PCMI 2024 Undergraduate Faculty Program

The Undergraduate Faculty Program (UFP) at PCMI is aimed at faculty members from all types of colleges and universities with a strong interest in undergraduate teaching and research.  Some participants are looking to rekindle their engagement with mathematical research and to interact with the broader mathematical community, while others are looking for new teaching approaches.  The focus of the UFP varies from year to year, although its mathematical content is always aligned with the main research theme of PCMI that summer. In some years it is run as a small group seminar focused on some interesting parts of mathematics, in a setting which emphasizes participants working on ideas and problems together; in other years some part of it is also devoted to pedagogical issues or curriculum development.  The UFP typically has around 10-12 participants, which allows for close and informal interactions among the participants.

UFP participants also interact with the participants and lecturers in the other programs, and in particular may choose to attend some of the Undergraduate and Graduate Summer School lecture series.  Members from all parts of PCMI may take part in the Experimental Math Lab, in which small groups of participants with close mentorship from a more senior mathematician investigate open-ended problems and report on their findings at the end of the three-week Summer Session.  Interaction among everyone at PCMI is fostered by various informal social activities open to all PCMI participants, as well as daily “cross-program activities’’ that include lectures and presentations on topics of general mathematical interest.

PCMI will be held July 7-27, 2024.

2024 Research Theme: Motivic Homotopy Theory

The lecturer for the Undergraduate Faculty Program in 2024 will be Michael Ching (Amherst College).

Operads in Algebra, Combinatorics, and Topology

Operads are gadgets which encode certain types of mathematical structure, such as monoids or Lie algebras. They have played a fundamental role in the development of homotopy theory by allowing topologists to capture succinctly operations which are associative (or commutative) up to homotopy, such as the concatenation of loops in a topological space. Operads are used now to describe structures in areas such as representation theory, (higher) category theory, and mathematical physics, among others.

This program will survey commonly used operads, their theory and applications, from algebraic combinatorics to homological algebra and topology. The focus will be on concrete constructions which can readily be investigated by undergraduates.