A chance to "sample" new mathematics: Mathcamp's colloquia are one hour talks each afternoon that almost all of camp attends. They're a chance to discover new branches of math or learn about unusual applications; colloquia have ranged over topics as wide as cognitive science, the Hodge conjecture, introductory topology, string theory, the mathematics of origami, and even "Why 37 is Evil."
A chance to meet exciting new visitors: Colloquia are how we introduce Mathcamp's visitors to the rest of camp. Visiting faculty will often present on their research or introduce their unique perspectives on mathematics. The colloquia often lead in to the classes they teach throughout the rest of the week.
And a chance to learn about the boundaries of mathematical knowledge: Often, regular Mathcamp instructors will offer colloquia introducing students to their own fields of research. At Mathcamp 2006, for instance, we held a colloquia series on the "seven millenial problems" throughout camp --- these are open questions in mathematics that have become central to modern research, and for whose solution the Clay Mathematics Foundation has offered a one million dollar prize.