Minerva Course Fall 2025

Here is a tentative plan for the Minerva Mini-Course at Princeton in Fall 2025. This page will be updated as the plan evolves. The goal of the lectures is to introduce some recent techniques in low-dimensional topology, and some famous old theorems and less famous new ones they can be used to prove.

The first talk will be at a gentle colloquium level; most of it should be accessible to second-year undergraduate mathematics majors (or other interested students or community members). The next three talks are aimed at advanced undergraduates or beginning graduate students interested in topology. The last three talks are intended for more advanced graduate students and postdocs.

Schedule

Talks are currently planned to be from 4:30 to 5:30 on (most) Mondays. Location TBD.

Abstracts

Here are abstracts for the first few talks; I will write abstracts for later talks once I see how far off schedule I’ve gotten.

Lecture 1. Overview.

This talk is a biased introduction to knot theory and low-dimensional topology, and an overview of the rest of the series. We will start by introducing knotted curves in 3-space and knotted surfaces in 4-space, and discuss how one can represent and (partly) visualize them. We will then discuss different notions of equivalence for knots and surfaces, and assert some theorems and examples related to the existence and uniqueness questions for surfaces. Time permitting, we will introduce the Jones polynomial, an important but still somewhat mysterious tool from the 1980s for sutdying knots. We will end by overviewing the rest of the series and mentioning some of the most famous (and active) open problems in the area.

Most of this talk should be accessible to undergraduate students who have taken a course that introduces metric spaces.

This lecture (only) will use slides (sorry).

Lecture 2. Khovanov homology and the $s$-invariant.

This talk is where the work begins. We will start by recalling the Jones polynomial of a knot, then explain its refinement to a chain complex, the Khovanov complex. We will then explain a variant of that construction that gives a number, Rasmussen’s $s$-invariant.

Lecture 3. Alexander polynomial 1 knots are slice.

We will introduce the Alexander polynomial, one of the most famous (and useful) classical invariants of knots, and then prove (modulo one or two hard black-boxed theorems) an important result of Freedman’s, that knots with Alexander polynomial 1 bound topological slice disks.

We will end with a poll of what the audience would like to hear next. In particular, as currently envisioned, the talks focus heavily on the existence question for surfaces; I will ask if the audience would like a detour to uniqueness, or to maintain our focus.

Lecture 4. Four-dimensional topology from Khovanov homology

We will apply Khovanov homology to the existence question for surfaces. We will start by showing the Rasmussen’s invariant obstructs knots from bounding low-genus surfaces, and outline some pain-free proofs that this obstruction is nontrivial. We will then combine these results with Freedman’s theorem from the previous talk (and a few related black-boxed results) to deduce that ${\mathbb{R}}^4$ admits nonstandard smooth structures.

Homework

I currently intend to suggest a few exercises in each talk, and will also post them here.

References

Some references cited / mentioned in the talks will be listed here, eventually.