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 FermatPell equations, continued fractions, polyhedra and polytopes, and hyperplane arrangements.
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 FermatPell equations, continued fractions, polyhedra and polytopes, and hyperplane arrangements.
Maxis and Minis
After the threeweek workshops, students express preferences for the direction of their mathematical activities for the rest of the summer. Each student selects one maxicourse (which meets 2.5 hours, six mornings per week, and in threehour problem sessions, five evenings per week) and two minicourses (consecutively, each 1.5 hours per day for seven days). A maxicourse covers material equivalent to a semesterlong undergraduate elective.
Some of the maxicourses that have been taught successfully in recent summers and are anticipated to be offered (with modifications) in subsequent summers are described below.
Some of the maxicourses that have been taught successfully in recent summers and are anticipated to be offered (with modifications) in subsequent summers are described below.

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 oldfashioned proofs and derivations to discover basins of attraction, unstable equilibria, preperiodic points (groupies), strange attractors, and chaos. You'll become familiar with unusual and surprising (and eventually notsosurprising) dynamics, and with strange and beautiful (and eventually evenmorebeautiful) patterns. We'll understand the mathematics underlying the Mandelbrot and Julia sets and other fractals.

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. A plethora of pretty problems and puzzling paradoxes are pondered.
The minicourses are more narrowly topicoriented. Recent topics include: nonEuclidean geometry, set theory, Lebesgue measure and integration, random walks, game theory, knot theory, generating functions, advanced number theory, reading recentlypublished papers, linear algebraic methods in combinatorics, dynamical systems, graph colorings, computational complexity, Ramanujan’s work, and number systems.