REUNION CLASSES — MATHCAMP 2010
Contents
9:10 AM
10:10 AM
11:10 AM
1:30 PM
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9:10 AM
Stuff that You’d Think Would Be Impossible but that Cryptographers Can Do Anyway. ( , Matthew Wright)
Cryptography lets us do a lot of surprising things, such as flipping a coin securely over the
telephone, proving to your friends that you’ve found Waldo without showing them where Waldo
is, communicating anonymously out in the open, and playing mental poker without being able to
cheat (if you have superhuman mental computation abilities, at least!). We’ll look at how to do all
of these things and more, without going into too much in the way of formal details.
Prerequisites: None
The Fringes of Chaos. (
, Zandra Vinegar)
Math through a kaleidoscope: http://www.fractal-recursions.com/. Beautiful, no?
This class will dive headfirst into the key concepts of Fractals including Symmetry, Expressible
Infinity, and Chaos. Specifically, we will take an in depth look at the Sierpinski Triangle (covering
the difference between fractal dimension and topological dimension), the Lorenz Water Wheel (illustrating the ideas of the Butterfly Effect and Strange Attractors), and the well-known Mandelbrot
Set. If you want to see mathematics from a completely alien perspective, this class is for you.
Prerequisites: None
Mathematical Knowledge for Teaching. ( , Dan Zaharopol)
Did you ever have a teacher where you felt that they didn’t know as much mathematics as they
should have? What is the mathematics that a good teacher needs to know — how do we know
what that is, and how do we teach it?
There’s a lot of very good research on this subject, done by both math educators and mathematicians. We’ll investigate the surprisingly deep knowledge needed for good teaching and see
what kind of training prospective teachers do (or don’t) get. We’re all very strong mathematically;
this is a chance to see how you can make a difference and what’s going on in education that affects
us all.
For the last ten or so minutes, we’ll also discuss what resources are needed and appropriate
specifically for talented students who are prepared to really excel in mathematics.
Prerequisites: None
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10:10 AM
, Jacob Steinhardt)
Robotics and Control Theory. (
There are many interesting math problems to be solved in the field of robotics. A robot can be
thought of as a system of differential equations, where the instantaneous derivatives can also be
affected by inputs from a controller. In this class, I’ll formulate the problem of controlling a robot
in a mathematical framework, go over some basic tools for controlling linear systems, and explain
how you can use a technique called “sum of squares optimization” in order to reduce a non-linear
problem to a collection of linear problems. Videos of robots doing cool stuff will be provided.
Prerequisites: You should know how to solve systems of linear differential equations. Failing that,
you should at least be comfortable with me writing down systems of differential equations on the
board (and be willing to take certain things on faith).
Fractal Dimensions. ( , Alfonso Gracia-Saz)
A line has dimension 1, a plane has dimension 2, and the space we live in has dimension 3. Can
you think of something of dimension 1.5? What does it mean to have dimension 1.5? Actually,
what does it even mean to have dimension 2?
In this lecture, I will give you one possible definition of dimension and we will compute the
dimension of a few objects, including some with non-integer dimensions.
Prerequisites: You need to understand logarithms and limits.
,
Teaching Computers Math: How To Symbolically Sum Mutual Recurrences. (
Berkeley Churchill)
P
It’s a well known property of the Fibonacci numbers that ni=0 Fi = Fn+2 − 1. Of course other
recurrence relations satisfy similar identities, albeit not as elegant. This talk addresses how a
computer algebra system might be able to discover identities for the sum of a symbol in a linear
recurrence on their own, just from the definition of the recurrence. This problem was mostly solved
for a single recurrence in a 2008 paper by Ravenscroft and Lamagna. I will describe the solution,
and then discuss my current research: how to solve this problem with mutual recurrences. A mutual
recurrence is just a “system” of recurrence relations. I will present an algorithm, why it is believed
to work, and steps being taken to prove it.
Prerequisites: Comfort with basic notions of vector space, basis, linear independence, linear transformations; ability to multiply matrices and put a matrix into row-reduced echelon form.
11:10 AM
, Adam Hesterberg)
Planar Graphs. (
Wagner’s Theorem states that a graph is planar unlesss it has K5 or K3,3 as a minor. Kuratowski’s
Theorem states that a graph is planar unlesss it has K5 or K3,3 as a subdivision of a subgraph.
We’ll define minors and subdivisions, prove both of them, and talk about why Wagner’s version is
the “right” one.
Prerequisites: Understand the terms “planar”, K5 , K3,3 , “connected,” and “subgraph”, and know
why K5 and K3,3 are not planar.
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Foundations of Probability Theory. (
, Jimmy Koppel)
What does “probability” mean? You are probably familiar with the “frequentist” definition of
probability as the ratio of successes to trials in an idealized experiment. Dressed in the rigor of
measure theory and presented as Venn diagrams, this gives an easily-graspable definition which is
unfortunately limited by its inability to ask questions such as “What is the probability the sun will
rise tomorrow,” “What is the probability the Riemann Hypothesis is true,” or “How likely is it the
suspect is guilty of murder?”
In this class, we’ll present Cox’s theorem. Cox’s theorem, loosely speaking, states that, if a
system for evaluating the plausibility of hypotheses satisfies a few common-sense criteria, then it
is isomorphic to the laws of probability as you know them. This lays the theoretical groundwork
for the Bayesian school of probability, under which many questions similar to the ones given above
may be rigorously evaluated.
Prerequisites: You should be comfortable with first-year calculus, especially with handling error
terms. There is some mulitivariable, but we’ll cover what’s needed in class.
Imaging at the Nanoscale. ( , Grace Gee)
Light waves are characteristically described by their intensity and phase. However, modern
microscopes can only detect the intensity. So how do we solve this phase problem? Mathematically,
there is a solution. By taking pictures of the x-derivatives and y-derivatives of the image, we can
use matrices to deduce the original phase. The class will go into more detail of how this process
works, what problems occured in the simulation and how we solved it, and what its applications
are. Come if you’re interested!
Prerequisites: Basic knowledge of inverses of matrices, matrix multiplication, and derivatives/integrals
is required.
False Proofs. ( , Meep Campbell)
Prove the impossible. Eff the ineffable. Let’s see how much wrongness can be crammed into one
hour as Meep proves that 0 = 1 = 2 = π = ∞ (don’t worry, she’ll also prove the natural numbers
don’t exist, but even if they do, the reals are countable, so it all works out in the end).
BE ADVISED: nothing Meep proves in this class will be true.
1:30 PM
The Busy Beaver Problem. (
, Daniel Briggs)
Given a positive integer n, how many 1s can an n-state Turing machine leave on a tape that
started with all 0s, and then halt? Tibor Radó asked this amusing question in 1962, and since then
it has been answered for n = 1, 2, 3, and 4. The 5-state problem is very possibly close to being
solved, and in this lecture I will explain how you can help bring it to completion.
More general results, dealing with incomputability, undecidability, and oracle machines will also
be discussed.
Prerequisites: Knowledge of theoretical computer science recommended but not required.
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Are the Laws of Physics Changing? ( , Charles Steinhardt)
A variety of theories, ranging from historical to modern, have proposed that the laws of physics
might be variable. We can investigate by considering whether the strengths of forces such as gravity
and electromagnetism might change over time, over space, or at different distance ranges. These
tests use a wide range of methods, even including atomic clocks, a two-billion year-old nuclear
reactor, aging spacecraft, and defects in the fabric of spacetime. This talk will cover some of the
highlights of these results, some of them surprising, and what this might mean for the laws of
physics. No background in physics or astronomy is needed.
Prerequisites: None
The Mathematics of Polygamy. (
, Mira Bernstein)
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 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.”
For 1800 years, this passage had baffled scholars: what could possibly be the logic behind the
Mishnah’s totally different ways of distributing the estate in the three cases? 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. The proof is very cool, based on an analogy with a simple physical system! See if
you can figure out this ancient puzzle for yourself, or come to class and find out.
Prerequisites: None
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