Weekly Bulletin

I will present my recent joint work with Samuel Kittle. We establish numerous novel explicit examples of absolutely continuous self-similar measures. In fact, we give the first inhomogenous examples in dimension 1 and 2 and construct examples for essentially any given rotations and translations, provided they have algebraic coefficients. Moreover we strengthen Varju's result for Bernoulli convolutions and Lindenstrauss-Varju's result in dimension >= 3. We also generalise Hochman's result to contracting on average measures and show that a separation condition weaker than exponential separation is sufficient.

There are two main recipes to associate to a Cohomological Field Theory (CohFT) an integrable hierarchy of hamiltonian PDEs: the first one was introduced by Dubrovin and Zhang (DZ, 2001), the second by Buryak (DR, 2015). It is interesting to notice that the latter relies on the geometric properties of the Double Ramification cycle — hence the name DR — to work. As soon as the second recipe was introduced, it was conjectured that the two had to be equivalent in some sense, and it was checked in a few examples. In the forthcoming years several papers followed, checking more examples of CohFTs, making the conjecture more precise, proving the conjecture in low genera, and eventually turning the statement of the conjecture in a purely intersection theoretic statement on the moduli spaces of stable curves. Lately, the conjecture was proved in its intersection theoretic form, employing virtual localisation techniques. (j.w.w. Blot, Rossi, Shadrin).

In many observational studies, researchers are often interested in studying the effects of multiple exposures on a single outcome. Standard approaches for high-dimensional data such as the lasso assume the associations between the exposures and the outcome are sparse. These methods, however, do not estimate the causal effects in the presence of unmeasured confounding. In this paper, we consider an alternative approach that assumes the causal effects in view are sparse. We show that with sparse causation, the causal effects are identifiable even with unmeasured confounding. At the core of our proposal is a novel device, called the synthetic instrument, that in contrast to standard instrumental variables, can be constructed using the observed exposures directly. We show that under linear structural equation models, the problem of causal effect estimation can be formulated as an ℓ0-penalization problem, and hence can be solved efficiently using off-the-shelf software. Simulations show that our approach outperforms state-of-art methods in both low-dimensional and high-dimensional settings. We further illustrate our method using a mouse obesity dataset.

Generalized Hamming Weights (GHW) have seen a big rise in popularity since Victor Wei described their many properties in 1991, linking them to code performance on the wire-tap channel of type 2. Many equivalent definitions have been proposed, including one relating them to Optimal Linear Anticodes by Ravagnani (2016): Anticodes (codes whose dimension is equal to the maximal weight) can be used as a family of test codes to determine the GHW (when the base field is not the binary field). The properties of GHW can then be inferred by the properties of the family of Anticodes.<br> In this talk, we further extend the approach to arbitrary families of test codes, focusing on a minimal set of assumptions yielding invariants with good duality properties (that is, similar to those proved by Wei for GHW). In doing so, we show that our approach is independent of the chosen metric: in particular, we recover in a unique result the duality of generalised weights in the Hamming and rank metrics. This level of generality also allows us to tackle the problem of duality of generalised weights in the sum-rank metric, by showing a first example of codes with nontrivial Hamming and rank metric parts for which the duality of generalised weights holds. Finally, we investigate the invariants obtained by using the family of Singleton-optimal codes (MDS/MRD codes) in place of Anticodes, highlighting similarities and differences between the two families that reflect on the properties of the obtained invariants. This is joint work with Elisa Gorla and Alberto Ravagnani.

In this talk I will explain how to invert matrices using an action of algebraic torus on certain algebraic varieties. Along the way, I will recall the construction of permutohedral toric variety and the space of complete quadrics, and explore the connection between them. As an application, I will present a recent polynomiality result for characteristic numbers of quadrics which was conjectured by Sturmfels and Uhler. Only very basic knowledge of algebraic geometry is needed.

SQIsign is the leading digital signature protocol based on isogenies and the only isogeny-based construction in the NIST standardisation process. In this talk, we introduce the SQIsign protocol, presenting its design and its connections to the most fundamental isogeny problems. We also discuss the many variants of SQIsign that have appeared over the last few years, with a particular focus on the recent changes that have become the round-2 NIST submission. Lastly, we discuss some recent results that provide a formal and complete proof of security of SQIsign, which fills a long-standing gap in the literature.

The talk discusses a class of Sch¨odinger operators the potentials of which are channels of a fixed profile, focusing on relations between the spectrum of such an operator and the channel geometry. We provide different sufficient conditions under which a non-straight but asymptotically straight channel gives rise to a non-empty discrete spectrum. We also address the groundstate optimalization problem in case of a loop-shaped configuration, and consider a modification of the model where the channel is replaced by an array of potential wells, each exhibiting a rotational symmetry.

In this talk, we consider large Boltzmann stable planar maps with index $\alpha\in(1,2)$. In recent joint work with Nicolas Curien and Grégory Miermont, we established that this model converges, in the scaling limit, to a random compact metric space that we construct explicitly. The goal of this presentation is to outline the main steps of our proof. We will also discuss various properties of the scaling limit, including its topology and geodesic structure.