Mathcamp's colloquia are one hour talks that happen periodically (or aperiodically) throughout camp. No other activities are scheduled opposite them, and the topics are selected to appeal to a broad audience, so most students attend.
Sample new mathematics: Colloquia are a chance to discover new branches of math or learn about unusual applications. They cover a wide range of topics, such as cognitive science, the Hodge conjecture, introductory topology, string theory, the mathematics of origami, and even "Why 37 is Evil."
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 tie into the classes they teach throughout the rest of the week.
Learn about the boundaries of mathematical knowledge: Often, regular Mathcamp instructors will offer colloquia introducing students to their own fields of research. One summer, 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.