2021 Graduate Summer School Course Descriptions

Below are the descriptions for each week of the Virtual PCMI 2021 Graduate Summer School Program.  Students may apply to one, two, or all three of the one-week sessions.

July 12-16 - Motivic Homotopy

July 19-23 - Illustrating Mathematics

July 26-30 - Number Theory Informed by Computation


Week 1 - July 12-16, Motivic Homotopy

The PCMI Workshop “An introduction to motivic homotopy theory and its applications” is being held as an online preparation for our planned PCMI Summer School on motivic homotopy theory and its applications.

Motivic homotopy theory arose out of the work of Morel and Voevodsky in the 1990s and since then has developed into both a powerful tool for understanding many arithmetic aspects in algebra and algebraic geometry, as well as being a fascinating generalisation of classical homotopy theory with an active development in its own right.

We will host five introductory talks of two lectures each by experts in motivic homotopy theory and  the fields to which it has been applied.  Two of the lectures will discuss the basics of motivic categories and motivic cohomology, while the remaining three will deal with areas of application: foundational questions on the arithmetic of fields and its relation to algebras, quadratic forms and cohomology classes, the closely related area of G-bundles in algebraic geometry, and the refinement of classical invariants of enumerative geometry to quadratic forms. Beside the online lectures, we will hold online discussion and question and answer sessions.

Frédéric Déglise (ENS de Lyon, UMPA)

Title: Motivic categories


Building on initial conjectures due to Beilinson, Voevodsky has initiated a rich variety of "motivic categories", the universal one being Morel-Voevodsky's homotopy category. This world, that is now called "motivic homotopy theory", has produced a wide range of results, settling older conjectures as well as opening new tracks to follow.

This lecture series will aim at giving a survey of this world, from the pure motivic origin, through the functoriality developments and then to some of the exciting open questions.

Philippe Gille (Université Claude Bernard Lyon 1)

Title: G-torsors over affine curves


We shall present the theory of G-torsors (or G-bundles) in algebraic geometry which includes for example vector bundles and quadratic bundles (Grothendieck-Serre, 1958). We focus on the case of an affine smooth connected curve firstly over an algebraically closed field k; we shall show that G-torsors are trivial for a semisimple k-group G. Secondly we will consider the case of a perfect field and discuss the important case of the affine line (Raghunathan-Ramanathan, 1984). This will be an opportunity to deal with étale cohomology and patching techniques.


V. Chernousov, P.  Gille, A. Pianzola, Three-point Lie algebras and Grothendieck's dessins d'enfants. Math. Res. Lett. 23 (2016), 81–104.

J.S. Milne,  Lectures on etale cohomology, https://www.jmilne.org/math/CourseNotes/LEC.pdf

M. S. Raghunathan, A. Ramanathan,  Principal bundles on the affine line. Proc. Indian Acad. Sci. Math. Sci. 93 (1984), 137–145.


For the background, I recommend to know basic algebraic geometry for example about flatness and smoothness and also a bit on linear algebraic groups.

For the first topic, the reference is Hartshorne (or Milne, Etale cohomology, the beginning). For the second one the reference is Milne's book "Algebraic groups''. 

Daniel Krashen (Rutgers University)

Title: Field arithmetic and the complexity of algebraic objects


A fundamental question in field arithmetic is how one can bound, in various senses, the complexity of algebraic objects such as algebras, quadratic forms, or cohomology classes. This question is intimately related to notions of essential dimension, symbol length, and also to the construction of generic splitting varieties. In these talks, I will describe some of the principal questions of this sort, and various methods by which they have been approached.

Matthew Morrow (IMJ-PRG)

Title: Aspects of motivic cohomology


Motivic cohomology offers, at least in certain situations, a geometric refinement of algebraic K-theory or its variants (G-theory, KH-theory, étale K-theory,…). We will overview some aspects of the subject, ranging from the original cycle complexes of Bloch, through Voevodsky’s work over fields, to more recent p-adic developments in the arithmetic context where perfectoid and prismatic techniques appear.

Prerequisites: Algebraic geometry, sheaf theory, cohomology. Comfort with derived techniques such as descent and the cotangent complex would be helpful. Casual familiarity with K-theory, cyclic homology, and their variants would be motivational. Infinity-categories and spectra will appear, though probably not in a very essential way.

Kirsten Wickelgren (Duke University)

Title: An introduction to A^1 homotopy theory using enumerative examples


Morel and Voevodsky's A^1 homotopy theory imports tools from algebraic topology into the study of schemes, or in other words, into the study of the solutions to polynomial equations. This theory produces greater understanding of arithmetic and geometric aspects these solutions. We will introduce some of this theory using as a guide questions such as "How many lines meet 4 lines in 3-space?"  

Background: some algebraic topology or algebraic geometry, for example as described in Hatcher's and Hartshorne's books

Week 2 - July 19-23, Illustrating Mathematics

The Illustrating Mathematics PCMI Graduate Summer School aims to erase the artificial divide between research and outreach in mathematics. It is impossible to “find” a mathematical idea without explaining it; exploration and exposition are two sides of the same coin. One striking example of this is the epochal work of William Thurston; often his theorems were accompanied by pictures and computer programs, illustrating the underlying ideas. 

We will bring together mathematicians from a range of fields, and practitioners from the digital arts (animation, 3D printing, laser cutting, CNC routing, virtual reality, computer games, etc) to share their expertise in mathematics and with the procedural tools used to illustrate mathematics. The school is targeted both to graduate students and to mathematics community members from all backgrounds. In addition to online lectures from a variety of mathematicians and practitioners, our school will consist of several workshops to train participants in a variety of digital media.

Click HERE for Illustrating Mathematics program details.

Plenary Speakers

Ingrid Daubechies

Vernelle Noel


Traditional CAD/CAM Software for Mathematical Models (Glen Whitney)

This workshop will introduce and provide a tutorial on the capabilities of traditional CAD/CAM drawing tools, a class of software that may be unfamiliar to many in the mathematics and mathematical illustration community. The first session will be joint with Edmund Harriss’
workshop on Mathematical CNC Toolpaths, so that participants can sample both aspects of CAD/CAM in mathematical contexts and select one track or the other for subsequent sessions.

The particular perspective of creating materials for mathematical outreach will serve as this workshop’s vehicle for introducing the tools. It will comprise (at least) one 2D project and one 3D project. In particular, participants will model a modular-origami-style unit and produce cut files to allow the automatic production of interlockable units on consumer machines (such as the Silhouette Cameo or Cricut Explore). Sample cut units of their designs will be sent to all participants following the workshop.

As a 3D project, participants will develop rigid hubs that can be used with drinking straws to assemble one or two polyhedra of their choice. (For example, the acute and obtuse golden rhomboids that can tile space aperiodically make a nice target.) Time permitting, we will add features such as internal magnets to allow the resulting models to attach to one another. The resulting hub designs can be exported in formats suitable for 3D printing or even injection molding.

The workshop will use only freely-available, open-source software. There are no prerequisites beyond high-school solid geometry; in particular, no familiarity with CAD/CAM or any specific software is assumed.

Mathematical CNC toolpaths (Edmund Harriss)

CNC has a rather high barrier to entry, but has a lot of mathematical potential. It is a fantastic example in Linear algebra (a machine comes with its own basis!) multivariate calculus and many advanced geometry courses. In addition as the objects created are made of wood (or brick, or metal, though that will not be covered here) they often become artworks in their own right becoming compelling paths into mathematical ideas.

Participants will get to create lines and paths and see the effects when these get cut into wood. On the first four days they will create a cut file that will be cut in the afternoon, so they can see the results the next day. On Friday there will be a virtual show of the completed work.

Drawing Julia sets (Arnaud Cheritaut)

We will review techniques for drawing Julia sets, including classical algorithms from the 80’s and more recent ones, and a few tricks from the teacher. We will focus on quality and the workshop will include some general advice about images. Prerequisites include knowledge of complex numbers, basics of programming. The goal is for students to be able to draw their own Julia sets, and we have specific projects in mind.

Blender for mathematicians: a beginner’s workshop (Remi Coulon)

Blender is an open source 3D graphics software, popular among digital artists for its organic virtual sculpting environment and advanced animation tools. This workshop is intended for beginners. No prerequisite in computer graphics is needed. During the sessions, participants will learn how to use native structures (meshes, bezier curves, nurbs, etc) along with Python scripts to build and modify objects. We will also cover the use of “nodes” to control features in Blender, allowing one to interact with models by dynamically changing control parameters. Finally, participants will learn how to export their models for 3D printing, animate a simple scene, and render it with lights and textures.

Our first meeting will be held jointly with the Grasshopper Workshop, so participants can understand the differences between these computer graphics packages.

Illustrating Mathematics with Grasshopper, a visual scripting platform for Rhino 3D (Dave Bachman)

Rhino 3D is the predominant CAD tool in the field of Architecture, as it allows one to build models using precise construction tools, as well as by scripting with several programming languages. For this reason, it has also become popular among Mathematicians for creating mathematical models. Grasshopper is an auxiliary program which allows one to build objects in Rhino by an easy to use “visual" scripting language. Grasshopper code has several advantages over traditional scripting. Most notably, it allows for the ability to isolate a specific element of the model being built and dynamically interact with and modify it in a large variety of ways. Participants in this workshop will learn the basics of model building with Grasshopper. Depending on time and interest, we may also explore more advanced topics such as popular plug-ins to enhance the functionality of Grasshopper (e.g. Anemone for looping, Kangaroo for physics simulation, etc.), interfacing with external scripts, optimizing code for parallel processing, etc.

Our first meeting will be held jointly with the Blender Workshop, so participants can understand the differences between these computer graphics packages.

Graphic Novels Can Teach Mathematics (Audrey Nasar)

In the United States, graphic novels are increasingly being used for scholastic purposes across the curriculum as supplements or replacements for traditional textbooks. In particular, there are a number of graphic novels that explore mathematical concepts in algebra, calculus, statistics, and even graduate level mathematics. Some examples include Apostolos Doxiadis and Christos H. Papadimitriou’s Logicomix: An Epic Search for Truth, Andrew Granville and Jennifer Granville’s Prime Suspects: The Anatomy of Integers and Permutations, Larry Gonick’s Cartoon Guide to series, and The Manga Guide to series.

In this minicourse, we will explore how graphic novels and comics can be used to teach mathematics. The sessions will begin with an introduction to the graphic medium followed by a study of several mathematically themed graphic novels, which participants will analyze according to their narrative approach, graphic style, and pedagogical content. In the sessions that follow, participants will choose mathematical concepts and then work on creating a storyline, developing characters, and drawing thumbnails for a short graphic novel (2-3 pages) to teach the concept. Participants will share and critique each others’ ideas and then revise and finalize their work.

Visualizing Mathematical Structures with Processing (Roger Antonsen)

We look at how mathematical structures, in particular discrete structures and sets of finite objects, can be visualized, and interactively experimented with, in Processing, a open-source, easily accessible computer programming language and environment based on Java. We start from scratch and look at how to implement and visualize structures like relations, functions, sets, equivalence classes, groups, and graphs.

Interactive videos for teaching and visualization (Yuri Sulyma)

Participants will learn to create videos that viewers can manipulate/interact with, with a focus on mathematical content. (See https://liqvidjs.org and https://epiplexis.xyz for examples.) Topics include: using TeX in videos, 2d graphs, 3d graphics, coding demos, animation, and other forms of interactivity.

Participants are strongly advised to learn as much as they can about HTML, CSS, frontend JavaScript, React, and TypeScript (in that order) in advance of the workshop. These topics will be reviewed, and help will be available for them, but the focus will be on using the Liqvid software which is built on top of them. We also recommend coming with a concrete idea (ideally even a rough script) for a video you want to work on throughout the week. Contact yuri_sulyma@brown.edu with questions.

Week 3 - July 26-30, Number Theory Informed by Computation

Tim Dokchitser, Unversity of Bristol

Title: Inverse Galois Problem

The Inverse Galois Problem asks whether every finite group occurs as a Galois group over Q. This question and its variant over Q(t) have driven a lot of research, and they are very interesting from a computational perspective as well. There are a variety of methods of constructing Galois groups, and they draw techniques from algebraic geometry, p-adic numbers, the theory of finite groups, Kummer theory, braid groups, group cohomology, and elliptic and modular curves, to name a few. In these lectures I will try to give an overview of the area, and explain where we stand. We will see how the various methods actually work for specific small groups, and get a feel for how far they can be pushed and what are the obstacles to solve the problem in general.

Standard courses on Galois theory and on algebraic number theory are essential, and familiarity with local fields and/or algebraic curves would be helpful. For the theoretical side of what we will cover, Serre's book "Topics in Galois theory" (available online) is an excellent source.

Lecture notes, codes, exercises and problem sets can be found here: https://people.maths.bris.ac.uk/~matyd/InvGal/

Kristin Lauter, Microsoft Research

Title: Supersingular isogeny graphs in cryptography

Abstract coming soon.

Hendrik Lenstra, Universiteit Leiden

Title: Algorithms in algebraic number theory
TA: Daan van Gent, Universiteit Leiden


This minicourse is about algorithms for computational problems that one encounters in algebraic number theory. The emphasis is not on actual computations, but on issues of mathematical interest that arise in the design of efficient algorithms; here "efficient" is generally taken to mean "polynomial-time". We shall discuss problems for which such algorithms are known, and techniques for circumventing problems for which no efficient algorithms are known. Special attention will be given to the computation of rings of integers, and to the computation of power residue symbols and Artin symbols. The virtual 2021 minicourse will cover introductory and preparatory material.

basic algebra: groups, rings, fields; elementary algebraic number theory: prime ideal factorization, class numbers, unit groups; understanding of algorithms and their complexity, polynomial-time algorithms; lattice basis reduction (which will possibly be covered in the 2022 course by Joe Silverman).