Courses, Usual and Unusual

Some Usual Courses


For the initial three weeks of the Summer Studies, participants are partitioned into workshops, each led by a faculty member and one or two talented math majors/graduate students. The workshops meet for four hours each morning, Monday through Friday, for two hours Saturday morning (after which the whole program comes together for a mathematical event), and for an evening problem session (every weekday).

The selection of specific topics varies among workshops and from year to year. The mathematical content is considerable: We commonly cover undergraduate material equivalent to most of an elementary number theory course, most of a combinatorics/graph theory course, half of a modern algebra course, and a third of one or two elective courses within the first three weeks. Other topics that have recently been discussed in workshop include complex analysis, topological surfaces, analysis of the Fermat-Pell equations, continued fractions, polyhedra and polytopes, and hyperplane arrangements.

Maxis and Minis

After the three-week workshops, students express preferences for the direction of their mathematical activities for the rest of the summer. Each student is sorted into one maxi-course (which meets 2.5 hours, six mornings per week, and in three-hour problem sessions, five evenings per week) and two mini-courses (consecutively, each 1.5 hours per day for seven days). A maxi-course covers material equivalent to a semester-long undergraduate elective.

Some of the maxi-courses that have been taught successfully in recent summers are described below.

One To/Two/Too Many: 
This maxi is on counting and enumerating too,
and this paragraph’s amounting to a list of things we’ll do.
We’ll use generating functions, exponentially of not,
to count graphs and trees and munchkins, which are dice of difference spot.
We’ll build on the lemma Burnside to enumerate like Polya,
and count checkerboards we’ll turn sideways and tile as we’ll show ya’.
We’ll meet Erdos, Schur and Ramsey, van der Waerden, Beck and Graham,
Fibonacci, sure you can see that this maxi’s not a scam.

Iteration, Fractals, Chaos, Iteration, Fractals, Chaos, … :
sin squared phi is odious to me.” – Carl Friedrich Gauss.
Even simple functions can exhibit varied and intricate behavior.  We’ll use calculators, computers, and old-fashioned proofs and derivations to discover basins of attraction, unstable equilibria, pre-periodic points (groupies), strange attractors, and chaos.  You’ll become familiar with unusual and surprising (and eventually not-so-surprising) dynamics, and with strange and beautiful (and eventually even-more-beautiful) patterns.  We’ll understand the mathematics underlying the Mandelbrot and Julia sets and other fractals.
Probability to the Limit:
An axiomatic approach is combined with computation and simulation; classical distributions are analyzed; laws of large numbers are formulated and proved; old and new paradoxes are pondered; random walks are investigated in n dimensions and on finite graphs; and Erdos’ probabilistic method is applied.  A plethora of pretty problems and puzzling paradoxes are pondered.

Here’s Looking at Euclid:
Oh, Bolyai, this will give your frontal Lobachevsky. The Flat Plane, the Curved Plane, and the Negatively Curved Plane? What in the world (or out of it) is this about? Find out what happens when we tinker around with Euclid’s Fifth in this Maxi on Geometry.

Considered to be one of the most essential branches of mathematics in all ages, Geometry is a beautiful subject that, although its origins stretch far back into history, still has the power to surprise. Find out how we can obtain equilateral triangles, quadrilaterals, pentagons, hexagons, and so on, all with right angles! Discover worlds where there are triangles with infinitely long sides, but whose areas stay finite.



The mini-courses are more narrowly topic-oriented. Recent topics include:

non-Euclidean geometry, Galois theory, Lebesgue measure and integration, random walks, game theory, knot theory, generating functions, peg solitaire, reading recently-published papers, linear algebraic methods in combinatorics, dynamical systems, graph colorings, computational complexity, model theory, Ramanujan’s work, diagram-chasing in group theory, open problems, mathematical music theory, cryptography, solving polynomials with origami, projective geometry, algebraic topology, …

For a better taste of these, we invite you to look at

Some Unusual Courses

Here are some mini-course descriptions taken from HCSSiM maxi-mini catalogs in recent years.

Projective Geometry: (Taught by 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.

How to Construct Rome in a Day: (Taught by Andy Soffer)
I used to be the best geometer in all the land.  I could draw some awesome stuff that would blow Escher’s mind!  But one day in my wanderings through the Desert Of Trisected Angles, I was robbed, left only with a compass, straightedge, a book on Field Theory, and $2.89 in dimes!  What was I to do?  I sat down and constructed a regular heptadecagon-shaped camel, and rode out of the desert back home.  Now I am here, and I will tell you about my travels.

Homology and Knot Theory: (Taught by Jeff Brown and 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.

Telepathy: (Taught by Alex Kunin and Max Levit)
You should be able to figure out the description of this mini.

G’raffes: Spotted, Striped, and Coloring: (Taught by Amber Verser)
Have you ever wondered what the colorings are for a wild g’raffe?  Or what the difference between a snark and a bridge is?  And just what was the big huff in the 1970s about the first Appel computer?  We will explore the world of assigning colors to graphs, visiting how to tell is a graph is planar by making it look pretty and the instructions Erdos gave about how to deal with visiting aliens.  We’ll create algorithms for coloring graphs, attempt to color the plane, and see how a number of problems can be solved with a small number of crayons.

Hyperfying: (Taught by 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!

Really Really Really Really Big Things: (Taught by Max Levit)
Mathematicians really like sets.  Everything from graphs and groups to probability and functional analysis is built out of sets of sets of sets of sets of…  But what about these sets?  What can you do with them all by themselves?  How do we build them?  And how big can we build them?  REALLY REALLY REALLY REALLY REALLY BIG!  In this mini we will talk about a few different types of really really really really big sets.  We will prove theorems about order, and show the equivalence of three seemingly unrelated statements (one of which is the infamous Axiom of Choice).You might like this mini if you like talking about definitions and abstractions without worrying about what they correspond to in real life or are interested in question such as: What’s the biggest set we can make without getting into trouble a la the barber paradox (does the set containing exactly those that don’t contain themselves contain itself?).
 Peg Solitaire: (Taught by Paul Phillips)
A good career, wealth sufficient for your every need, and good friends can all be yours even if you take this mini.  In the game of peg solitaire, the goal is (like golf) to get the lowest score possible; in particular, by making a series of “jumps” with the pegs, to remove all but one peg from the board.  We will investigate the patterns and mathematics that underlie this seemingly simple game with its nearly endless variants.  By the end of this mini, you should be able to solve many of the standard versions and have acquired some useful concepts that can be applied to your other mathematical studies.

Gal-Whaaaa?: (Taught by Amber Verser)
Ever wonder why Euclid couldn’t trisect an angle, other than the fact that he was too old to have paper to fold?  Or why he couldn’t double a cube?  Or perhaps why people claim that quintics are “unsolvable”? Wait no longer! We’ll explore the magical world of polynomials, how their roots can answer those nagging questions in the back of your skull, explain why heptadecagons are but heptagons are not constructible, learn a lot about the hidden structure of groups (and just how much they can tell you about polynomials), and talk about the myth of the man who was killed in a duel at age 20 who is responsible for this math.