PCMI 2026 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 June 28-July 8, 2026.
2026 Research Theme: Knotted Surfaces in Four-Manifolds
The lecturer for the Undergraduate Faculty Program in 2026 will be Jeffrey Meier (Western Washington University).
Bridge trisections of knotted surfaces in four-manifolds
The study of knotted surfaces in four-space is the natural four-dimensional analogue to the classical study of knotted curves in three-space. A key strength of knot theory in dimension three has been its robust diagrammatic framework, which underlies diverse aspects of the theory from the creation of knot tables and the definition of invariants such as the Jones polynomial to the descriptive efficacy of Kirby calculus and computational programs such as SnapPy.
In contrast to the three-dimensional theory, which has been richly developed and explored over the last 150 years, the four-dimensional theory remains less well-charted, and many basic questions remain unanswered. In this course, we will introduce and explore the diagrammatic framework for studying of knotted surfaces via the theory of bridge trisections. Through the lens of this nascent theory, we will investigate a diverse array of aspects of knotted surface theory such as invariants of surfaces in four-space, branched coverings of four-manifolds , and surgery on surfaces in four-manifolds. The approach throughout will be heavily diagrammatic and combinatorial, and emphasis will be placed on open problems – both within the theory of trisections and pertaining to the study of knotted surfaces in general – and drawing connections to techniques from three-dimensional knot theory.
