Courses

Workshops

For the initial three weeks of HCSSiM, 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.

Maxis

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.

Minis

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, …