This post is my third report from ICM 2018. You may remember that I had planned to live-blog on the last four days of the congress but on Monday evening I realized that this was an unrealistic task and decided instead to blog only on a single day – Monday. A little later I realized this was also unrealistic and decided to limit my blogging to a single lecture by Peter Kronheimer and Tomasz Mrowka on Knots, three-manifolds and instantons. But in the end I did not live-blog at all and in this post I will briefly described Tuesday’s morning lectures and give a belated report on Peter and Tomasz’s lecture. I actually arrived in Rio very early on Saturday morning and attended many talks on Saturday, but this was not part of my planned blogging and I was relaxed and did not take lecture notes. Sunday was a day off and I had a great time with friends.

I greatly enjoyed all of Monday’s morning lectures. The first very inspiring lecture by my friend and colleague Rafi (Ronald) Coifman was entitled Harmonic analytic geometry in high dimensions – Empirical models (click for the video). Rafi’s research spans across a wide range of areas many of which he himself created and goes from the very applied (e.g., applications of harmonic analysis to pluming, biology, and finance) to the very pure (e.g., applications of wavelets to classical problems in harmonic analysis). The lecture covered a lot of ground, starting with Fourier’s original ideas and his perception that he had discovered the “language of nature” and continuing with wide applications to structural and multi-scale analysis of high dimensional data, and to the possibility, pushing Fourier’s vision one step further, of automatically learning the laws of physics from data.

Toward the third lecture on the history of mathematics by Catherine Goldstein I thought that I could relax and listen to a historical lecture that does not require much mathematical efforts. To my surprise, it was very demanding for me (but fully worth the effort) to follow the mathematics itself. The historical discussion and insights were great. The title of the lecture was Long-term history and ephemeral configurations (click for the video) and it started with a famous quote of Poincaré: Mathematics is the art of giving the same name to different things (Poincaré gave the examples of “groups” and “uniform convergence”.) At the center of the talk was Charles Hermite and the lecture dealt, among other things, with the very interesting question: Is mathematics a natural science? For Hermite the answer was: Yes! Altogether there were a lot of great insights and great lines. (Pictures from these two lectures at the end of the post.)

## Kronheimer and Mrowka: Knots, three-manifolds and instantons

Knots, three manifolds and instantons

The talk was fantastic, it had great results, the slides were great, the presentation was great, thoughtful, with a lot of food for thought, both for the large audience and (I think) also for experts. A main famous theorem by the speakers is:

**Theorem:** Knots with vanishing Instanton Floer homology and (therefore) also knots with trivial Khovanov’s homology are unknots.

Khovanov’s homology are invariants of groups that refine the famous Jones polynomials and, of course, two problems naturally arise. First, is it the case that the Jones polynomial itself determines unknots? (This is a famous open problem.) And also does Khovanov’s homology or Floer’s homology distinguish different knots? (Maybe the answer for the second bold question is known to be negative…) The lecture had four parts

### Part I (Tomasz): knots, Papakyriakopoulos, and the main theorem.

I was surprised that I had the feeling that I understood everything in the first part. It started with a quick pictorial introduction to what knots are, then looking at the complement of a knot, followed by Dehn’s lemma that was proved by Papakyriakopoulos. (I think but am not sure that Papakyriakopoulos’s proof is still needed for all the stronger results that follow.)

So Papakyriakopoulos’ theorem tells you that the fundamental group of the complement of a non-trivial knot is not Abelian, but could we say something stronger? Peter mentioned that for most, but not all non-trivial small knots the fundamental group maps onto a dihedral group. And the main result is that for all non trivial knots the fundamental group maps onto SO (3). There were two delicate yet important points that were mentioned. The first is that often SO(3) can or should be replaced by its double cover SU(2), and the second is that there is also a crucial condition (that makes the theorem stronger) about the images of “meridians” (small circles around a point on the knot in its complement).

### Part II (Peter): Floer’s instanton homology and many mathematical ideas and tools

The second part was about ideas, notions and tools needed for the proof of the main theorem. Naturally it was more difficult and for various things I only just pleasantly got a general impression together with some pointers on notions that I should (finally) learn. Connections, flat connections, Chern-Simons functionals, Young-Mills equations and their solutions called “instantons”, and the Floer’s (instanton) homology, … . As you can see from the fourth slide the list of tools that are actually needed for the proof extends even further and Peter and Tomasz also mentioned connections with Ozsváth and Szabo theory of Heegard Floer homology.

### Part III (Tomasz): Khovanov homology, and skein relations

Surprisingly, the third part dealt with notions that were somewhat easier for me than those of the second part. The Khovanov homology is a refinement of another famous knot-invariant the Jones polynomial.

I remember hearing a few talks about Khovanov homology in the early 2000. Dror Bar Nathan showed how they appear very naturally and how it is a straight forward matter to compute them (alas, not efficiently). In a different talk some years later David Kazhdan showed how, taking a different point of view, those invariants depend on a sequence of amazing miracles. In any case, the Khovanov homology groups are finite dimensional and the Jones polynomial are just the alternating sums of their dimensions (or Euler characteristics). Like the Jones polynomial themselves there is also some connection (“skein relations“) between the Khovanov homologies of knots when you apply two simple operations on the knots.

The skein relations for Khovanov homology are given in terms of a long exact sequence, and similar relations hold for the Floer homology. Moreover there is some relation (a spectral sequence) between these two exact sequences which shows that when Khovanov homology is not trivial then Floer (instanton) homology is also non trivial and hence from what we already know about Floer homology the knot is not trivial.

and when part IV came I expected that the discussion will be aimed at real experts in the audience and that I could relax and think about other things. However this was not the case. Below the fold I will tell you about the surprising fourth part, and then proceed to talk about various other really interesting things. Statistics tell me that only about a third of the readers read below the fold but this time I truly recommend it.

### Part IV (Peter): A new approach to the four color theorem!

Next Peter talked about spacial graphs, trivalent graphs to be precise. And when he talked about Tait’s coloring I could see that the aim is to give a new approach for proving the four color theorem. Tait’s colorings are coloring of the edges of trivalent graphs so that the three edges incident to every vertex are colored with the three different colors. (In other words, no two edges that share a vertex have the same color.) A reformulation of the 4CT is that every planar bridgeless trivalent graph has a Tait coloring. In view of the non vanishing theorems of Peter and Tomasz (extended to spacial graphs), for a proof of the four color theorem what is left to be proved is a simple equality:

**Conjecture (Kronheimer and Mrowka):** for (bridgeless) graphs embedded in the plane (hence planar) the dimension of the Floer instanton homology (actually a certain variant called **J**) is equal to the number of Tait colorings.

This is a very nice approach. K&M proved that for spacial graphs embedded in the plane the number of Tait’s coloring is at most the dimension of **J. **This inequality is expected by them to extend to all spacial graphs.

Two remarks on the conjecture:

a) This is an equality, if true it should be provable, no? ( 🙂 ) Of course the conjecture may well be false. (This conjectural equality implies the 4CT but as far as we know is not a consequence of it, which is also a good sign for the approach.)

b) This equality is proven for bipartite trivalent graphs which reflects a natural test for 4CT approaches. (Can it be proved for bridgeless trivalent Hamiltonian planar graphs?)

c) For more on this part of the lecture see P&T’s papers Tait colorings, and an instanton homology for webs and foams; and Exact triangles for SO(3) instanton homology of webs.

More comments and links:

a) The Kronheimer-Mrowka theorem was used by Greg Kuperberg in his 2012 proof that under GRH, telling if a knot is non-trivial is in NP. An earlier 2002 (completely different) proof was described by Ian Agol. (This is a great result in computational complexity and as far as I can see the two proofs represent a rare meeting point of “two cultures” in 3-dimensional topology.) See this post. Joel Hass, Jeff Lagarias, and Bill Thurston proved that telling if a knot is trivial is in NP. (Update: in 2016 Mark Lackenby gave an unconditional proof that telling if a knot is knotted is in NP. )

b) The four color theorem. Both before and after the amazing 1976 proof by Appel and Haken of the 4CT, the four color theorem was surely on the minds of combinatorialists and not only them. Let me come back to the 4CT in some future post but here are some related earlier posts on coloring simple polytopes and triangulations, another on coloring circles and pseudocircles (I, II), an MO question generalization of the 4CT and a GLL post. **Update:** I forgot that one of the answers to my MO question by Ian Agol is actually about Peter and Tomasz’ Instanton homology conjecture.

c) Christos Dimitriou Papakyriakopoulos (papa), was a great mathematical hero. John Milnor wrote a famous poem

**The perfidious lemma of Dehn**

**Was every topologist’s bane **

**‘Til Christos D. Pap- **

**akyriakop-**

**oulos proved it without any strain.**

According to this post, Gina’s cat was named after Papakyriakopoulos. (Gina was the hero of a book that I wrote.)

d) In the first part Tomasz mentioned dihedral representations (pursued by Fox). The fundamental group of most small knots maps onto a dihedral group (with only 22 out of 3000 exceptions). It would be nice to understand it for some model of random knots. (See the paper models of random knots by Chaim Even-Zohar, and references there including to the recent “Pataluma model” of Even-Zohar, Joel Hass, Nati Linial, and Tahl Nowik.)

e) Peter Guthrie Tait (from Tait colorings) conjectured that every planar trivalent bridgeless graph is Hamiltonian which was disproved by Tutte. (This conjecture easily implies the 4CT.) He also made several exciting and important conjectures about knots that were all proved.

## Pictures from Coifman and Goldstein’s lectures

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thank you