Weekly Bulletin

I will discuss recent results on the problem of determining a Riemannian metric on a compact manifold with boundary from the set of lengths of geodesics with endpoints on the boundary and their tangent vectors at the boundary (the so-called lens data). Based on joint works with Bonthonneau, Cekic, Lefeuvre, Jezequel.

Spectral invariants defined via Embedded Contact Homology (ECH)or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics and C^0 symplectic geometry. For example, it plays a key role in the proof of the smooth closing lemma for three-dimensional Reeb flows and area preserving surface diffeomorphisms, and in the proof of the simplicity conjecture. ECH and PFH are highly sophisticated theories whose construction in particular relies on Seiberg-Witten theory. I will explain how one can use much more elementary methods (no Floer or Gauge theory) to define spectral invariants satisfying an analogous Weyl law with similar applications. I hope that this elementary perspective makes the underlying mechanisms of the symplectic Weyl law more transparent. This is based on joint work with Michael Hutchings.

The structure of the dispersion relation is one of the central aspects to the study of periodic Schroedinger operators. Besides the intrinsic interest from the viewpoint of several complex variables, the dispersion relation also carries relevant information for the spectral theory of periodic media. In particular, for the structure of spectral boundaries, isospectrality, and existence of eigenvalues for locally perturbed operators. I will discuss some of these connections as well as recent irreducibility theorems for the Bloch and Fermi varieties, focusing on the joint work with Jake Fillman and Wencai Liu (arXiv:2107.06447, J. Funct. Anal. 2022). This recent paper covers a wide class of lattice geometries in arbitrary dimension and verifies the discrete version of a conjecture of Kuchment for various models. Time allowing, I will also comment on future directions and ongoing work.

Due to the presence of the exponential nonlinearity the Liouville equation in dimension three and higher is supercritical. In particular it admits several singular solutions. We will talk about asymptotic behaviour of a family of stationary solutions and how to use it to obtain partial regularity results

We will explain how the fractional quantum Hall effect on a Riemann surface of genus g can be studied using algebraic geometry. The wave functions of charged particles have a semi-phenomenological description by Laughlin states. These states form a holomorphic vector bundle over a Picard group of the Riemann surface. The Chern characters of this vector bundle can be computed by the Grothendieck-Riemann-Roch formula. The mathematical part of the talk involves Grothendieck-Riemann-Roch computations for a universal line bundle on the symmetric power of a smooth curve over its Picard group. This is joint work with Semyon Klevtsov.

In classical combinatorics, polymatroids have been introduced as an extension of the concept of matroid. There are many known connections between linear codes and matroids and many invariant in coding theory are also matroid invariants. $q$-Matroids and $q$-polymatroids are the $q$-analogue of matroids and polymatroids. These objects have gained a lot of interest among an increasing number of researchers, especially in the last few years, due to their connection with rank-metric codes. In particular, it has been shown that to an $\mathbb{F}_{q}$-linear rank metric code $\mathcal{C}\leq \mathbb{F}_{q}^{n\times m}$ it can be associated a $q$-polymatroid $M_\mathcal{C}$ and when $\mathcal{C}$ is also $\mathbb{F}_{q^m}$-linear, $M_\mathcal{C}$ is a $q$-matroid. In this talk we will show some recent results on the invariants of $q$-polymatroids and rank-metric codes. One of these results lead to the solution of the $q$-analogue of the classical Critical Problem, proposed by Crapo and Rota. We will make use of the characteristic polynomial of a $q$-polymatroid as a basic tool for this result. Finally, we will provide the coding theoretic interpretation and we will partially solve it for \emph{maximum rank distance} codes. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

Minimal linear codes were first introduced by Cohen and Lempel over the binary field with the name of linear intersecting codes. They later gained interest due to their use in secret sharing schemes proposed by Massey. Recently, it has been shown that k-dimensional minimal linear codes are in one-to-one correspondence with strong blocking sets in a (k-1)-dimensional projective space. Strong blocking sets are sets of points such that their intersection with each hyperplane generates the hyperplane itself, and they were originally introduced as a tool for constructing covering codes. In this talk we propose a new general method to construct small strong blocking sets - and hence short minimal linear codes - starting from a set of points in a finite projective space and a graph with large vertex integrity. In particular, we explore how one can get explicit constructions of families of asymptotically good minimal linear codes, combining Ramanujan graphs and families of asymptotically good linear codes. This is a joint work with Noga Alon, Anurag Bishnoi and Shagnik Das. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

Consider the percolation problem induced by the level sets of the Gaussian free field on a supercritical Galton-Watson tree. It was proved by Abächerli and Sznitman that the associated critical parameter is positive as long as the mean offspring distribution m is at least two. We extend this result to all supercritical Galton-Watson trees, that is m>1. I will explain why the positivity of the critical parameter is typically harder to obtain in the regime 1<m<2 by comparing it to independent percolation, and present some ideas of the proof. Of particular interest will be a new exploration process of Galton-Watson trees via random walks, and its link with random interlacements and the Gaussian free field. Joint work with Alexandre Drewitz and Gioele Gallo.

In this talk I will present a rigorous construction of examples of black hole formation which are exactly isometric to extremal Reissner--Nordström after finite time. In particular, our result can be viewed as a definitive disproof of the ``third law of black hole thermodynamics.''

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead we use a block decomposition. Approximating the small submatrices by low-rank matrices and agglomerating them into a new, coarser block decomposition, we obtain a recursive method. The required computational work is O(rnm) where r is the desired rank and nxm is the size of the matrix. We discuss error estimates for A-B and M-A where A is the result of the recursive trunction applied to M, while B is the best rank-r approximation. Numerical tests show that the approximate trunction is very close to the best one.

A famous theorem of Gross-Kohnen-Zagier states that the generating series of Heegner divisors on a modular curve is a weight 3/2 modular form with values in the first Chow group. An analogous result for special divisors on orthogonal Shimura varieties was proved by Borcherds, and for higher codimension special cycles by Zhang, Raum and myself. After recalling some of these results, we report on possible extensions to special cycles on toroidal compactifications of orthogonal Shimura varieties.

Ehrhart polynomials are counting functions for integer lattice points in dilates of polyhedral objects. In this talk, I'll explain how these polynomials arise when computing tautological intersection numbers on the moduli space of pointed stable curves. In particular, it turns out that tautological intersection numbers can be organized into evaluations of Ehrhart polynomials of partial polytopal complexes. The proof of this result primarily relies on a theorem of Breuer, which classifies Ehrhart polynomials of partial polytopal complexes. At the end of the talk, I'll discuss various ways one can try to generalize this Ehrhart phenomenon.