Unusual Classes
Here are some mini-course descriptions taken from HCSSiM maxi-mini catalogs in recent years.
Hyperfying (Cynthia Vinzant)
Feeling too limited by the restrictions of graph theory? Ever wish edges could be sets of any number of vertices rather than just two? Well, just hyperfy your problems away. Learn about quirky kinds of hypergraphs, such as “linear spaces” and Steiner systems (like the Fano plane and the game SET). Learn how to properly color your very own hypergraph! “And what about Ramsey theory,” you ask? No, you won’t even need to give up Ramsey theory. Free yourself from the world of 2-uniform hypergraphs!
Projective Geometry (Cary Malkiewich)
Child: Mommy, where do eleven-line conics come from?
Mother: Well, uh, you see... you see, there’s this great big bird, the
stork, and it... umm... sort of drops the eleven-line conic down the chimney...
Child: ...but the kids at school told me that the eleven-line conic arises
from a simple isomorphism between two real projective lines!
Mother: Oh boy...
Projective geometry is the study of old math movies (geometry) run on projectors (projective). As an immediate corollary, always pole and polar are the same color. But seriously, projective geometry is a weird/cool subject that will totally rock your world. Let’s take the complex plane and add a point at infinity (CP1). Better yet, let’s take the real plane and add a line at infinity (RP2) Three-dimensional space with a plane at infinity (RP3)?! Oh yeah. Non-commutative geometry (HP1)?!?!?! Oh, come on. That’s just silly.
Trash those textbooks that just throw axioms at you like they don’t even care. We’ll look at the real projective plane in a more descriptive, analytic way. We’ll be talking about duality in all of its awesomeness. There will be conics galore. And then there’s finite projective geometry, which includes a preview of Samuel L. Jackson’s upcoming hit, Snakes on a Fano Plane (Z2P2). Prepare to rock.
Contract This: Fractals! (Lawrence Valby and Cynthia Vinzant)
GLAUCON: (seeing a tree branch) Look! A fractal. The branch splits into smaller
branches, each of which looks like the larger branch.
SOCRATES: So, would you say that a fractal is anything that is self-similar?
G: Well, yes.
S: Do you consider the real line a fractal?
G: Why no!
S: Is the real line self-similar?
G: Well, I guess so. If you zoom in, it looks the same.
S: Clearly, you need a better definition of fractals. Would you say that a
fractal is a fixed point of a contraction mapping in some Hausdorff metric
space, whose box dimension is fractional?
G: Hmm... yes, that sounds better.
S: Are you sure?
Telepathy (Josh Greene and Aaron Magid)
You should be able to figure out the description.
The Principle of Solid Reality (Ari Turner)
Dear Mathematics Students,
I have a theory which I have been showing to numerous people from much esteemed
Universities, but such people cannot get beyond the rigid constraint of universally
accepted but not true theories. I hope that you will be able to see my theories
with a fresh mind. I have a principle, of Solid Reality, which says that a
solid object is solid. Scientists are very often described as rational and
reasonable people, but I would like to prove this wrong: Scientists believe
that a solid gold sphere may be cut up into five pieces and reassembled into
two solid gold spheres (the Banach Tarski Paradox). My principle of Solid Reality
proves them insane. Scientists believe that atoms and molecules may pass through
solid walls and my Principle proves them to be bonkers. They believe that squeezing
a solid object can cause it to liquefy and my Principle proves them to be fantastically
unhinged. Scientists believe that yellow pigs evolved into human beings, and
even the idea of a Yellow Pig is an abomination. Science and math are not the
study of common sense—so they are certainly wrong! I shall present many
strange scientific dogmas (from topology, measure theory, quantum mechanics,
and?...) so that you will see how humorously scientists have misled themselves.
Sincerely,
Ari Turner
manager of Bank of Narrowton
Linear Algebra Methods in Combinatorics (Jonah Blasiak)
In this mini we will begin by reading How to Stop Worrying and Start Multiplying Large Matrices by Hand. After that we will watch The Matrix a bunch of times. If this sounds scary to you, you are in luck, because we will also learn that linear algebra is much more than multiplying matrices, and has cool connections with geometry (including 4-line conics), probability, algebra, and combinatorics.
It is not difficult to learn and prove many basic linear algebra theorems,
but it is difficult to grasp the importance and wide applicability of its methods.
Linear algebra solves many problems effortlessly where other methods make no
headway. We will discuss basic linear algebra and survey some of its applications
in combinatorics. We will constrain regular graphs of girth 5, bound the number
of points in Euclidean space with only 2
separating distances, learn several cool ways of computing determinants, and
more!
All Is Empty Set: Dr. Mengenlehreliebe, or How I Learned to Stop Worrying and Love Zorn’s Lemma (Jason Murcko)
A tortoise questions a hare as they stand together at the base of a stone
stairway.
TORTOISE: What is the sound of one set being bijectively mapped to its power
set?
HARE: It is the same sound made by the set of all ordinal numbers.
TORTOISE: Mmm... and how many Zermelo-Fraenkel axioms can stand on the head
of a pin?
HARE: If we account for the Axiom of Comprehension, the number is endless.
TORTOISE: I see. One final question: does the continuum have aleph1-nature?
HARE: (Does not exist.)
NARRATOR: I'm sorry to interrupt you two, but we have decided that GEB-style
dialogues are passé. Consequently, it has been determined that you no
longer exist.
Homology and Knot Theory (Jeff Brown + Wing Mui)
If Romeo and Juliet were both steam-rolled and had to live the rest of their lives in two dimensions, you could choose to ruin said lives by taking a pair of scissors to the piece of neoprene they lived on, thereby destroying any hopes they had of reuniting. Your friends would think you were a jerk. After taking our class, and buying a Glue Stick™, you would know how to rearrange their landscape so that the next time they met, Juliet had become left handed and dyslexic, and they were suddenly evenly matched on their (linear-screened) Game Boys™, leading to a lifetime of fulfillment. Your friends would think you were pretty cool.
In this mini we will study homology, a beautiful group structure on a certain well-behaved subset of all n-dimensional shapes, that leads to equations like “square + triangle + triangle = nothing”, or “walk to the dentist’s + walk back home = the border of Tunisia”. We will also study knot theory, particularly how to color them (for a viewer who picks a nice spot and then DOESN’T MOVE! ) which we feel is a reasonable amount of discipline to request in exchange for good art. We will prove the deceptively simple Jordan Curve Theorem, that every closed curve divides the plane into an inside and an outside. By the end of the course you will have acquired a deadly fear of shoelaces, and thereby have permanently increased the excitement level of your reality. If that doesn’t sound worth doing, what does?
There will knot be puns in this mini.
