PCMI 2022 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 17-August 6, 2022.

Research Theme: Number Theory Informed by Computation.

In 2022 there will be one daily lecture series at 1pm, given by Christelle Vincent (University of Vermont), and the morning sessions will involve experimental mathematics component with open-ended problems and computational work.  Here is a rough outline of Professor Vincent's lectures:

Title: Introduction to mathematical cryptography

Course description: This course is an introduction to mathematical cryptography as well as a short primer on the history of cryptography, its possible future, and some societal issues related to cryptography. Through lectures and problem sets, we will delve into the mathematical intricacies of the discrete logarithm problem and lattice-based cryptography. In addition, the instructor will give general-audience lectures on topics such as how TLS security works, the NSA and backdoors, how the PlayStation3 was hacked, and the impact of quantum computers on cryptography.

All admitted students will be asked to commit to affirmatively work towards a joyful and inclusive atmosphere at the USS, and to center the needs of the most marginalized members of our program.

The prerequisites for this course are linear algebra and either abstract algebra or number theory.  At least one of these courses should be proof-based.