PCMI 2024 Undergraduate Summer School
The Undergraduate Summer School (USS) at PCMI provides a unique opportunity for undergraduate students to learn some fascinating mathematical ideas in a setting that allows them to interact with mathematicians at all levels. The program itself is typically centered around lecture series delivered by leading experts on topics related to the main research theme of PCMI that summer. These lectures generally present material not usually part of an undergraduate curriculum, allowing students to become familiar with key ideas and techniques in the field, and often leading toward further research. The program is structured so that students at different levels will have many opportunities to learn new things.
The USS is not like a typical REU in several ways. The focus of the USS is more on the specialized lecture series and most importantly, the ability to interact informally with the many graduate students and researchers attending other parts of PCMI. Students can get to know mathematicians who have pursued a wide variety of career paths, and they can get a sense of which of these paths may be most appealing to them. Many USS participants report making connections that strongly influence their choice of graduate school. Interactions are fostered by the various informal social activities open to all PCMI participants, as well as daily "cross-program activities," which include lectures and presentations on topics of general mathematical interest. 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.
The USS is open to undergraduate students at all levels, including those who have just completed their undergraduate studies. Participants are expected to be in residence for the entire three weeks of PCMI.
The PCMI Summer Session will be held July 7-27, 2024.
Research Theme: Motivic Homotopy Theory
In 2024 there will be one daily lecture series at 1pm, given by Anna Marie Bohmann (Vanderbilt University), and the morning sessions will involve experimental mathematics component with open-ended problems and computational work. Here is a rough outline of Professor Bohmann lectures:
Title: Homotopy theory: towards stability and beyond!
Description: Homotopy theory is about understanding the features of a space that don't change when you deform it in a continuous way. Using tools from algebra, we will create invariants of spaces up to such deformations, including the higher homotopy groups (discovered by Hurewicz in the mid 1930's) that lie at the core of modern algebraic topology. We will then explain how the remarkable notion of stabilization (introduced by Freudenthal in the late 1930's) leads to new, more tractable invariants, revealing rich connections with algebra, geometry and more.
Prerequisites: Linear algebra, at least one semester of abstract algebra (including some group theory), and at least one semester of either real analysis or point-set topology (including the study of continuity, compactness and connectedness using open sets).
All admitted students will be expected to contribute toward an inclusive and collegial atmosphere in this program. USS students come from many different backgrounds, and with different strengths, and we ask that all USS students commit to working together positively and noncompetitively.