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 selects 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 and are anticipated to be offered (with modifications) in subsequent summers are described below.
One To/Two/Too Many: We'll take combinatorial excursions  not directly related to counting, including Ramsey theory (which states that complete disorder is impossible) van der Waerdan's theorem, king chicken theorems, non-standard dice, the chromatic number of the plane problem, Sylvester's theorem (which states that if you have at least 2 points in the plane, not all on a single line, then you can find a line that passes through exactly two of the points) and possibly Schur's lemma
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.
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.