CLASS PROPOSALS—WEEK 5, MATHCAMP 2012
Why is this so long?
These are all the possible Week 5 classes. Since there are too many to all happen
we need you to vote on them.
How do I vote?
Go to http://tinyurl.com/ckdpcgv. We’re using approval voting so you can vote
for as many or as few classes as you want. However: Voting for every class is
equivalent to voting for none, so pick only the ones you want. And since staff will
be sad without students, especially near the end of camp, we’re asking you to ask
yourself the following question to test whether or not you should vote for a class:
“Would I get up and go to this class even at 9am on Friday?” Your votes will be
used not only to decide which classes will run, but also to decide which other classes
they conflict with. We’ll do our best to minimize conflicts. Voting closes on W4
Tuesday at 11:59pm.
What about these teaching projects?
These are classes taught by campers. They begin with infinitely many votes, so
they are guaranteed to run, but you should still vote for them if you want to attend
so that they will be less likely to conflict with other classes you want to take.
Contents
Teaching Projects
Algebraic Logic. (
, Riley Thornton, 2 Day)
Algebraic Logic is the branch of mathematics that studies logics (yeah, that’s the plural of logic) by building
algebras out of them. So why do we care? Algebraic logicians care because they can generalize the algebrization
to process to build semantics for particularly thorny logics. We care because we get throw posets at things
and watch cool stuff happen! This class will cover a few basic constructions and prove things like “ϕ ` ϕ” and
“Modal Logic is secretly Topology.”
Prerequisites: Be comfortable with posets; It will help to have seen Boolean algebra before
Related to (but not required for): Set Theory, Continuum Hypothesis, Model Theory, Computability Theory
Everything Must Have a Limit. (
, Florin Feier, 1 day)
Calculating the limit of a sequence whose general term we know in closed-form is usually straightforward. But
what if we don’t know that form? We can attempt to find it, but this might be very hard or even impossible!
What can we do then?
In this class, we will discuss a theorem that allows us to calculate the limit of a sequence without knowing its
closed-form. Rather than give the details of the proof, we will talk about the main ideas and focus on examples.
Prerequisites: Basic familiarity with limits.
1
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Staged Self-Assembly. ( , Rohil Prasad & Jonathan Tidor, 1 Day)
Say you want to make a dinosaur. The only problem is that the Zome Lounge is closed, and the only thing
you can find to make your dinosaur out of are a bunch of nano-sized squares. That’s okay, you can make a really,
really tiny dinosaur instead. Except that your fingers are a bit larger than the squares; that makes it kinda
hard to arrange them the right way. Fortunately, the squares stick to each other and follow certain rules while
doing so. Maybe if you mix the right ones together they’ll assemble into a dinosaur on their own. Unfortunately
they tend to just form a blob. Making dinosaurs is pretty tricky.
In this class, we’ll introduce the staged self-assembly model which describes exactly how these squares attach
to each other. Then we’ll let you figure out how to make some useful and interesting shapes. By the end of
class,you’ll know exactly how to make a dinosaur and why you’ll be waiting for couple of hours even if you work
as fast as possible (hint: it’s because dinosaurs are complicated).
Prerequisites: No formal prerequisites, but some familiarity with dinosaurs will be helpful.
Visitors & JC’s
, Dan Zaharopol, 2 days)
Computational Complexity Theory. (
Have you ever worked on a problem and found it easy, or hard? How might you classify a type of problem as
being easy or difficult? It turns out to be possible to define different complexity classes of problems that capture
the inherent difficulty of questions you might pose. Some of the difficulties are obvious: Do they take a lot of
time to solve? Do they need a lot of storage space to compute? Others might be less obvious; for example, the
class of problems where verifying an answer is easy but finding an answer is (possibly) hard is one you may
have heard of, called NP.
We will study various complexity classes and the kinds of computational resources that you might allow
(or not). This will develop a rich structure for understanding problem difficulty that is remarkably robust—
it holds up over virtually any kind of computational device (with the possible, but unproven exception of
quantum computers) and under any reasonable restating of the problem in different language. In the end,
you’ll understand how computer scientists discuss complexity and you’ll understand the million-dollar P vs. NP
question.
Prerequisites: None
Isometries of Euclidean Space. ( , Dan Zaharopol, 1 day)
An isometry is a transformation that preserves distances. The following remarkable theorem is both beautiful
and simple: every isometry of Rn can be realized by composing at most n+1 reflections. We’ll prove it, touching
upon some group theory and getting more comfort working in higher dimensions.
Prerequisites: None
From the Trenches: Math Education for the Talented but Underserved. ( , Dan Zaharopol, 1 day)
Over the past three weeks, I’ve been running a summer math program for rising 8th graders with talent in
math who attend schools where 75% or more of the student body receives free lunch. I’ll tell you about their
mathematical preparation (or lack thereof), and our efforts to prepare these students to eventually attend
high-level math circles and summer programs, maybe even Mathcamp.
Prerequisites: None
What’s Blocking Equity? ( , Dan Zaharopol, 1 day)
Have you noticed that most of the students at Mathcamp come from a relatively high socio-economic background? (Parents have college degrees, maybe are academics, reasonably well-off?) Maybe you’ve seen that
most of the students are White or Asian, or that there are a lot more coming from some places (the Bay Area,
Boston, NYC) than from others. Why is this?
We don’t know, but I’ll share some research on the challenges that people are facing and some of my own
experiences working with students from underserved backgrounds. In the end, I hope that you’ll walk away
with a broader understanding of the challenges—and opportunities—that we should all be addressing.
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3
Prerequisites: None
, Don, 1-2 Days)
The Category of Finite Sets. (
The category of finite sets is quite similar to the category of sets; you restrict so that instead of looking at all
sets, you’re only looking at the finite ones. In this class, we’ll show that the category of finite sets has many
of the same nice properties that Waffle showed existed in the Category of Sets, and then define a collection of
categories whose objects are all essentially finite sets, and show that category theory is useful by proving things
about these using category theory. If there’s time, we might even show that the category of finite sets is a topos.
Prerequisites: The Category of Sets
Related to (but not required for): Schemes, Set Theory
Squaring the square. (
, Julian Gilbey, 2 days)
The square is one of those perfect shapes, a thing of beauty. So who would dare cut it up? We will! And not
only will we cut it up, but we’ll aim to chop it into smaller squares, each of a different size.
Go on, I challenge you to do this without coming to the class!
Prerequisites: None
Aztec Diamonds. (
, Julian Gilbey, 2 days)
An Aztec diamond is a diamond shape made of squares, as shown in the left hand diagram for a diamond of
order 4. It can be covered in dominoes as shown in the right hand diagram.
How many ways are there of covering such a shape with dominoes? And what would a random such tiling
look like on average? These seemingly hard questions have been investigated in the past twenty years and have
revealed some beautiful mathematics (as well as pictures).
In this class, we will explore some of the hidden secrets of the Aztecs; there will be a few proofs and sketch
proofs, but the emphasis will be on the known results.
Prerequisites: None
When integration goes wrong. (
, Julian Gilbey, 3–4 days)
We know how to integrate a function such as x2 , but how does one go about defining such an integral
precisely? And in what situations does the fundamental theorem of calculus break down? (Recall that this
theorem
states, roughly speaking, that integration and differentiation are inverses of each other. So, for example,
R 2
d 1 3
x dx = 31 x3 + c and dx
( 3 x + c) = x2 .)
As an example to chew over, consider the following function:
(
x2 sin(1/x2 ) for x > 0
F (x) =
0
when x = 0.
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This function can be differentiated on [0, ∞) to get a function f , but f is far too badly behaved at zero to be
able to integrate it on [0, a] for any a > 0.
How can we rescue the fundamental theorem in cases like these?
We will begin with the formal definition of Riemann integration, and then tweak it to create a new definition
of integration which provides for the most powerful possible version of the fundamental theorem. (For those
who have heard of the Lebesgue integral, the integral we will learn in this class includes the Lebesgue one as a
special case! In spite of this, it is even easier to define and understand.)
Prerequisites: Some analysis, in particular some familiarity with ε–δ arguments. You will get much more out of
this class if you have already seen standard Riemann integration from an analytical perspective.
Rational Trigonometry and Universal Geometry. (
, Julian Gilbey, 2 days)
Know the intersecting chords theorem? (See below if you don’t.) Nice result. At least on the Euclidean plane,
where we measure distances with real numbers. But what would happen if we were to measure “distances” with
integers modulo 7, or the complex numbers? Can we still do geometry? What would the statement of the
theorem even mean? And could it still, in some sense, be true?
And is there a way of doing some trigonometry (triangle measuring questions) in this bizarre setting?
Come and learn how to do trigonometry without a calculator, and then how it can be extended to wild and
wonderful scenarios!
(The intersecting chords theorem: if AB and CD are two chords of a circle which intersect at P, then AP.PB
= CP.PD.)
Prerequisites: Modular arithmetic; complex numbers are useful but not essential
Rubik’s Cube. ( , Lucas, 3 days)
Did you buy a cube and/or want to spend all of TAU playing with a Rubik’s Cube with the excuse of calling
it ”work”? Of course you do! But more seriously, the Rubik’s Cube is a great tool for understanding ideas
in subjects from combinatorics to group theory. This will be a survey of various aspects of the Rubik’s Cube,
including:
• The group structure of a Rubik’s Cube.
• Parity, permutations, and the number of states.
• ”How to solve any twistypuzzle”: commutators and conjugates.
• The number of moves needed to solve a Rubik’s Cube (God’s number, which is 20 moves).
• Practical tips to start speedcubing.
According to class interest, other topics might include how to solve a cube blindfolded, speed cubing history
and lore, conjugacy classes, Sylow subgroups, puzzle scrambling, and computer algorithms for solving the cube.
Prerequisites: None. (Some familiarity with the Rubik’s cube, or with permutations and parity, is helpful but
not necessary.)
Why Math Is The Way It Is. ( , Lucas, 2 days)
Mathematics can be awfully meta – there are a lot of ways to discuss mathematics itself using mathematics.
But sometimes it is nice to take a step out and think about where mathematics through the eyes of philosophy
and history.
When we’re working with numbers and mathematical objects, it might seem obvious that we are working
with “real” concepts. But, to steal a quip: “When was the last time you stubbed your toe on a 7?” Going
further, what ”seven-ness” connects a collection of 7 piffles to the numeral 7? Philosophers have proposed ways
of interpreting the existence (i.e. ontology) of mathematical objects.
This is also tied to the idea of mathematical truth. What role do proofs play in establishing inherent truths,
and does it need to be that way? And if we can prove that the nonexistence of something is impossible, how can
we immediately know that that it exists? Many of the standpoints are associated with important mathematicians
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and mathematical ideas, and to understand how our practice of mathematics has evolved into the present day
it is instructive to look at early 20th century people for important meta-mathematical groundwork.
I plan to run this class as a semi-moderated discussion seminar. I will begin with the questions of the existence
of mathematical objects and the nature to mathematical truth by presenting various historical viewpoints.
From there, we will touch on 20th century foundational mathematics (Hilbert’s foundationalism vs. Brouwer’s
intuitionism), the Quine-Putnam indispensability argument and empiricism, nominalism, Gdel’s incompleteness
theorem and the Continuum hypothesis, infinity, the Church-Turing thesis, and computability.
Prerequisites: A willingness to think and talk about the nature of mathematics.
Related to (but not required for): Everything
Two Games and a Code. (
, Mira, 2 - 4 days)
Game 1: 20 Questions with a Liar. I’m thinking of a number from 1 to 16. You are trying to guess my
number by asking me Yes/No questions about it. The catch is that, for exactly one of your questions, I am
allowed (but not required) to lie. How many questions do you need to ask if you want to be sure of guessing
my number? (It’s pretty easy to see that you can always do it with 9 questions – but can you do better?)
Game 2: The Coolest Hat Game of Them All. This game is played by a team of 7 players. Each person is
randomly assigned a red or blue hat. Everyone can see everyone else’s hats, but not their own. However, the
hat assignments are independent, so seeing everyone else’s colors gives a player no information about her own.
At a signal from the host, all the players simultaneously either guess their own hat color or say ”pass”. The
team wins if there is at least one correct guess and no incorrect guesses (passes are OK). What guessing strategy
should the players adopt to maximize their probability of winning? (It’s pretty easy to see how they can play
to win with probability 1/2 – but can they do better? Try it for 3 players first.)
What do these problems have in common? It turns out that the perfect strategies for both games involve a
beautiful mathematical structure called the Hamming Code, invented by Richard Hamming in 1947 to “correct”
computer errors caused by (literal) bugs. The Hamming Code seems to pop up everywhere, from digital
communication to projective geometry. In this class, we’ll solve both of the above games (and play them!). We’ll
also talk a little more generally about error-correcting codes. Nim – the universal game from combinatorial
game theory – may also make a brief guest appearance.
Prerequisites: None. If you are familiar with the Hamming code, you should figure out how to play these games
on your own (a fun challenge) and skip the class.
Mathematical Counseling for Spousal Pairings. (Mira,
, 3 - 4 days)
You may think that marriage is outside the scope of applied mathematics. That is true. Once you are married,
mathematics is of no possible use. But there is a superficial model for a simpler problem – getting married in
the first place...
– Gilbert Strang, Introduction to Applied Mathematics
Want to know how many people you should date to maximize your chance of eventually finding true love?
Ever wonder who has the better deal in 19th century novels: the men, because they get to propose, or the
women, because they get to choose? Believe it or not, there are actual mathematical answers to these questions!
Sure, the model of marriage they assume is somewhat simplified, but it’s not entirely divorced from reality.
And the math turns out to model other situations too, with practical applications that extend far beyond the
problem of choosing a spouse.
Nor does the mathematics end once you get married, whatever Professor Strang might claim. Here is a
passage from the Mishnah – the 2nd century codex of Jewish law:
“A man has three wives; he dies owing one of them 100 [silver pieces], one of them 200, and one of them
300. If his total estate is 100, they should split it equally. If the estate is 200, then the first wife gets
50 and the other two get 75 each. If the estate is 300, then the first wife gets 50, the second one 100,
and the third one 150. Similarly, any joint investment with three unequal initial contributions should
be divided up in the same way.”
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For 1800 years, this passage has baffled scholars: what could possibly be the logic behind the Mishnah’s
distribution of the estate? Then, in 1985, a pair of mathematical economists produced a beautifully simple
explanation based on ideas from game theory. They showed that for any number of creditors and for any estate
size, there is a unique distribution that satisfies certain criteria – and it turns out to be exactly the distribution
proposed in the Mishnah. See if you can figure out the logic for yourself, or come to the class and find out.
Prerequisites: None. Also, all the days of the class will be independent of each other – come to all or just one.
Information Theory and the Redundancy of English. (
, Mira, 4 days)
NWSFLSH: NGLSH S RDNDNT!! (DN’T TLL YR NGLSH TCHR
SD THT)
The redundancy of English (or any other language) is what allows you to decipher the above sentence. It’s
also what allows you to decipher bad handwriting or to have a conversation in a crowded room. The redundancy
is a kind of error-correcting code: even if you miss part of what was said, you can recover the rest.
How redundant is English? There are two ways to interpret this question:
• How much information is conveyed by a single letter of English text, relative to how much could
theoreticaly be conveyed? (But what is information? How do you measure it?)
• How much can we compress English text? If we encode it using a really clever encoding scheme, can we
reduce the length of the message by a factor of 2? 10? 100? (But how will we ever know if our encoding
is the cleverest possible one?)
Fortunately, the two interpretations are related. In this class, we will first derive a mathematical definition
of information, based on our intuitive notions of what this word should mean. Then we’ll prove the Noiseless
Coding Theorem: the degree to which a piece of text (or any other data stream) can be compressed is governed
by the actual amount of information that it contains. We’ll also talk about Huffman codes: the optimal way
of compressing data if you know enough about its source. (That’s a big “if”, but it’s still a very cool method.)
Finally, we’ll answer our original question – how redundant is English? – in the way that Claude Shannon
originally answered it: by playing a game I call Shannon’s Hangman and using it as a way of communicating
with our imaginary identical clones!
The class is 4 days long, but you can skip some of the days and still come to the others. Here’s how it works:
Day 1
: Introduction and definition of information. Required for the rest of the class, unless you’ve
seen some information theory before.
Days 2, 3
: Noiseless coding and Huffman codes. The mathematical heart of the class, where we’ll
prove the Noiseless Coding Theorem.
Day 4
: Shannon’s Hangman and the redundancy of English. You can come to this class even if you
don’t come on Days 2 and 3 – you just need the material from Day 1.
Prerequisites: None.
Homework: Recommended.
Dominoes on a Chessboard. (
, Mira, 4 days)
Innocent Question: How many ways are there to cover an M × N chessboard with non-overlapping dominoes?
Answer:
M Y
N
Y
(4 cos2
m=1 n=1
mπ 1/4
mπ
+ 4 cos2
)
M +1
N +1
“Wait”, you say, “that’s insane!!! That doesn’t even look like an integer!” You’re right – but it’s true.
Come find out where all those cosines come from. If time allows, we’ll also talk about domino tilings on cylinders
and/or on hexagonal grids (a totally different ball game).
Prerequisites: Eigenvectors and eigenvalues (Linear Algebra from Week 1 or equivalent)
Related to (but not required for): Graph theory – but you don’t need to know any
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SVD and its Amazing Applications. (
7
, Mira, 4 days)
In this class, we’ll focus on a powerful linear algebra technique called singular value decomposition (a.k.a.
SVD). You can think of SVD as a way of finding eigenvectors for matrices that don’t have any. (Actually, you
find the eigenvectors for a different matrix, which is closely related to the original one.)
Applications of SVD are everywhere. After explaining the theory, we’ll look at examples from (some subset
of) image processing, genetics, web search, psychology, and the Mathcamp Week 5 schedule. In the homework
assignments, you’ll get to try some of these applications for yourself: expect to spend at least some of TAU
working with Matlab in the computer lab. (Don’t worry, I’ll teach you the basics of Matlab first.)
Homework: Highly recommended (and fun!)
Prerequisites: Enough linear algebra to understand the statement of the Spectral Theorem: “Let A be an n × n
matrix. A is symmetric iff there exists an orthonormal basis of Rn consisting of eigenvectors of A”. The
Mathcamp linear algebra class from Week 1 should be sufficient, assuming you followed it all the way to the
end and remember it well.
The Semicircle Law. (
, Roxana, 2 days)
Random matrix theory studies the eigenvalues and eigenvectors of large-dimensional matrices whose entries
are sampled according to known probability densities. The semicircle law is the first important result in this
field, established by Eugene Wigner in the 1950s when researching a problem in nuclear physics. It describes
the distribution of the eigenvalues of a symmetric random matrix whose entries are sampled independently from
the same probability distribution. If you tried to compute the eigenvalues of some large matrices in Matlab,
and then drew a histogram of all of them, you would get a picture similar to this one:
In this class, we will go over a combinatorial proof of this result, similar to the one originally given by Wigner.
If there is time, I will explain how the semicircle law for random matrices is essentially an analogue of the central
limit theorem for scalar random variables.
Prerequisites: Linear algebra and real analysis; some familiarity with random variables and/or probability distributions recommended
Alfonso
The Redfield-Polya Theorem. (
, Alfonso, 1 day)
The Redfiled-Polya Theorem is like Burnside Lemma on steroids.
If you have taken a group theory class, or maybe some combinatorics class, you are familiar with the Burnside’s
Lemma (a.k.a. many other things): If G is a group acting on a set X, then the number of orbits of the action
is given by
1
(1)
|Fix(g)|
|G|
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It is a nice result, but not enough. For instance, how many different necklaces can you build with 20 beads out
of very large supplies of red, green, and blue beads? With the help of (??), it will take you less than 5 minutes
to calculate that the answer is 87230157 with just pen and paper. But what if I ask you to tell me how many
such necklaces are there with R red beads, B blue beads, and G green beads, for each value of R, B, and G? If
you think there is no way to avoid using brute force to count in this problem, think again! You can still answer
it in less than 5 minutes, but you will need the full force of Redfileld-Polya. Come receive it!
Prerequisites: You need to understand (??) and its proof.
The Banach-Tarski Paradox. (
, Alfonso, 1 day)
You may have heard it before. We can take a sphere, divide it into five pieces, rearrange them, and get two
spheres of the same size as the original one. Nifty trick, but how does it work? Come learn it!
PS: What is a good anagram of “Banach-Tarski”? “Banach-Tarski-Banach-Tarski.”
Prerequisites: You need to be comfortable with group theory. Specifically, you need to know what the free group
with two generators is. You also need to be comfortable with matrices and rotations.
The Stable Marriage Problem. ( , Alfonso, 1 day)
N single men and N single women want to pair up and get married. These are their names and preferences:
•
•
•
•
•
Alfonso: Marisa, Ruthi, Jamin, Susan, Helin.
David: Helin, Marisa, Susan, Ruthi, Jamin.
Kevin: Helin, Susan, Ruthi, Marisa, Jamin.
Nic: Jamin, Ruthi, Susan, Helin, Marisa.
Paddy: Jamin, Helin, Marisa, Susan, Ruthi.
•
•
•
•
•
Helin: Alfonso, Nic, Paddy, Kevin, David.
Jamin: Kevin, David, Alfonso, Paddy, Nic.
Marisa: Kevin, Nic, Paddy, David, Alfonso.
Ruthi: Kevin, Paddy, Alfonso, David, Nic.
Susan: Kevin, Nic, David, Paddy, Alfonso.
Is it possible to make everybody happy? Obviously not since almost everybody wants to marry Kevin. But
is it possible to at least create a stable situation? For instance, it is a bad idea for Kevin to marry Ruthi and
for Susan to marry Nic, because then Kevin and Susan would prefer each other rather than staying with their
partners, so they will run away together. How can we at least avoid having a run-away couple? Is there more
than one way to do it? What is the best way to do it? What if we move to Canada, where Ruthi and Susan
can forget about Kevin and marry each other?
Prerequisites: None
Unsolvability of the quintic polynomial equation. (
, Alfonso, 4 days)
To solve the equation ax2 + bx + c = 0 you simply use the quadratic formula:
√
−b ± b2 − 4ac
x=
2a
There are similar ways to solve an arbitrary cubic or quartic polynomial equation, but it is impossible to solve the
quintic equation. It is actually worse than that. Not a single one of the solutions to the equation x5 + 3x + 6 = 0
can even be written in terms of rational numbers, the four operations, nested roots, and combinations of those.
Think about this last statement. It is crazy? How could one go about proving it? I will show you how. We
will be using lots of hardcore algebra and invoking the name of Galois on a daily basis.
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Prerequisites: You need to fully understand the statement of the Fundamental Theorem of Galois Theory and
what the Galois correspondance for a field extension is (the two weeks of Mark’s Field Extensions class will do
for this). Of course, this means you are comfortable with groups, rings, fields, and fields extensions. You also
need to be comfortable with polynomials, in particular with basic results about factorization and irreduciblity
of polynomials (or being willing to take things on faith).
Sequence the traitor. (
, Alfonso, 4 days)
You know what a topological space is. You also know how to define convergence of a sequence in an arbitrary
topology. As it turns out, sequences are very sneaky objects. They pretend they are your friend, they tell you
that all your topological problems can be solved with them, but they are lying.
For example consider the following definition. Given a sequence (xn ) in a topological space X, a point a ∈ X
is an accumulation point of the sequence if, for every open set U such that a ∈ U , there are infinitely many
points xn in the sequence such that xn ∈ U . At first sight, it looks like being an accumulation point of the
sequence is the same as being a limit of a subsequence. Don’t be fooled! Those are the treacherous sequences
trying to deceive you! Accumulation point is not the same as limit of a subsequence.
Or consider the following notion. A map f : X → Y between topological spaces is sequentially continuous
when for every sequence (xn ) in X convergent to a point a ∈ X, the image sequence f (xn ) in Y converges to
the point f (a) ∈ Y . Sequences would like you to believe that sequentially continuous is the same as continuous,
but this is just another of their lies.
It turns out sequences are a flawed notion. They should be done away with and replaced with any of the
much more reliable notions of nets, filters, or waffles (yes, that is the technical term).
Prerequisites: Point-set topology. Specifically, you need to be able to follow this abstract.
Covering spaces, monodromy, and square dancing. (
, Alfonso, 1 day)
A covering space of a topological space (for instance, a surface or a curve) is what you get when you “unfold”
it. For instance, you can unfold a circle entirely and get a line, or unfold it partially and get... another circle.
You could also unfold a torus, and get another torus, a cylinder, or a plane. You can unfold almost anything,
like a Klein bottle or GL(n), but you cannot unfold a Hawaiian ring. Interestingly, when we unfold a topological
space, paths that started and ended at the same point end up wandering in space, creating something called
monodromy.
Covering spaces have many applications in daily life, such as Lie groups, quantum field theory, or square
dancing. What does square dancing have to do with covering spaces? Usual square dances 1have 8 dancers,
but there is a 12-dancer variant called “hexagon squares” (sorry, I did not name it). SD callers usually go
through a lot of trouble to explain the rules for hexagon squares, and are often at a loss to figure out whether
a choreography that resolves in regular squares will resolve in hexagon squares. Their lives would be so much
simpler if they simply said “hexagon squares are a triple cover of the quotient of regular square dancing by a
Z/2Z symmetry and a choreography that resolves in regular squares also resolves in hexagon squares if and only
the path of every boy composed with the inverse of the path of his girl has winding number around the center
equal to 0 mod 3.” This talk will be illustrated with shiny animations, courtesy of former JC Ryan Hendrickson.
No topology or dancing knowledge will be asumed.
Prerequisites: None
How to Teach. ( , Alfonso & Ari, 4 Days)
See Ari’s blurb!
Homework: Required
MC2012 ◦ W5 ◦ Classes
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Ari
How to Teach. ( , Alfonso & Ari, 4 Days)
Teaching is not just for professional teachers. Have you ever wanted to present a solution to a class? Show a
friend something cool you’ve learned? Or maybe explain to a non-mathy person what’s so interesting about what
you do? In this class, we will explore the essential skill of communicating mathematics face-to-face with speech
and writing. Great teachers make teaching seem entirely natural; so natural, in fact, that their students often
overlook the skills that go into a successful presentation. In this class, we’ll get into the details of organizing
material, board technique, verbal communication, classroom management, pacing, and more. Sound like a lot
to handle? Relax – even the best teachers rarely get every detail right. Our goal is to give you some awareness
of these issues so that you can start improving and take your first steps on the road of teaching. The format
will be part lecture and part workshop, in which you will prepare and deliver miniature lectures for the class.
Homework: Required
Asilata
, Asilata, 1 or 2 days)
The Hairy Ball Theorem. (
The Hairy Ball Theorem states that if you have a hairy ball, then the hair on it cannot be combed flat
continuously without some of it standing up straight. On the other hand, it is easy to see that a hairy torus is
continuously combable. We will try to sketch a proof of the hairy ball theorem, while simultaneously exploring
what makes hair on surfaces combable. The answers to these questions will lead us towards nice geometric
invariants such as degree and Euler characteristic.
Prerequisites: Familiarity with calculus and preferably some basic topology
Related to (but not required for): Classification of surfaces
The Ring Swap-out Cop-out. (
, Asilata, 2 days)
You may have heard of the Cayley-Hamilton theorem, which states that any square matrix satisfies its own
characteristic polynomial. This theorem is true for matrices with real or complex entries. But in fact, it is also
true for matrices with entries over any commutative ring with identity. Normally, a statement like this might be
proved using some abstract properties of rings. Instead, I will show you a proof that appears to be a cop-out at
first glance, but secretly turns out to be awesome. There will be some rings, much linear algebra, and perhaps
surprisingly, some epsilons and deltas.
Prerequisites: Linear Algebra, familiarity with continuous functions and the definition of a ring.
Related to (but not required for): Linear Algebra
The Hoffman-Singleton Theorem. (
, Asilata, 1 day)
Consider a simple undirected graph with the following properties:
• Every vertex has the same degree (number of edges emanating from it), denoted by d.
• The graph has exactly d2 + 1 vertices.
• The length of the shortest cycle in the graph is 5.
If we set d = 1, then the “line segment” graph satisfies all conditions. For d = 2, the 5-cycle satisfies all the
conditions. You have probably also seen an example for d = 3. Can you think of it? What about d = 4 or
more? The Hoffman-Singleton theorem provides a funny answer to this question: it turns out that the only
values that d can take are 2, 3, 7 or 57. We’ll prove this theorem using little more than some linear algebra.
Prerequisites: Linear Algebra
MC2012 ◦ W5 ◦ Classes
11
Representation Theory of the Symmetric Group. (
, Asilata, 3 days)
The representation theory of finite groups has been extensively studied, and it is well-known that the number of
distinct irreducible representations of a finite group is equal to the number of conjugacy classes of the group. If
you have seen the proof however, you may recall that this equality of numbers is proved by an indirect method.
In general, there is no direct correspondence between the two numbers. But if we focus just on the symmetric
group Sn , we get a very beautiful classification of their representations, as well as a nice correspondence of
irreducible representations to conjugacy classes. There is a wealth of algebra and combinatorics to be studied
in this subject. We will barely skim the surface of this topic, but nevertheless we will be able to work out some
very pretty results.
Prerequisites: Representation Theory
, Asilata, 1 days)
Line Bundles on a Circle. (
A vector bundle on a topological space is a consistent, continuous way of associating a vector space with each
point of the space. For example, a line bundle on a space assigns a line to each point of the space. Vector
bundles are ubiquitous and very useful in geometry. For example, you may have heard of tangent bundles.
In this class, we will try to systematically develop what it means to classify vector bundles on a space. Our
main example will be line bundles on a circle. We will figure out what the tangent bundle of a circle looks like,
and find out how many essentially different types of line bundles can exist on a circle.
Prerequisites: Basic point-set topology. Familiarity with vector spaces helpful but not required
Hannah
, Hannah, 2 Days)
Fibonacci Sums and Differences. (
The Zeckendorf theorem says that every natural number has a unique Zeckendorf representation: a way to write
it as a sum of distinct non-consecutive Fibonacci numbers. Furthermore, there is no shorter way to write a
natural number as a sum of Fibonacci numbers.
Hannah’s awesome Fibonacci theorem says that every integer has a unique far-difference representation: a
way to write it as a sum of ±Fibonacci numbers such that every two of the same sign are at least 4 apart in
index, and every two of different sign are at least 3 apart in index. Furthermore, there is no shorter way to
write an integer as a sum of ±Fibonacci numbers.
We’ll prove the Zeckendorf theorem on the first day, and Hannah’s awesome Fibonacci theorem on the second
day.
Prerequisites: None.
Five Intersecting Tetrahedra. (
, Hannah, 2 Days)
A regular dodecahedron has 20 vertices. Five regular tetrahedra together have 20 vertices. Did you know
you can arrange five intersecting tetrahedra so that their vertices are the vertices of a regular dodecahedron?
On the first day, we’ll build five intersecting tetrahedra out of origami—a confusing and illuminating task. On
the second day, we’ll use our origami model to figure out the group of rotations of a dodecahedron.
Prerequisites: Knowledge of what a group is.
Large Obstacle Number. (
, Hannah, 2 Days)
To get an obstacle representation of a graph G, first draw G in the plane with (possibly intersecting) straight
edges, and then add polygonal obstacles disjoint from the edges such that whenever two vertices of G are not
adjacent, the straight line segment between them intersects an obstacle. The obstacle number of G is the least
possible number of obstacles, over all obstacle representations of G. It is not obvious that there exist graphs
with obstacle number greater than 1.
The Happy Ending Theorem is a theorem in Ramsey theory that says that if we draw enough points in the
plane (no three collinear), then some 6 of them form a convex hexagon—and correspondingly for other values
of 6. We can use this theorem to show that there exist graphs with arbitrarily large obstacle number.
12
MC2012 ◦ W5 ◦ Classes
On the first day, we’ll assume Ramsey’s theorem without proof and use it to prove the Happy Ending
Theorem. On the second day, we’ll construct enormous graphs with large obstacle number.
Prerequisites: Intro to Graph Theory. (In particular, Ramsey Theory and Obstacle Numbers of Graphs are
both helpful but not required.)
Related to (but not required for): Ramsey Theory, Obstacle Numbers of Graphs
Kevin
Skittle Packing. ( , Kevin & Mo, 1 Day)
Ever wondered how to estimate the number of skittles in a jar? Well don’t wonder . . . instead, figure it out
yourself! In this class, we will pack various objects each with the exact same number of skittles. Your job
is to use your cunning estimation skills, together with mathematics and the wonderful wonderful internet to
determine the number of skittles packed in each object . . . and you only have 50 minutes to do it!!
Prerequisites: None
, Kevin, 2-4 days)
Distributive Lattices. (
Some posets are better than others. The best are called distributive lattices, which come equipped with two
operations in addition to the usual partial order. For example, the poset of subsets of {1, 2, . . . , n} is a distributive
lattice whose operations are union and intersection.
The Fundamental Theorem of Distributive Lattices is a gorgeous theorem which gives a bijection between
finite distributive lattices and regular old finite posets. After proving this theorem, we’ll see several examples
of intricate distributive lattices whose underlying posets are strikingly simple.
Prerequisites: None.
Summing Series: Complex Analysis Edition. (
, Kevin, 2-4 days)
Let xk be the kth positive solution of the equation x = tan x. Consider the sum
∞
X
1
.
x2k
k=1
1
Is it transcendental? Maybe with some πs or es or something? No. It’s 10
.
P∞
We’ll do things like that. And we’ll find the value of the Riemann zeta function ζ(s) = n=1
And 4. And all other positive even numbers.
1
ns
at s = 2.
Prerequisites: Complex Variables; if there’s enough interest, I will run an evening presentation of the necessary
theorems, without proof.
Related to (but not required for): Complex Variables, Summing Series
Marisa
Zero-Knowledge Proofs. (
, Marisa & Pesto, 1 or 2 days)
Let’s say that the purpose of a proof is for you (the ‘prover’) to convince me (the ‘verifer’), beyond reasonable
doubt, that a fact is true. In math, we usually accomplish this feat by you showing me a series of logical steps
that demonstrate why your fact is true. In a zero-knowledge proof, your goal is to convince me that a fact is true
without leaking any information about how you know it: at the end of the conversation, I should be completely
sure of the fact, but unable to prove it to anyone else! In this class, we’ll meet zero-knowledge proofs about
philosophy, graph theory, cryptography, and Rubik’s cubes.
Prerequisites: None
MC2012 ◦ W5 ◦ Classes
13
Mark
Elliptic Functions. (
, Mark, 4 days)
If you like periodic functions, such as cos and sin, then you should like elliptic functions even better: They
have two independent (complex) periods, as well as a variety of nice properties that are not that hard to prove.
Despite the name, elliptic functions don’t have much to do with ellipses. (The name is a kind of historical
accident; people were interested in the arc length of an ellipse, which comes up in the study of planetary
motion; this led to so-called elliptic integrals, and elliptic functions were first defined as inverse functions of
those integrals.) Instead, elliptic functions parametrize elliptic (cubic) curves, and they are also related to
modular forms. If time permits, we’ll use some of this material to prove the remarkable identity
σ7 (n) = σ3 (n) + 120
n−1
X
σ3 (k)σ3 (n − k) ,
k=1
where σi (k) stands for the sum of the i-th powers of the divisors of k . (For example, for n = 4 this comes
down to
1 + 27 + 47 = 1 + 23 + 43 + 120[1(1 + 33 ) + (1 + 23 )(1 + 23 ) + (1 + 33 )1] ,
which you can certainly check if you have nothing better to do . . . .)
Prerequisites: Week 2 of Complex Variables
Related to (but not required for): Congruent Numbers and Elliptic Curves
Quadratic reciprocity. (
, Mark, 2 days)
Let p and q be distinct primes. What, if anything, is the relation between the answers to the following two
questions?
Question 1: “Is q a square modulo p ?”
Question 2: “Is p a square modulo q ?”
In this class you’ll find out; the relation is an important and surprising result which took Gauss a year to
prove, and for which he eventually gave six different proofs. If all goes well you’ll get to see one particularly nice
proof, part of which is due to one of Gauss’s best students, Eisenstein. And next time someone asks you whether
101 is a square mod 9973, you’ll be able to answer a lot more quickly, whether or not you use technology.
Prerequisites: A bit of number theory, specifically Fermat’s little theorem.
Multiplicative Functions. (
, Mark, 2 days)
Many number-theoretic functions, including the Euler phi-function (which you don’t have to have seen before
this class) and the number of divisors, share the property that
f (mn) = f (m)f (n) whenever gcd(m, n) = 1 .
There is an interesting operation, related to multiplication of series, on the set of such multiplicative functions,
which gives that set a nice structure. If this sounds like fun and/or if you’d like to be able to find the sum of
the 10th powers of all the divisors of 686, 000, 000, 000 (by hand) in a minute or so, do consider this class.
Prerequisites: No fear of summation notation, and a bit of number theory.
Related to (but not required for): Dirichlet series (but I don’t think we’ve had a class on those)
14
MC2012 ◦ W5 ◦ Classes
Primitive Roots. (
, Mark, 1 day)
Suppose you are working modulo n and you start with some integer a and multiply it by itself repeatedly. For
instance, if n = 17 and a = 2 you get 2, 4, 8, 16, 15, 13, 9, 1 and then you’re back where you started. Note that
on the way we haven’t seen all the nonzero integers mod 17; however, if we had used a = 3 instead we would
have gotten 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6, 1 and cycled through all the nonzero integers mod 17. In
general we can ask when (that is, for what values of n) you can find an a such that every integer mod n that’s
relatively prime to n shows up as a power of a (such an a is called a primitive root mod n). We may not get
much beyond the case that n is prime, but even in that case the analysis is interesting. In particular, we’ll be
able to show that a exists in that case without having any idea of how to find a, other than the flat-footed
method of trying 2, 3, . . . , p − 1 until you find a primitive root.
Prerequisites: Modular arithmetic; knowing a bit more number theory may be helpful
Why All Conic Sections Are The Same, And Beyond. (
, Mark, 2 days)
Except for special cases, conic sections can be classified as ellipses, parabolas, and hyperbolas; however, in
this class we’ll see that given the proper geometric context, these three types are really all “the same”. (We’ll
also see, looking at intersections of conic sections, that there are two points that all circles have in common,
again in an appropriate context). Once we’ve recovered from this, we’ll move on to cubic curves, where not all
types are the same, and on the other hand (again, except for special cases) the points on any single curve have
a remarkable structure (they form a commutative group). Studying intersections of cubic curves (and assuming
an altogether plausible theorem about them) should help us see why this should be true and also lead us back
to conic sections and the following beautiful fact: If A, B, C, D, E, F are points on a conic section, then the
three points you get by intersecting AE and BD , AF and CD , and finally BF and CE are always collinear!
Prerequisites: Having seen conic sections before
The Riemann Zeta Function. (
, Mark, 3 days)
Many highly qualified people believe that the most important open question in pure mathematics is the
Riemann hypothesis, a conjecture about the zeros of the Riemann zeta function; the conjecture had its 150th
birthday properly celebrated in 2009. So what’s the zeta function, and what’s the conjecture? By the end of
this class you should have a pretty good idea. You’ll also have seen a variety of related cool things, such as the
probability that a “random” positive integer is not divisible by a perfect square (beyond 1) and the reason that
−691/2730 is a useful and interesting number.
Prerequisites: Single-variable calculus, including infinite series. Complex variables would help a bit but is not
required.
Rescuing Divergent Series. (
, Mark, 1 day)
Consider the infinite series 1 − 1 + 1 − 1 + 1 − 1 + . . . . What is its sum? Maybe (1 − 1) + (1 − 1) + · · · = 0 ,
maybe 1 − (1 − 1) − (1 − 1) · · · = 1 . At one time mathematicians were quite perplexed by this, and one even
thought the issue had theological significance. Now presumably it’s nonsense to think that the “real” answer
1
is , just because the answers 0 and 1 seem equally good, right? After all, how could the sum of a series of
2
integers be anything other than an integer?
Prerequisites: A bit of experience with the idea of convergence
15
MC2012 ◦ W5 ◦ Classes
The Delta “Function”. (
, Mark, 1 day)
In introductory books on quantum mechanics you can find “definitions” of a “function”, called the delta
function, which has two apparently contradictory properties: Its value is zero for all nonzero x , and yet its
integral over any interval containing 0 is 1 . This “function” was introduced by the theoretical physicist Dirac
and it turns out to be quite useful in physics, but how can we make any mathematical sense of such a creature?
Prerequisites: Integration by parts
A Tour of Hensel’s World: p-Adic Numbers. (
, Mark, 2 days)
Suppose we start with the formula for the sum of a geometric series:
1
1 + x + x2 + x3 + · · · =
1−x
and we substitute 3 for x to get
1
1 + 3 + 9 + 27 + · · · = − .
2
Now imagine a number system in which this actually makes sense. And imagine a geometry in which all triangles
are isosceles. Then decide whether you like what you’re imagining, and whether you’d like this class to happen.
Prerequisites: Modular arithmetic (a bit more number theory might help); a bit of experience with the idea of
convergence
Mike
Shock Waves. (
, Mike, 3-4 days)
What do cars on the road, gas in a pipe, and campers in a large piffle accelerator have in common? They form
shock waves! To describe shock waves, we’ll use the shock wave equation, but just writing down the equation
won’t tell us everything we want to know, because it usually can’t be solved. That is, unless we’re willing to
loosen our standards a bit about what exactly a solution is. In this class we’ll investigate solutions of the shock
wave equation, including some exotic notions of solution (with good reason for existing).
Prerequisites: Multivariable Calc Crash Course
Related to (but not required for): Vibrations, Waves, and Diffusion
Distribution Theory. (
-
, Mike, 3-4 days)
When is a function not a function? Well, maybe when it’s measure, you might say. Ah, but then, when is a
measure not a measure?? When it’s a distribution!
Distributions arise naturally when you start with nice functions and begin looking at different ways in which
sequences may converge. In this class we’ll discuss a number of examples of distributions, also known as
generalized functions. The most familiar will probably be the dirac delta ”function” (and its derivatives!), but
we’ll meet a number of other distributions, such as 1/(x + i0) and limt→∞ sin(tx)
. (Or perhaps the latter is
x
actually something familiar?)
Prerequisites: None
Related to (but not required for): Asymptotics, Complex Variables, Geometric Mapping Theory
Exp. ( , Mike, 1-2 days)
√
At some point in school, you happily began writing down things like 2 2 and eπ − π e , but did they actually
define what it means to take irrational exponents? Maybe you worked really hard to get all that epsilon stuff
down so that you could write down infinite sums or prove uniqueness of solutions to differential equations and
feel confident about what you were doing, but all you really needed was the arithmetic mean-geometric mean
inequality and a nice cup of sup.
MC2012 ◦ W5 ◦ Classes
16
Prerequisites: None
Related to (but not required for): Epsilon the Enemy
Period 3 Implies Chaos. (
, Mike, 1-2 days)
A fixed point of a mapping f is a point x such that f (x) = x. If there aren’t any fixed points, or we just
don’t like the ones we found, we might still look for other fixed points of f ◦ f , which form periodic sequences
under iteration. More generally, we can ask for periodic orbits of any other period, for example points x such
that f (f (f (f (f (f (f (f (f (f (f (f (x)))))))))))) = x (but f (x), f (f (x)), f (f (f (f (x))))f (f (f (f (f (f (x)))))) 6= x).
Amazingly, when f is a continuous function from R to R, the existence of a single period 3 orbit implies the
existence of orbits of all other periods. The rediscovery of (a stronger version of) this result, which was originally
discovered by Sharkovsky, led to the modern usage of the word ’chaos’.
Prerequisites: None
Related to (but not required for): Epsilon the Enemy
Wallpaper. ( , Mike, 4 days)
Painters and architects over the ages were able to identify 17 possible symmetry patterns for a wallpaper type
design (consisting of a regular repeating pattern). Why 17, and what does it have to do with the classification
of surfaces? Come to this class and find out!
Prerequisites: None
Homework: Optional
Related to (but not required for): Musical Orbifolds
Mo
Skittle Packing. ( , Kevin & Mo, 1 Day)
See Kevin’s blurb!
Prerequisites: None
, Mo, 1 Day)
A Chapter of Mo’s Thesis. (
I will give an outline of the theory I developed in one of the chapters of my Ph.D. thesis. It will be epic. Topics
that will come up include algebraic geometry, ring theory, and some optimization theory.
Prerequisites: Lots of math.
Problem Solving: Putnam. (
, Mo, 3 Days)
Many problem solving classes at the high school level restrict themselves to problems based on precalculus
material. With a focus on problems coming from the Putnam Math Competition (a continental undergraduate
math contest), this course will play with problem solving that requires material such as calculus, analysis, linear
algebra, and even some abstract algebra.
Prerequisites: Comfortability with first year university mathematics (calculus, analysis, linear algebra) is preferred, but not required.
MC2012 ◦ W5 ◦ Classes
17
Zoealogy: A Course on Anagrams. ( , Mo, 1 Day)
Some English names for numbers can be anagrammed to get other numbers. For instance, “sixty seven” is an
anagram of “seventy six”, and “one hundred twelve” is an anagram of “two hundred eleven”. But these are
relatively trivial, because the respective numbers are obtained by anagramming (67 to 76; 112 to 211). Are
there non-trivial examples? What if we are allowed to use other languages?
Another question: Given an n letter word, and a permutation π ∈ Sn , is there an English word with letters
w1 w2 · · · wn in that order such that wπ(1) wπ(2) · · · wπ(n) is also an English word (we call this the anagram
afforded by π)? For example, this is true for n = 5 with the permutation π = (12)(345): MANGO maps to
AMONG. In fact, every permutation in S3 has this property:
(Think about why (132) is omitted.) These questions will be investigated in this class, with lots of camper
participation!
Prerequisites: The definition of “zoea” must be looked up before class.
Problem Solving (or Creation!): How to Create a Power Round. (
, Mo, 2 Days)
This class has two goals, each the focus of a day in the class:
Day 1: Giving campers the inside scoop on how Power Rounds (in competitions like ARML) are created.
Day 2: Getting campers to create their own Power Round!
Prerequisites: It will be helpful to have seen a Power Round before.
Nic
Bézout’s Theorem and Intersecting Plane Curves. (
, Nic, 3 days)
Bézout’s Theorem is a classic result in algebraic geometry from the 19th century. It says that if I have two
polynomials in two variables, one of degree d and the other of degree e, then there are exactly de points in the
plane where both polynomials are zero. As I just stated it, this result is very false — unless you interpret the
words “point” and “plane” in the right way! This theorem is a great example of one of a common mathematical
practice: find out what should be true, and then keep changing your definitions until it becomes true.
We’ll start by exploring what needs to change about our definitions in order for Bézout’s Theorem to hold,
and then we’ll explore some exciting geometric consequences. We won’t be able to go into the proof, but the
discussion around the theorem will give you an exciting taste of the kinds of things that algebraic geometers
like to think about.
Prerequisites: None
More Haskell. (
, Nic, 3 days)
If you took “Five Programming Languages in Ten Days,” then you’ve probably surmised that I’m a big fan of
Haskell, and I’m a little sad that we only had two days to spend with it. So if you feel like I do, come join me for
one more little taste. We’ll go deeper into the mathier, more abstract features of Haskell — things like monads,
monoids, and applicative functors — and learn how their design was informed by mathematical objects that
have the same names.
I’m going to want to use the small amount of Haskell that we did in the previous class as a jumping-off point,
so if you want to take this and you weren’t in the class, you should try to get a copy of the notes from me,
Asilata, or someone who took the class.
Prerequisites: Some exposure to Haskell
Related to (but not required for): Five Programming Languages in Ten Days, An Introduction To Malbolge
18
MC2012 ◦ W5 ◦ Classes
Homology. (
, Nic, 4 days)
Homology is a construction in algebraic topology that attaches a bunch of abelian groups to a topological space.
In some sense, the groups measure the extent to which a hollow sphere sitting inside the space can be filled in
to form a solid ball: the more ways there are to find a sphere which can’t be filled in, the bigger the groups
get. In this class, we’ll give a definition of homology that works for some special spaces and explore (but, in
many cases, not prove) some of its basic properties. After we’re able to compute some homology groups of some
specific spaces, we’ll get a bunch of cool topological consequences.
Prerequisites: Enough topology, at least about subsets of Rn , to be okay with the words “open set,” “continuous,” and maybe “compact.” Also some basic group theory, up through the ideas of the kernel of a group
homomorphism and the quotient of a group by a subgroup. The corresponding concepts from linear algebra
will do in a pinch.
Related to (but not required for): Topology Your Friend, The Fundamental Group
Paddy
, Paddy, 1 day)
An Introduction to Malbolge. (
Malbolge, named after the eighth circle of Hell in Dante’s Inferno, is a language specifically designed to be
impossible to write useful programs in. It took two years before anyone discovered how to write ”Hello, World1”
in it: furthermore, this wasn’t even done by humans. (A beam-search algorithm2 was used to generate the
program.)
You may have noticed that Nic and Asilata’s class “5 Programming Languages in 10 Days” somehow forgot
to talk about this language! Let’s fix that.
Prerequisites: Prior experience with some programming language.
NP-Completeness and Wang Tilings. (
, Paddy, 2-3 days)
Take a n × n chessboard, and arbitrarily many copies of the following seven tiles:
1
2
2
2 1 3
3
2
4
4 4
1
3
4 4
2
2
2 3
3
2
4 5
1
1
2 1
2
2
4 5
4
1
3 1
3
1
3 2
2
3
2
4
5 5 5 5 5 4
2
2
1
A tiling of this chessboard with these tiles is a way to place each tile (without rotating them) onto the squares
of a chessboard without overlap, so that whenever two tiles are adjacent along a given edge, they share the same
color along that edge.
Given some set of tiles, for what values of n can you cover a n × n chessboard? In this class, we’re going
to study this problem as a specific instance of a NP-complete problem: i.e. a question that is simultaneously
“hard” to solve (all currently known algorithms take an absurdly long time to run) while being “easy” to check
(given any solution, just check all of the edges where two tiles meet: if they agree, it’s a solution.)
In this class, we’ll study this problem alongside several other examples of NP-complete problems, and hopefully give you an idea of what “P versus NP” is all about.
Prerequisites: Being comfortable with the idea of “algorithm.”
1Or, more accurately, “HEllO WORld”.
2Roughly speaking, this search takes a program, generates a number of possible “successors” to it by adding random little bits
to the end of them, picks a handful that it thinks are likely to work out at the end because they’re at least printing out something,
and then repeats this search on the successors. Basically a miniaturized version of evolution.
19
MC2012 ◦ W5 ◦ Classes
The Unit Distance Graph and the Axiom of Choice. (
, Paddy, 1 day)
Consider the following graph: V = R2 , E = {(x, y) : x, y ∈ R2 , d(x, y) = 1}.
• Fun fact 1: the chromatic number of this graph is completely unknown!
• Fun fact 2: the chromatic number of this graph likely depends on the axiom of choice!
We’ll talk about why this is true in this class.
Prerequisites: Graph theory of some sort.
The Bridges of Königsberg: A LARP. ( , Paddy, 1 day)
Character sheet: You are Euler! Your mission, should you choose to accept it, is the following: starting from
any point in the city of Königsberg, can you find a way to cross every bridge exactly once?
In this roughly 50-minute LARP, we will roam over the scenic quads of UPS, attempting to solve a series of
problems like the above.
Prerequisites: Graph theory of some sort.
The Only Random Graph. ( , Paddy, 1 day)
Claim 1: This graph is pretty.
0
1
2
3
4
5
6
7
8
Claim 2: This graph is all the graphs. And not a potato! Vote for this class to find out why.
Prerequisites: Graph theory of some sort.
Pesto
Premodern cryptography. ( , Pesto, 2 days)
UI UOHH JFHQI SGXEKFNGMT ELZZHIJ HOWI KRI FDIJ KRMK XFL TONRK AODP OD M DIUJEMEIG, AOGJK CX KGOMH MDP IGGFG, KRID CX KISRDOBLIJ KRMK YLJK IVKIGTODMKI KRIT.
Then we’ll look at some of the premodern code-writers’ attempts at strengthened versions, and see why they’re
still breakable.
MC2012 ◦ W5 ◦ Classes
20
Prerequisites: None.
Mini Computational Systems’ Properties. (
, Pesto, 2 days)
We’ll talk about some ways of defining computation too weak for Steve’s class. For instance:
(1) A finite-state automaton reads straight through an input string, keeping track of a fixed finite amount
of stuff (its states), then outputs ”true” or ”false” depending on that finite amount of stuff.
(2) A regular expression is a shorthand for common text formats: for instance, email addresses are usually
of the form
(a|b|...|z|0|...|9)*@(a|...|z)*.(com|org|net),
whose parts’ meanings you can figure out from that it describes email addresses.
Actually, those two are the same. We’ll play with them, prove that they’re the same, and then repeat for a
slightly stronger computational system.
Prerequisites: None.
Related to (but not required for): Many Campers Sort Piles, Computability Theory
, Pesto, 2 days)
Maximum flows. (
A bear appears at the uphill entrance to University. Many Campers Scatter Panickedly, trying to reach the
safety of a the office to sign in. Some run straight through the main lounge, but a traffic jam develops at the
Smith-University door, and only 17 campers per minute can get through; some run out the side door, which 30
campers per minute can get to (and those 30 can easily take another entrance to Smith); some run upstairs to
avoid the main lounge, but still get caught in the crush at the door; in the end, only 47 campers can escape per
minute.
We’ll prove that if only 47 campers can escape per minute, then there’s some set of bottlenecks (like the
Smith-University door and the main lounge side door) that only 47 people can pass through. We’ll do so by
making a graph out of the problem, but we’ll prove that it doesn’t really matter how we make the graph; we’ll
get similar results in many ways.
Prerequisites: Intro to Graph Theory, or the Dragon Marriage Condition, or talk to Pesto.
Related to (but not required for): Nowhere-zero flows
Zero-Knowledge Proofs. (
See Marisa’s blurbs!
Prerequisites: None
, Marisa and Pesto, 3-4 days)
Nowhere-zero flows. (
, Pesto, 2 days)
Planar graphs like to paint their faces: we can color them with lots of colors, making no two of the same color
adjacent. (In fact, we can do so with four colors.) This makes graphs not drawn on a surface like the plane
feel left out: only their vertices and edges can be colored. We’ll cheer them up by defining something just as
good as a coloring of a planar graph’s faces that works for them too: a nowhere-zero flow, a way of pushing
piffles around the graph that uses every edge. We’ll find out what that has to do with colorings, then forget the
colorings and just talk about the flows.
Prerequisites: Intro to Graph Theory, or talk to Pesto.
Related to (but not required for): Four-color Theorem, Maximum Flows
The Hoffman-Singleton theorem. (
, Pesto, 1 day)
There are three known graphs in which
(1) every vertex has the same degree,
(2) the shortest cycle has length 5, and
(3) between every two vertices there’s a path of length at most 2.
In those three graphs, every vertex has degree 2, 3, or 7, respectively. It’s known that if any other such graphs
exist, then every vertex has degree 57. We’ll prove it.
MC2012 ◦ W5 ◦ Classes
21
Prerequisites: Linear algebra: eigenvectors.
Related to (but not required for): Intro to Graph Theory
Ruthi
, Ruthi, about 4 days)
Modular functions and forms. (
The study of modular forms is a key part of modern number theory. Roughly, they are a particular class of
functions on C statisfying certain conditions relating to growth, as well as certain transformations of C. In
this class we will study the action of SL(2, Z) on C, discuss some of the basics of modular forms, and perhaps
introduce a bit of how they relate to some of our favorite number theoretic objects.
Prerequisites: Complex Analysis, some familiarity with matrices
Homework: Optional.
Related to (but not required for): Congruent numbers and elliptic curves, Unique factorization and Fermat’s
Last Theorem
, Ruthi, about 4 days)
p-adics. (
For p a prime, and f a polynomial with integer coefficients, Hensel’s Lemma tells us when a solution to f mod p
generalizes to a unique solution mod pn for any n ≥ 1. This compatible set of solutions gives rise to the notion
of a p-adic number, a number system with strange properties different from those you are familiar with. For
example, the following is a p-adic number:
∞
X
pi
i=0
Prerequisites: None. Metric spaces would be helpful.
Homework: Recommended.
Dynamics of the Rationals. (
, Ruthi, 1 to 4 days)
Let f be some polynomials with rational coefficients. What kind of things can we say about the sequences
q, f (q), f ◦ f (q), f ◦ f ◦ f (q), . . .
for any rational number q? This question falls within the topic of arithemetic dynamics. We will discuss
preperiodic and periodic points and why dynamics over the rationals is so unique.
Prerequisites: None.
Homework: Optional.
Steve
Gödel’s Incompleteness Theorems. (
, Steve, 2 days)
In 1931, Kurt Gödel shocked the mathematical world by proving that there were true sentences of arithmetic
which could not be proven. At the time, however, his results were poorly understood, and even now they
are frequently misused. In this class, we’ll prove (with one brief moment of handwaving) Gödel’s (First)
Incompleteness Theorem, and talk about what it does - and does not - mean.
Prerequisites: None
Related to (but not required for): Model Theory, Set Theory, Computability Theory
Set Theories Which Are Terrible. (
, Steve, 2 to 3 days)
The standard axioms of set theory - called ZF (or ZFC, depending on how you feel about Choice) - are incredibly
useful and interesting, but they are not the only way mathematicians could have chosen to describe the universe
of sets. There are, in all probability, equally beautiful and powerful axiom systems which paint an entirely
different picture. In this class, though, we won’t talk about them. Instead, we’ll look at some of the truly awful
set theories which have been proposed over the years, try to understand the universes they describe, and then
laugh at them.
MC2012 ◦ W5 ◦ Classes
22
Prerequisites: Some set theory
Related to (but not required for): Model Theory, Set Theory
Alan Turing’s Life and Times. ( , Steve, 1 day)
2012 is the 100th anniversary of the birth of Alan Turing! In honor of this, I’ll talk about his role in creating
modern mathematical logic. This class won’t just talk about Turing, and there will be serious mathematics,
but we will be looking at everything with a bird’s-eye view.
Prerequisites: None
Related to (but not required for): Computability Theory
, Steve, 3 days)
Reverse Mathematics. (
Reverse mathematics is the study of what axioms are needed to prove what theorems. More precisely, Reverse
Mathematics asks how much information (in the computability-theoretic sense) is implicit in different mathematical constructions. Suppose we have the magical power to find a nontrivial proper ideal in any commutative
ring with unity which is not a field; then it turns out we can find a finite subcover of any open cover of the unit
interval [0, 1], but we can’t find the range of an arbitrary function of natural numbers. In this class, we’ll cover
the Big Five systems of axioms which most theorems seem to be equivalent to one of, and then we’ll look at
the ever-growing number of theorems which thumb their noses at the Big Five and go off to do crazy, terrible
things in the corner. Although this subject is deeply intertwined with computability theory, this class will not
assume any computability theory.
Prerequisites: None
Related to (but not required for): Model Theory, Computability Theory
Susan
, Susan’s Nose, 1 day)
Wedderburn’s Cute Little Theorem. (
Wedderburn’s Little Theorem states that any finite domain is a field. So! Finite domains with zero divisors?
Nope! Finite noncommutative division rings? Not gonna happen! The proof of this adorable theorem will be
presented by an appropriately adorable entity: Susan’s nose fuction.
Prerequisites: Ring Theory
More Forcing! (
, Susan, 1 day)
Know how to force? Want to do some more of it? Sweet, let’s do it!
Prerequisites: Continuum Hypothesis
Ur Multiplication A Splode. (
, Susan, 2 days)
Okay, so maybe you’ve seen polynomials. They look something like this:
n
X
ai xi
i=0
And maybe you’ve seen formal power series. They look something like this:
∞
X
ai xi
i=0
And maybe–maybe you’ve even seen the formal Laurent series. They’re the ones that look like this:
∞
X
i=−n
ai xi
MC2012 ◦ W5 ◦ Classes
23
But I bet you haven’t played with the things that look like this:
∞
X
ai xi
i=−∞
And with good reason–these things don’t play nice. In this class we’ll see how the multiplication on these
objects a splodes, and see how much we can salvage from the wreck.
Prerequisites: None
, Susan, 2 days)
How To Make Rings That Do Terrible Things. (
Suppose I ask you to give me a ring that has a left zero divisor that is not a right zero divisor? Or a ring that
has a left unit that is not a right unit? Or a ring R so that R2 is equal to R3 ? Or a noncommutative domain
that is not embeddable into a division ring? Well your first instinct, of course, is to try a quotient of ZhXi for
some appropriate collection of coefficients X. For example, in ZhXi/(xy), we can see that x is a left zero divisor
that is not a right zero divisor. But how do we know? How can we be sure that in the process of modding
out by our ideal, we didn’t accidentally set a bunch of elements equal to each other, thus collapsing the entire
ring to zero, and along with it civilization as we know it. Oh noes! In certain cases, we have a result called the
diamond lemma, which allows us find a unique expression for each element, thus guaranteeing that our ring,
and civilization, is safe.
Prerequisites: Ring theory
Hacking Heads Off Hydras. (
, Susan, 1 day)
The Oxford English Dictionary defines a hydra as “a fabulous many-headed snake of the marshes of Lerna,
whose heads grew again as fast as they were cut off: said to have been at length killed by Hercules.”
Mathcamp Lore defines a hydra as “a monster that Susan likes to kill for fun.” Come and find out how you,
too, can kill hydras, using just a couple of graphs and the ordinal numbers.
Prerequisites: None
Related to (but not required for): Graph theory, set theory
Stupid Games On Uncountable Sets. (
, Susan, 2 days)
Let’s play a game. You name a countable ordinal number. And then I name a bigger countable ordinal. We’ll
keep doing this forever. When we’re all done, we’ll see who wins. In this class we’ll be discussing strategies for
winning an in
nite game played on ω1 . In particular, we’ll talk about how to set up the game so that at any point, neither
player has a winning strategy.
Prerequisites: None
Waffle
, Waffle, 1 day)
Transfinite Induction and Calculus. (
Induction is a very powerful proof technique for proving statements about natural numbers. If you took my set
theory class, you saw that transfinite induction is a very powerful proof technique for proving statements about
infinite cardinals. In this class, we’ll use transfinite induction to prove theorems in about the real numbers.
These will be basic theorems you might see in a calculus class, such as the Intermediate Value Theorem, or the
fact that if a function has derivative 0 everywhere then it is constant. If you don’t understand how you could
possibly prove a theorem like this by induction, come and be enlightened!
Prerequisites: Week 1 of Set Theory, or familiarity with transfinite induction, or being gullible enough to believe
that transfinite induction works when I tell you it does. Only minimal calculus knowledge will be needed: you
should be comfortable with the definitions of continuous functions and derivatives.
Homework: Optional
MC2012 ◦ W5 ◦ Classes
24
Related to (but not required for): Epsilon the Enemy
, Waffle, 4 days)
Schemes: Part 2. (
In the last five minutes of the Schemes class in week 3, I defined what a scheme is. In this class, we’ll see more
examples of schemes, talk about maps of schemes, and possibly talk about other cool things like products of
schemes.
Prerequisites: Schemes
Homework: Recommended
, Waffle, 1 day)
Zω . (
Abelian groups are very nice mathematical objects. They pop up all over the place in math, and it turns out
that there’s a very nice classification of finitely generated abelian groups (see the blurb for “The Classification
of Finitely Generated Abelian Groups”!).
What about infinitely generated abelian groups? Can we say nice things about them? In this class, we will
learn that in general, the answer to this question is no. In particular, we’ll study one particular abelian group
that has some surprisingly nasty properties. This group is Zω , a direct product of infinitely many copies of the
integers. In particular, we will prove that this group is not a direct sum of cyclic groups (or any particularly nice
and simple abelian groups), and that despite this group being huge and uncountable, it is surprisingly difficult
to define a homomorphism out of it.
Prerequisites: You should be comfortable with thinking about abelian groups.
, Waffle, 2 days)
The Classification of Finitely Generated Abelian Groups. (
One of the basic questions of group theory is to classify all finite groups up to isomorphism. This question is
universally considered to be impossibly hard, and one of the great achievements of 20th-century mathematics
was the classification of just the finite simple groups (a group is simple if it has no nontrivial normal subgroups;
this roughly means that it can’t be broken down into smaller pieces).
On the other hand, for abelian groups, this question is much nicer and easier to answer, and in fact we can
answer it more generally for groups that are just finitely generated, not necessarily finite. The answer is that
every finite abelian group is isomorphic to an essentially unique direct product of cyclic groups. This result
is extremely useful throughout mathematics, as whenever you have some finitely generated abelian group, you
automatically have a good idea of what it has to look like. In this class, we will prove this classification of
finitely generated abelian groups and possibly also sketch some ways in which it generalizes to other areas of
abstract algebra.
Prerequisites: Group theory
Homework: Optional
The Weierstrass Approximation Theorem. (
, Waffle, 1 day)
In calculus, you learn how to write some functions as Taylor series. What this really means is that you’re writing
a function as a limit of polynomials. However, only very special kinds of functions have Taylor series—there are
infinitely differentiable functions whose Taylor series do not converge to the function.
On the other hand, the Weierstrass Approximation Theorem says that any continuous function can be
approximated by polynomials. More precisely, any continuous real-valued function on a closed bounded interval
is a uniform limit of a sequence of polynomials. We’ll prove this, and talk some about how the proof generalizes
to let us approximate arbitrary continuous functions by other kinds of functions as well.
Prerequisites: Epsilon the Enemy