# Some Usual Courses

### Workshops

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.**

**Information on Information:**

Can you define the amount of information in a mathematical way, and is it useful? We will use Shannon’s definition of information to find how much longer newspapers have to be if mice randomly chew 15% of the letters out (this is how satellites encode pictures of Jupiter!) and to understand hwy ti si psosbile to reda smotehnig evne hwen tehre si a lot fo nitrefrenece nad tehre rae msipirtns all ovre teh plce. On the way to finding codes that protect information we will use finite fields and study packing of spheres in many dimensions (learning how different higher dimensions are).

**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.

**What is a Number? Get Set 2 Be Surreal:**

Have you ever wanted a number system that contained infinity in a principled way? Is one infinity just not enough for you? Can’t find a small enough rational positive? End up in this Maxi, and you will know about a number system that would inspire jealousy and admiration even in a Dali.

In this Maxi we will build the more numbers than you thought you could ever need, learn how to play games you never knew existed and then use those games to say things about numbers, and work with more pluses and minuses than you ever dreamed of, all from scratch, which is to say from literally nothing.

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

non-Euclidean geometry, Galois theory, random walks, game theory, knot theory, generating functions, peg solitaire, historical precursors to calculus in the non-Western world, algebraic combinatorics, dynamical systems, graph colorings, computational complexity, elliptic curves, conic sections, neural networks, tropical geometry, linear algebra, the Yoneda Lemma, 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.

**Make Literally Anything in Desmos:** (Taught by Dylan Lee)

Drivable cars! Funky screen effects! 3D render engines! Knot generators! Water simulations! The entire backrooms! Origami that folds itself before your very eyes! Recreations of beloved board games! Procedurally generated cities! Loved ones lost to time! Desmos is a free online graphing tool commonly used for plotting 2D points and equations. But it can be so much more.

In this Mini, you will learn how to: bully desmos into graphing in 3D; manipulate lists of numbers, points, and polygons; bully desmos into doing recursion using regressions and actions; create some of the most cursed functions known to mankind; build cool stuff that you can show your friends, family, and pets — and so much more!

**Geometry for the Voracious:** (Taught by David Tu)

Your entire life you have been told that doing geometry on a doughnut is weird and difficult. This is a lie perpetuated by Big Doughnut to protect their glazed, sugary secrets. We’ll do geometry not just on doughnuts, but doughnut holes, Pringles, Fruit Roll-Ups, and various other foods. This culinary journey will be seasoned with exotic spices we find along the way, like lines that intersect themselves infinitely many times, and triangles with infinitely long sides, no vertices, and finite area.

We’ll also figure out why, aside from their edibility, a coffee mug is not the same as a doughnut, and why we can still be friends with the people who think coffee mugs and doughnuts are the same. Warning: This Mini only contains fun math and will not help you with geometry problems on math competitions.

**No Apologies for My Topology:** (Taught by Jeff Brown and Joy Hamlin)

Mobius strips! Klein bottles! Projective space! You might have heard of these strange spaces, but do you really understand what they are? Those who are wise in the ways of topology can analyze them and many more besides, using neighborhoods, groups, and loops that shouldn’t vanish but do anyway. But before any of that, we need to answer a much more fundamental question — what even is a continuous function, when we’re no longer in a line or a plane?

We’ll also figure out why a coffee mug is the same as a doughnut, and why we can still be friends with the poor, deluded souls who claim to see a difference between the two. Warning: This Mini might leave you paranoid for the rest of your life that even after you’ve closed the door to your room, it’s somehow still open.

**Designs & Codes:** (Taught by Jessie Feng)

Let’s say you’re a small ice cream maker trying to test out 17 new flavors on 2 victims paid volunteers. Unfortunately, they are both lactose intolerant and can only taste up to 3 flavors before they are “unavailable” to eat more ice cream. Is it possible to get all 17 flavors tested? How many people would we need to effectively and efficiently test all the flavors?

One method of answering this kind of question is by using experimental designs. These configurations are a combination of graph theory, modular counting, linear algebra, and real life applications. Among other things, we’ll learn what “Barry Very Regularly Kicks Llamas” means, discover the Deathly Hallows Design, and prove the LinkedIn Theorem.

**Telepathy:** (Taught by Alex Kunin and Max Levit)

You should be able to figure out the description of this mini.

**Graph Theory, Brought to You by Crayola:** (Taught by Cece Henderson)

Are you tired of bland and boring graphs? Do your eyes water with dreariness whenever you see vertices drawn in plain old expo marker on the board? Can’t stand dull edges rendered in greyscale on your problem sets? Take your graphs from drab to fab in this multi-hued Mini! Bust out your highlighters and arm yourself with crayons as we dive into the rainbow of graph coloring. If you take this Mini, be prepared: life is about to get a lot more colorful.

**Puzzling Patterns:** (Taught by Amber Verser)

Come on a wild adventure through the twisting and intersecting worlds of algebra, geometry, and topology, where we’ll answer pressing questions you never knew you had like — How can you ensure that your golden stakes are protected using just a guard dog and a long circular leash? How would Greek mosaics look different if they were glued together with twists? How much should wallpaper cost? And what would your interior design options be if your living room were a Klein bottle? We might even investigate what happens when you pile 30 million grains of rice on a single point in Germany.

Be prepared for using algebra to connect pretty things to each other, to find (and prove the existence of!) many surprising 17s, and to never be able to look at bathroom tiles the same way ever again. This mini may be “just algebra,” but there will be no shortage of beautiful images and fun puzzles!

**Snakes on a Plane:** (Taught by Milo Brandt)

Once upon a time, in the far away kingdom of Incidence, there lived a geometer named Oinklid. Oinklid had a problem: Her clumsy hooves could not draw straight lines, nor measure angles nor lengths. One day, Oinklid enlisted all of her (finitely many) snake friends (who conveniently happened to be infinite curves stretching all the way across a plane) and bequeathed an axiom unto them: As pairs of lines must intersect exactly once, so too must pairs of snakes. The resulting theory of snake geometry was a lot like normal geometry, but without requiring dexterous diagramatization. Oinklid lived happily ever after, asking questions of snake-arrangements: “Can snakes (lines?) divide the plane into only triangular regions?”, “Will there always be somewhere where exactly two snakes meet?”, “Can I always straighten out an arrangement of bendy snakes?”, and “Wow, this theory has something to do with tilings and also polynomial inequalities!?” (The End)

**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.