PCMI 2021 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.  In normal years, participants are expected to be in residence for the entire three weeks of PCMI; in 2021, due to COVID-19, all activities are online, and the participants are expected to attend every lecture.

The 30th Annual PCMI Summer Session will be held virtually July 11 - July 31, 2021.

Research Theme: Quadratic forms, Milnor K-theory, and arithmetic

In 2021 there will be one daily lecture series at 1pm, given by Dustin Clausen (University of Copenhagen) and Akhil Mathew (University of Chicago), and the morning sessions will involve experimental mathematics component with open-ended problems and computational work. 

Course Description

In this course, we will begin by developing the general theory of quadratic forms, which is related to a variety of topics across arithmetic and algebraic geometry. After some foundational results, we will specialize to the case of the rational numbers. We will aim to cover aspects of three significant results:

The Hasse-Minkowski theorem

The quadratic reciprocity law, and its formulation via Milnor K-theory

The Siegel mass formula


The primary prerequisite is abstract algebra. Some familiarity with notions of general analysis and topology (e.g., metric spaces, completions) will be helpful when we discuss the p-adic numbers.