly one parallel. The fifth axiom

turns out to be equivalent to requiring through a point a unique parallel to a line not containing

the point; with the five axioms we have Euclidean Geometry. When the fifth axiom is replaced by

an axiom allowing through a point two or more lines parallel to a given line not containing the

point, we have Hyperbolic geometry. Therefore, Euclidean Geometry and Hyperbolic Geometry

are specializations of Neutral geometry. Thus all results in Neutral geometry are valid is both

Euclidean and Hyperbolic geometries, but not conversely.

The course enables the student to understand the nature of both Euclidean and Hyperbolic

geometries. Although Euclid’s five postulates are initially used in the course, their modern

modification by Hilbert is addressed. At the completion of the course, the student will also

understand that the Cartesian Plane is only a model of Euclidean Geometry and that even that

model is obtained only with the addition of an additional axiom. What axiom? We will see in the

course. 3 stars

Number Theory I, Dr.V (Sam Vandervelde)

Number theory has delighted young mathematicians throughout the years due to the accessibility

of its ideas and the ingenuity of its techniques. This introductory level breakout presents the

foundations of the subject, starting with divisibility and moving on to the Euclidean Algorithm,

primes, factoring, counting divisors, perfect numbers, and relatively prime integers. We will

conclude with a first look at congruences and modular arithmetic. 1 star

Geometric Machines, Dr. Z (Paul Zeitz)

In this breakout, you will learn how to build, or at least contemplate, several different geometric

"machines" that expand or contract space, turn lines into circles, and more. Some machines can

be built with everyday materials. Others require a robust imagination. We will use physical

models but also lots of pencil and paper and Geogebra. If time permits, we will also add scissors

to our toolkit and explore the mathematics of dissection. 1-2 stars

Week 1, Afternoon

Basic Counting (Combinatorics), Dr. Jane (Jane Butterfield)

For students who know a little about how to count, but want to know more and get better. We

start with the basics: sum rule, product rule, permutations, combinations, the binomial theorem.

Then we learn about “combinatorial arguments”, sometimes called proof by story. Time

permitting, we will look at inclusion-exclusion, and pigeon-hole arguments. This course is a good

prerequisite for various later courses that involve counting. 1 star

Cryptology, Sunil (Sunil Chetty)

Have you ever made up a code to send a secret code to a friend, or solved the cryptogram in the

newspaper? It turns out there’s an entire area of math dedicated to this: cryptology is the study of

making and breaking codes. We’ll use modular arithmetic to cover classical codes, including shift

ciphers, Vigenere ciphers, and substitution ciphers in general. By the end of the week, we’ll also

cover modern cryptosystems, which are used to keep our information secure online.

Prerequisite: Number Theory I. 2 stars

Elementary Logic, Prof D (Matt DeLong)

If you attend MathPath, you cannot be a cow. Bessie is a cow. Most of us would correctly deduce

that Bessie cannot attend MathPath, but how do we know this, and how can we prove it? Logic

serves as the foundation for our reasoning, setting out formally the rules we understand

intuitively. Much as algebra allows us to generalize relationships among numbers by using

variables to represent quantities, we will use symbols to represent propositions, demonstrating the

form or structure of an argument (premises leading to a conclusion). Then we will formally prove

these arguments using rules such as Modus Ponendo Ponens, Modus Tollendo Tollens, and

Reductio Ad Absurdum. We will explore truth tables, semantic trees, symbolic logic proofs, and

potentially predicate logic. Also, there are paradoxes which seem to defy common sense. We

shall see how putting some paradoxical examples into the regimented forms demanded by logic

causes the paradoxical element to disappear. 1-2 stars

Heavenly Mathematics, Glen (Glen Van Brummelen)

How were the ancient astronomers able to find their way around the heavens without anything

even as sophisticated as a telescope? With some clever observations and a little math, it’s

amazing how much you can infer. Following the footsteps of the ancient Greeks, we will

eventually determine the distance from the Earth to the Moon...using only our brains and a meter

stick. Along the way, we will develop the fundamentals of the subject the Greeks invented for this

purpose, now a part of the school mathematics curriculum: trigonometry. Scientific calculators

are encouraged. 2 stars

Elegant Area, Dr.V (Sam Vandervelde)

There exists a remarkably diverse collection of formulas for computing the area of various

regions within the Euclidean plane. We are all familiar with one-half base times height; during

this breakout we will encounter at least ten triangle area formulas, along with a selection of other

formulas that give the areas of quadrilaterals and beyond. We will derive some of the formulas,

see novel applications of others, and create our own original area problems. Students taking this

breakout should be comfortable with algebra and have completed a basic introduction to

geometry. 3 stars

Discretizing Differential Equations, Sarah (Sarah Vesneske)

In this class, we will develop methods to solve Ordinary Differential Equations, which are

equations that relate a function to its rate of change (like y=dy/dx), without using Calculus!

Along the way we'll approximate pi, race hot air balloons, and try to predict the future. (No

coding experience is required, and this class will not cover any, but I will provide optional labs

that go with homeworks.) 2-3 stars

Advanced Problem Solving, Dr. Z (Paul Zeitz)

We will work with a variety of "contest math" topics, with an emphasis on slowness. In other

words, we won't focus on short-time-limit MATHCOUNTS questions, but instead work with

questions that require logical arguments for their resolution. Our week will culminate in a "Math

Wrangle" team contest, a hybrid of a debate contest and a wrestling match (minus the violence).

4 stars

Week 2, Morning

Eulerian Graphs, Dr. Jane (Jane Butterfield)

What is the best route to take through my neighbourhood this Halloween? Can I walk through this

building, locking each door behind me, without getting stuck? How can I search this game's level

without missing any treasure chests? Can I embroider this pattern so that it looks good on both

sides of the fabric? In this breakout session, we will learn how to model each of these questions

with graph theory. Then we will learn -- and prove -- a powerful theorem credited to Euler, as

well as some extensions that we will need in order to answer all of these (and more) questions!

Prerequisite: some previous experience with graph theory, such as from taking the Elementary

Graph Theory Breakout. 1 star

Knot Theory, Prof Cahn (Patricia Cahn)

Knots have fascinated farmers, sailors, artists, and myth-makers since prehistoric times, both for

their practical usefulness and for their aesthetic symbolism. For over 200 years, mathematicians

have intensely studied knots, both for their own intrinsic mathematical interest and for their

potential application to chemistry, physics and biology. Today, Knot Theory is an active field of

mathematical research with many important applications. It is visual, computational and hands-

on, and there are many easily stated open problems. This class will give an introduction to the

fundamental questions of Knot Theory. It will take students from simple activities with strings to

open problems in the field. It will answer such questions as, “what is the difference between a

knot in the mathematical sense and a knot in the everyday sense?,” “how do I tell two knots

apart?,” “how can I tell whether a knot can be untangled?,” and “how many different knots are

there?” 2 stars

Error-Correcting Codes, Dr. Clark (David Clark)

If you've ever heard bad static in a phone call or seen strange blocks appear on your TV, then you

needed an error-correcting code! A code is not a secret -- at least not the kind of code that we'll be

playing with. (For that sort of coding, take the Cryptology course.) Our codes are exactly the

opposite: A way to write a message that makes it easier to understand and protects it from

mistakes. In this breakout, we'll transmit messages to each other and take turns playing the part of

static: the evil corrupter of messages. We'll learn how modular arithmetic, matrices, and even

geometry can help make your message understood. Number theory I recommended as a

prerequisite but not required. 2 stars

Number Theory II, Asia (Asia Matthews)

Suppose a fruit stand owner wants to arrange oranges neatly. If the oranges are arranged in rows

of 5 then there are 2 left over, if arranged in rows of 6 then there is 1 left over, and if arranged in

rows of 7 there are 3 left over. How many oranges could the owner have? Is there a unique

solution? Answering this question requires a deeper understanding of modular arithmetic than we

achieve in NT1. So in NT2 we discuss inverses mod m, the Chinese Remainder Theorem,

Fermat's Little Theorem, Wilson's Theorem, the Euler phi-function and Euler's extension of

Fermat's Little Theorem. These results will allow us to answer other questions as well, such as:

How many ways can 1 be factored? What happens when one repeatedly multiplies a number by

itself? And because the answers to such questions are so interesting, we can all enjoy quick

divisibility tricks. 2 stars

Compositions, Partitions and Fibonacci Numbers, Kathryn (Kathryn Nyman)

How many ways can you write 4 as a sum of positive integers? If order matters, so that we

consider 3+1 and 1+3 to be different, then the answer (8) is an easy application of basic

combinatorics. But if we only care about the (unordered) *set* of addends, the answer is 5. This

simple question leads to a rich, elegant and surprisingly challenging mathematical area on the

intersection of combinatorics and number theory. We will look at integer partitions from several

vantage points and take detours to visit the Fibonacci numbers. Along the way we will encounter

beautiful bijections, clever counting, and tricky tilings. For this breakout, some previous

experience with elementary combinatorics will be helpful. 3 stars

Fair Division, Prof Su (Francis Su)

How do you cut a cake (or other things) among several people fairly? Well, we have to make

sense of the words 'how', 'cut', 'cake', and 'fair' and math can help us do all these things. Modeling

social behavior leads to some interesting mathematics. We'll learn about some procedures for

dividing things in ways that feel fair and can reduce envy, while learning about some neat

theorems that can help us. 2 stars

To Infinity and Beyond, Dr.V (Sam Vandervelde)

Which set is larger: the set of positive integers {1,2,3,4,…} or the set of all integers {…,-2,-

1,0,1,2,…}? Does it make sense to compute the sum 1/1+1/2+1/3+1/4+…? Is it possible to

completely cover the plane with non-overlapping segments? These sorts of questions, drawn from

such distinct branches of mathematics as set theory, analysis, and geometry, all force us to

confront the notion of infinity and the often counterintuitive phenomena that arise once infinite

sets or sums are considered. The ideas and mathematical tools that we will need for this breakout

should be accessible to any student with a basic background in algebra; our goal will be to enjoy

and explore various topics in which infinity crops up and makes life interesting. 2 stars

Week 2, Afternoon

Graph Colouring, Dr. Jane (Jane Butterfield)

Graphs can be coloured by assigning colours to the edge set, or to the vertex set, or even

sometimes to both. What makes a colouring proper, and how many colours do you need? In this

breakout, we will explore some of those questions. We will focus particularly on a class of graphs

called planar graphs, and will analyze several attempts by mathematicians through the ages to

prove the famous Four Colour Theorem. Pre-requisites: some previous experience with graph

theory, such as from taking the Elementary Graph Theory Breakout. Some of the proofs we will

explore will use induction, so it would help to be familiar with induction. 3 stars.

Induction, Prof B (Owen Byer)

If you line up infinitely many dominoes on their ends, with each one close enough to the previous

one, and knock over the first, then all infinitely many fall down. This is the essence of

mathematical induction, the main proof technique when you have infinitely many statements to

prove indexed by the integers, such as “1-ring Towers of Hanoi can be won. 2-ring Towers of

Hanoi can be won…. 487-ring Towers of Hanoi can be won….”

Every budding mathematician needs to know mathematical induction, and it's a great proof

technique to learn first, because it has a standard template (unlike most proof techniques) and yet

leaves room for an infinite amount of variety and ingenuity. Thus this course is one of MathPath's

foundation courses.

But don’t take my word for it. Here is what a MathPath student wrote on an AoPS MathPath

forum:

I took induction last year, and I knew induction before the class. But it was very well

taught, and I learned how to write proofs by induction, which was very valuable on the

USAJMO. I'd recommend this class to anyone who wants to learn about induction,

whether you know it at the beginning or not.

The point: Even if you have done induction before, you don’t really know induction, because it

can be used in so many ways in so many parts of mathematics. Every mathematician will

probably do 1000 inductions in his/her life. Get 50 under your belt in this course. 2 stars

Guessing Games, Dr. Clark (David Clark)

I'm thinking of a secret whole number from 1 to 10. How many yes-or-no questions do you need

to ask in order to figure it out? What if my number is 1 to 1,000,000? How about 1 to n -- and

how do you know that you've asked the fewest questions possible? In this breakout, we will start

with those problems and then go much farther. For example: What if I'm allowed to lie -- but only

once? Twice? What if you have to ask all of your questions before I give any answers?

Prerequisites include comfort with simple proofs and Basic Counting (Combinatorics). We will

be thinking and counting in some unusual ways, so be ready to struggle! Homework will

regularly involve playing guessing games and implementing a strategy we've learned in class as

you do so. Come prepared to play! 3 stars

Complex numbers, Asia (Asia Matthews)

Complex numbers are just (linear) combinations of real and imaginary numbers! First, I’ll tell

you a story about how imaginary numbers arose naturally from studying geometry… and then

algebra… and then functions, and how they were completely disregarded because of how

unnatural they are. Then, we will see how many years of playing around with polynomials and

looking for solutions led to an interesting realization: if we represent these imaginary things by

some symbol, we can keep doing algebra and everything works out nicely. A few people decided

to embrace the nonsense and they invented some imaginary geometry, and thus were born

complex numbers! It turns out that really ugly looking algebraic representations of real systems

(e.g. electrical circuits) are easily solvable by passing into the imaginary realm, doing calculations

there, and then using only the real results. In this course we will play around with complex

numbers. We will do some algebra with complex numbers, and see how they are related to

vectors, to circles and triangles, to the number e, and to the sine and cosine functions. Some

experience with polynomials will be helpful, but not necessary. It may sound really difficult, but

it’s just complex. ☺ 1-2 stars

Spherical Trigonometry, Glen (Glen Van Brummelen)

To find your way through the heavens, along the earth, or across the oceans, you need

mathematics. But the math you learn in school mostly takes place on a flat surface --- not the

sphere of the heavens, or the earth. To navigate properly, we develop a completely new

trigonometry that allows us to find our way around a sphere. We shall discover surprising

symmetries and a rich world of theorems, many of which are beautiful and unexpected extensions

of some of the most familiar geometric theorems we have learned in school. Scientific calculators

are encouraged. 3 stars

Taming the Torus, Dr.V (Sam Vandervelde)

Torus is to bagel as sphere is to cantaloupe. The torus presents a splendid variety of mathematical

aspects for our examination. We will discover the geometric formulas associated with a torus,

catalog the topological properties of a torus, investigate graph embeddings on a torus, even

explore the possibilities for art and puzzles with a torus. 1 star

Intermediate Contest Problem Solving, April (April Verser)

In this course, we will be discussing strategies and problem-solving techniques for the AMC 10 &

12. We will focus on problem-solving, while also discussing tricks and strategies for approaching

the AMC contests. Topics that will be covered include: AMC strategy, number theory, algebra

(including polynomials and functions), geometry, trigonometry, counting and probability, etc. We

will also solve challenge problems and work through previous contest problems for practice. 2-3

stars

Week 3, Morning

Hyperbolic Geometry, Dr. Kent (Deborah Kent)

This course explores connections between mathematics and the nature of truth. We'll start with

foundational geometrical results that can be developed from a list of four basic assumptions.

Adding a familiar definition of parallel lines leads to comfortable facts about Euclidean triangles.

But changing what it means for lines to be parallel results in the wildly different reality of non-

Euclidean geometry. We'll explore some of these mind-bending results using several models for

hyperbolic geometry. Pre-requisites: Trigonometry. 3 stars

Origami, Asia (Asia Matthews)

The mathematical worlds that we choose to create and explore sometimes lead to surprising

results. For example, if our world is made up of only circles and lines, the Greeks showed that

while we can draw and bisect any angle we want (is this even true?), it is NOT possible to trisect

an arbitrary angle. But strangely, if we begin with a square piece of paper, it IS possible to

construct a trisection of an arbitrary angle. What is going on here? What do we have with the

square piece of paper that we do not have with lines and circles? In this breakout session, we will

play with paper folding to investigate this problem and other geometric results, we will fold

checkerboards, discuss the fold-and-cut problem (proved less than 20 years ago) and we will

discuss the intersection of mathematics and art. Come ready to apply yourself to the logical

deductive process. 2 stars

Catalan Numbers, Professor Pudwell (Lara Pudwell)

How many ways can you arrange 3 pairs of parentheses legally? The answer (5) can be found by

careful listing: ((())), ()(()), (())(), (()()), ()()(). How many ways can you divide a pentagon into

triangles by connecting pairs of non-neighboring vertices with line segments? The answer is 5

again! The fact that we got the same answer here is not an accident. The Catalan numbers are

the answer to "how many ways can you arrange n pairs of parentheses legally?", "how many

ways can you divide a regular (n+2)-sided polygon into triangles?", and literally hundreds of

other interesting counting problems. We'll look at the Catalan numbers from several points of

view. We'll learn multiple ways to compute the nth Catalan number and see a variety of

surprisingly different places where they appear in mathematics. For this breakout, some previous

experience with basic combinatorics will be helpful.3 stars

The Shape of Space, Prof Rogness (Jon Rogness)

This breakout will explore ideas in the area of math known as topology, where a donut (torus) and

coffee cup are equivalent objects. It will provide a nice followup for any students who have done

the previous breakouts this year which covered the torus, although those courses are not

prerequisites for this one. We'll start by learning a few basic rules in topology. Then we'll look at

the torus, either as a review or brief introduction depending on whether students were in those

previous breakouts. Then we'll look at how to construct other two dimensional surfaces, like the

Mobius strip and Klein Bottle. As it turns out, in one of the triumphs of mathematics, we can

describe how to build all of the so-called "closed surfaces" by sewing together spheres, Moebius

strips and donuts. After we deal with surfaces, we'll have the necessary skills to move on to three

dimensions. That means we can think about different possible shapes for the universe -- i.e. the

Shape of Space -- and explain why the picture below with the dodecahedra might be a model of

how our universe is put together. There won't be much in the way of computations or algebra in

this breakout, but you should like thinking about three dimensional shapes. (For example, if you

like looking at nets and figuring out what the resulting shape is, we'll be doing more complicated

versions of that process.) 2 stars

Probability through Games, Prof S (Daniel Showalter)

In this course, we will explore some concepts of probability underlying two rather simple games,

one involving dice and the other involving playing cards. For the dice game, you and your

opponent each have k tokens that you must place on the integers 2 to 12. You then take turns

rolling a pair of dice. If the sum of the dice matches a number that you have at least one token on,

you remove one token from that stack. The winner is the first player to remove all their tokens.

We will use probability to figure out the optimal placement of k tokens, assuming that you have

no information about your opponent’s placement. We will then tackle more difficult problems

such as how the optimal strategy might vary based on information about your opponent’s

placement or different types of costs and rewards for playing the game.

Studying the probability behind these two games more in depth will (hopefully) help prompt

similar types of questions and explorations related to your favorite games and more general

situations involving probability. A basic understanding of probability will be helpful, but not

necessary. 2 stars

Mental Math, Prof Su (Francis Su)

Mental math is the ability to do some kinds of calculations in your head. One reason to learn how

to do mental math is to impress your friends, but that's not the only reason. Another is to avoid

relying on a calculator. But perhaps the most important one is to see how ideas from algebra can

help us understand why these mental tricks work. 3 stars

Frobenius numbers and chicken nuggets, Cornelia (Cornelia Van Cott)

You may think that a course about chicken nuggets doesn't sound all that enticing, especially if

you're vegetarian or vegan. But think again! This topic has attracted interest from mathematicians

of all food preferences since the 1800's, and mathematicians are still working on the problem

today. Here is how the problem begins: McDonald’s sells chicken nuggets in packs of 6, 9, and

20. What is the largest number of chicken nuggets that one cannot purchase? The generalizations

of this question lead to a series of elegant results and open questions. Prerequisites: Number

Theory I recommended but not required. 2-3 stars

Week 3, Afternoon

Wallpaper Patterns and Life on the Klein Bottle, Prof Cahn (Patricia Cahn)

What is it like to live on a Donut? A Mobius Strip? A Klein Bottle? We'll think about this using

wallpaper patterns—repeating patterns in the plane with nice symmetries. We'll learn how

mathematicians classify these patterns, and practice telling two wallpaper patterns apart with a

quick glance. Then we'll fold the patterns up to produce lots of interesting surfaces, called

orbifolds, and study life in these spaces. There are no specific prerequisites for this class. 3 stars

Visual Group Theory, Asia (Asia Matthews)

A GROUP describes symmetries in objects. The Higgs Boson was predicted to exist before it

was observed because the known particles interacted as a group, except that a part of the group

seemed to be missing. This is very exciting that physicists can use this information to answer

questions in the natural world. But from the perspective of pure mathematics, who cares about

the applications? The internal structures themselves are so beautiful that we are interested in

them in their own right. Look forward to learning visually pleasing representations that help us to

classify different finite groups. 2 stars

Combinatorial Games, Kathryn (Kathryn Nyman)

Do you like to play games? In this breakout, you’ll play a variety of simple 2-player games and

we’ll use mathematical tools to think carefully about whether there are winning strategies.

(Sometimes, we won’t even have to play a game to know who can win!) As we play together

with these ideas, you’ll develop a deeper understanding of some mathematics of combinatorial

games and strategic thinking. Relevant mathematical topics include introductory elements of

combinatorics, number theory, and graph theory. 2 stars

Proof by Story, Professor Pudwell (Lara Pudwell)

Armed with a bit of practice and a proper understanding of what certain well-known numbers

mean, it becomes possible to prove intimidating identities involving these numbers simply by

telling the right story! For instance, in this breakout we will discover that Fibonacci numbers have

a combinatorial meaning: they count the number of ways to accomplish a certain task. This view

of a Fibonacci number, which goes well beyond the basic algorithm for obtaining a Fibonacci

number as the sum of the two previous, enables us to establish a surprisingly large number of

clever relationships among them. 2-3 stars

Cryptology, Prof Rogness (Jon Rogness)

See description under Week 1 afternoon.

It Slices, It Dices, Cornelia (Cornelia Van Cott)

We'll be slicing, dicing, and dissecting all kinds of things in this class, beginning with familiar

things (like pizza!) and then moving on to other objects. With each new challenge, we will try to

do our dissection optimally, which will make the challenge difficult but fun. Let's give an

example to get you started. What is the maximum number of pieces that you can cut a pizza into

with n straight line cuts? We call such numbers pizza numbers. As we'll find out, pizza numbers

are related to tons of other mathematical objects. This will be just the beginning of our adventure,

as we will discuss new slicing and dicing challenges every day. 2 stars

Graphs on Donuts and Donutholes (and extra-holey donuts), April (April Verser)

In this course, we will look specifically at the properties of planar graphs (i.e. graphs that can be

drawn in the plane with no edge crossings). The first 2-3 days we will focus on planar graphs in

2-dimensions or on a sphere, including Euler's Formula and some interesting related results. The

last 1-2 days we will shift and look at embeddings on different topological surfaces, such as on

tori, and their properties. Prerequisite: Some experience with Graph Theory, such as taking the

Elementary Graph Theory breakout or having done some previous Graph Theory in a different

context. 3 stars.

Week 4, Morning

The Binomial and Poisson Distributions, Prof B (Owen Byer)

The Poisson Distribution is used to find the probabilities of the number of occurrences of some

event in a fixed interval of space or time. Typical questions include the following.

1. What is the probability that there are two typographical errors on a random page of a

given novel?

2. What is the probability that three different calls are made to an Emergency Response

Unit in a given hour?

We will derive the formula for the Poisson probability distribution from beginning principles,

discovering the binomial distribution and the number e along the way. We will examine many

applications to real world phenomena. Students taking this course should have experience with

basic probability and familiarity with combinations and permutations will be very helpful. 2-3

stars

Special Relativity: The Mathematics of Paradox, Silas (Silas Johnson)

Einstein's Special Theory of Relativity, at heart, is not about physics so much as the geometry of

the 4-dimensional space-time we live in. Motivated by the paradoxes that plague older ways of

thinking about physics, we'll discover the equations that define this geometry. We'll finish by

using these all-important equations to resolve the paradoxes we explored earlier in the week. If

we have extra time, we might take a look at why faster-than-light travel is impossible. 3 stars

Introduction to Programming in Python, Hannah (Hannah Leaman)

Students will explore programming logic including declaring and manipulating variables, if-

statements, loops, and basic functions using Python. They will complete a series of projects and

challenges that encourage creativity and problem-solving. Students will emerge from this course

with the ability to code simple projects in Python on their own. 1 star

Single Variable Optimization, Prof. M (Lidia Mrad)

Many of the choices we make are driven by our desire to make the "best" decision: How far from

a painting or movie screen would you want to be for the best possible view? What is the most

economical shape of a can? These and similar questions can be answered using optimization. In

this breakout, we will explore some interesting and practical techniques for optimizing functions

of one variable, using only function evaluations. These include Search by Golden Section,

Fibonacci Search, and the Secant Method. To learn these methods, we will rely on graphical

interpretations of functions and computer-assisted calculations. 2 stars

Generating Functions and Sequences, Professor Pudwell (Lara Pudwell)

A sequence is a list of numbers, and the sequences that we are interested in are those with some

pattern. For example: consider the sequence 1, 2, 3, 4, 5, 6,... There are many ways to describe

this sequence. One (implicit) description is "every number is the previous number plus one".

One (explicit) description is "the 100th term is 100". Can you find an implicit or explicit formula

for the sequence 1, 5, 14, 30, 55, 91? For some sequences, this question is easy, but usually it's

difficult. Sometimes finding an implicit description is easier than finding an explicit one

(consider, for example, the Fibonacci sequence). While there aren't techniques that always work,

in this course, we will learn a technique that finds an implicit formula for some very nice cases of

sequences. This technique involves infinite polynomials (known as generating functions). We'll

do some analysis (maybe some long division, perhaps partial fractions) and learn a lot more about

these implicit formulas. This course is heavy on algebra, so come prepared to play around with

abstract symbols. 3-4 stars

Integer Hokey Pokey, Cornelia (Cornelia Van Cott)

Let's look for numbers that can "turn themselves around” in interesting ways. For instance,

multiply 1089 by 9, and you get 9801. Notice that the digits of the answer are in reverse order

from the number we started with. Let's give this behavior a name. We'll say 1089 is a 9-flip.

Some families of integers require tough mathematical machinery to understand them deeply, but

numbers like 1089 that flip around are not so high maintenance. We can learn a lot about them

with basic tools from number theory and a good dose of curiosity. 2 stars

The Fabulous Fibonacci Array, Dr.V (Sam Vandervelde)

It may not come as a surprise to learn that there are many Fibonacci-like sequences, generated in

the same manner as the famous Fibonacci sequence, but starting with different initial values.

What is less familiar, and much more remarkable, is the fact that it is possible to assemble all

these Fibonacci sequences into a single array with so many nice properties and cool patterns that

it will blow your mind. In this breakout we will lay the groundwork for discovering this array by

learning about Beatty sequences and practicing with the golden ratio, then find and prove as many

of those properties and patterns as time permits. 2-3 stars

Week 4, Afternoon

Math and Music, Silas (Silas Johnson)

Why do so many people who love math also love music? Perhaps not surprisingly, the two are

connected on a deep level in many different ways, and you can "hear" each in the other if you

know what to listen for. In this breakout, we'll learn a little bit of math, a little bit of music, and a

lot about how they relate to each other. Mathematical topics may include number theory, abstract

algebra, trigonometry, combinatorics, and more. On the musical side, we'll try to answer

questions like: What is a note, and when are two notes the same? What makes certain

combinations of notes sound good together? Why is the 12-tone scale nearly universal in

Western music, and what problems is it designed to solve? 1 star

Game Theory, Dr. Kent (Deborah Kent)

This class will start with a game of Hex and end with a proof of Nash's Equilibrium Theorem, one

of the triumphs of classical game theory. In between, we'll explore a nice blend of of continuous

and combinatorial mathematics, including the elegant and useful Sperner's Lemma. Class will

also incorporate a dash of topology with some definitions and an introduction to Brouwer's Fixed

Point Theorem. We'll be able to draw and visualize deep mathematical ideas in a 2D-context and

discuss generalizations to higher dimensions. If there's time, we can even put it all together to

determine a fair way to divide a cake! 4 Stars

Applications of Graph Theory, Maciej (Maciej Piwowarczyk)

This breakout aims to explore some of the many applications of Graph Theory using the many

lenses mathematicians have studied the field through. We will explore classic problems like the

Travelling Salesman problem, more contemporary problems like the usefulness of random graphs

and walks in real world situations, and even dip our toes into how we can encode information

about graphs using some algebra and matrices. Pre-requisites: Students should be familiar with

elementary graph theory and basic matrix operations. 2 stars

Counting Permutations, Professor Pudwell (Lara Pudwell)

A permutation is a list of the numbers 1, 2,..., n where the order matters. There are 6

permutations of 1, 2, 3 (can you list them?), and in general there are n!=n*(n-1)*(n-2)....*2*1

permutations of 1, 2,...,n. That's the warmup problem. In this class we'll look at helpful ways to

visualize permutations (other than writing them as lists) and consider questions like "how many

permutations of 1, 2, ..., n have k descents?" and "what does it mean for a larger permutation to

have a smaller permutation inside of it?". Familiarity with basic combinatorics (permutations,

combinations, and recurrences) will be helpful. 4 stars

Rigidity Theory, Maddy (Maddy Ritter)

This breakout will use interactive materials, Cinderella software, linear algebra, and graph theory

to look at rigid frameworks in 2-space. We will explore motions, Henneberg moves, and how to

express graphs in matrix form to further our understanding of them. Prerequisites: Students

should be familiar with matrices and basic graph theory. 3 stars

Metrics on the plane, Cornelia (Cornelia Van Cott)

How far is point A from point B? We usually define the distance to be the length of the straight

line connecting the points. This is the Euclidean metric. But, who says it must be done this way?

We will investigate a variety of other options: the taxicab metric, the elevator metric, the post

office metric, and more. In the process of our investigation, we will discover new geometries on

the plane and understand the familiar Euclidean geometry even better. 2-3 stars

Inversion, Dr. Z (Paul Zeitz)

What does it mean to "reflect about a circle?" This innocent question has a very definite answer,

and takes us into a wonderland of new insights about the unity of circles and lines. We will use

pencil and paper as well as Geogebra, and consider easy, hard, classical, and olympiad-style

problems involving inversion. Note that the geometric machines breakout of week 1 has a teaser

machine which we will contemplate in this breakout (but you don't have to take the geometric

machines breakout to take this one). 2-3 stars

–end–