Random 17 fact:

There are seventeen distinct wallpaper groups.

Usual Classes

The academic core of HCSSiM is the 37 weekly hours of small, highly interactive classes and problem sessions. These are described here; not described here are the frequent occurrences of mathematics on dinner napkins, in the swimming pool, and on shopping expeditions. Never again do most HCSSiM participants have access to so much interaction with and attention from faculty. Students connect with peers and staff members who share their interest in mathematics, and maintain these connections for years into the future.

The 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.

Maxi-courses and Mini-courses

After the three-week workshops, students express preferences for the direction of their mathematical activities for the rest of the summer. Each student selects one maxi-course (meeting 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 and are anticipated to be offered (with modifications) in subsequent summers are described below.

Graphs on Surfaces: Within topological graph theory, students learn about graphs and graph coloring, topological surfaces, types of graph embeddings, Euler's Formula on topological surfaces, graph genus, and the Heawood bound. Students become aware of the current state of knowledge and some become ready to begin research in the area.

Mathematical Origamist's Toolkit: Topics include modular origami and how this models the creation of polyhedra and coloring of graphs, comparison of origami-axiomatic constructions to straight-edge and compass constructions, the combinatorics of possible crease patterns, the mathematics of origami design (circle packing, optimization), matrix models for paperfolding, spherical geometry, Descartes’ Theorem, and Gaussian curvature.

Probability: 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.

Polytopes and More: Convex and combinatorial geometry are studied in general dimension with primary examples taken from the classical polyhedra and special classes of polytopes (e.g. regular, simplicial, cyclic, Coxeter-Petrie, neighborly, etc.). Accessible recent results are surveyed.

The mini-courses are more narrowly topic-oriented. Recent topics include: non-Euclidean geometry, set theory, Lebesgue measure and integration, random walks, game theory, knot theory, generating functions, advanced number theory, reading recently-published papers, linear algebraic methods in combinatorics, dynamical systems, graph colorings, computational complexity, Ramanujan's work, and number systems.

The Prime Time Theorem

The entire program comes together for the afternoon Prime Time Theorems (PTTs), which are informal lectures that lead to the communal understanding of a proof of a theorem. Summer Studies faculty and visiting mathematicians deliver the PTTs.

Recent talks given by visitors to the Summer Studies include: