## To cheer you up in difficult times 22: some mathematical news! (Part 1)

To cheer you up, in these difficult times, here are (in two parts) some mathematical news that I heard in personal communications or on social media. (Maybe I will write later, in more details,  about few of them that are closer to combinatorics.)

## Mathematics and the real world

### Zvi Artstein  The pendulum under vibrations revisited

Zvi (Zvika) was one of the first people that I met in 1971 at the university as a high school student and he gave me a lot of good advice and help. Here he shed light on an old mystery!

Abstract: A simple intuitive physical explanation is offered, of the stability of the inverted pendulum under fast violent vibrations. The direct description allows to analyze, both intuitively and rigorously, the effect of vibrations in similar, and in more general, situations. The rigorous derivations in the paper follow a singular perturbations model of mixed slow and fast dynamics. The approach allows applications beyond the classical inverted pendulum model.

I first heard about the stability of the inverted pendulum under fast violent vibrations from Sylvia Serfaty in connection with my study of “forehead-football.” See this post. Let me mention with pride that Endre Szemeredi, who is a fan of football in general (and a capable player), is also a fan of my forehead-football idea.

### Imre Bárány, William Steiger, and Sivan Toledo the cocked hat

Abstract (arXive): We revisit the cocked hat — an old problem from navigation — and examine under what conditions its old solution is valid.

This is a great story, read it in The Journal of Navigation, or on the arXive.

## Rigidity

### Katie Clinch, Bill Jackson, Shin-ichi Tanigawa: Maximal Matroid Problem on Graphs

In mid January, we had a great lecture by Shin-ichi Tanigawa (University of Tokyo). I certainly plan to blog about it. Here is the abstract.

The problem of characterizing the 3-dimensional generic rigidity of graphs is one of the major open problems in graph rigidity theory. Walter Whiteley conjectured that the 3-dimensional generic matroid coincides with a matroid studied in the context of bivariate splines.  In this talk I will show a solution to the characterization problem for the latter matroid.

I will explain the idea of our characterization from the view point of constructing maximal matroids on complete graphs. Specifically, for a graph H, a matroid on the edge set of a complete graph is called an H-matroid if every edge set of each subgraph isomorphic to H is a circuit. A main theme of my talk will be about identifying and constructing a maximal H-matroid with respect to the weak order. This talk is based on a joint work with Bill Jackson and Katie Clinch.

Here is the video

The papers on the arxive

## Polytopes

### Ardila and Escobar:  The harmonic polytope

Abstract: We study the harmonic polytope, which arose in Ardila, Denham, and Huh’s work on the Lagrangian geometry of matroids. We show that it is a (2n2)-dimensional polytope with $latex (n!)^2 (1+1/2++1/n)$ vertices and $latex 3^n3$ facets. We give a formula for its volume: it is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

These polytopes look truly great!

Federico Ardila, Laura Escobar, The harmonic polytope

## Percolation

I am thankful to Itai Benjamini who told me about these results.

### At last: Rotation invariance theorem for planar percolation for the square grid.

Rotational invariance in critical planar lattice models

This is very big news: twenty years after Smirnov’s result for the triangular grid, finally rotational invariance is proved for critical percolation on the square grid.

Authors: Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Mendes Oulamara

Abstract:  In this paper, we prove that the large scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight 1q4 exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce from this result that the correlations of the Potts models with q{2,3,4} colors and of the six-vertex height function with Δ[1,1/2] are rotationally invariant at large scales.

### Noise sensitivity for planar percolation without Fourier

Endre Szemeredi sometimes say that we should try to avoid the use of his regularity lemma “at all costs”. Similarly, we should try to avoid Fourier tools if we can. Vincent and Hugo could!

Noise sensitivity of percolation via differential inequalities, Vincent Tassion and Hugo Vanneuville

Abstract: Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten’s proof of scaling relations. We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals 2/3 a.s.

### Plaquette Percolation on the Torus by  Paul Duncan, Matthew Kahle, and Benjamin Schweinhart

Harry Kesten famously proved that the critical probability for planar percolation is 1/2. Planar duality is crucial.  Many people expected or speculated that a similar statement for $2d$-dimensions holds if we replace “connectivity” by some statement about $d$ dimensional homology. This is what the new paper does.

(Next part: Topology, graph theory, algebra, Boolean functions, and Mathematics over the media.)

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