Weekly Bulletin

Entov-Polterovich's celebrated symplectic big fiber theorem says that any smooth map from a closed symplectic manifold to an Euclidean space with Poisson commuting components has at least one Hamiltonian non-displaceable fiber. I will discuss contact analogues of this theorem that we proved with Yuhan Sun and Igor Uljarevic using symplectic cohomology with support. Unlike the symplectic case, the validity of the statements require conditions on the closed contact manifold. One such condition is to admit a Liouville filling with non-zero symplectic cohomology. In the case of Boothby-Wang (pre-quantization) contact manifolds, we prove the result under the condition that the Euler class of the circle bundle is not an invertible element in the quantum cohomology of the base symplectic manifold. I will also explain how to obtain Givental's Legendrian rigidity result in the standard contact real projective spaces as an application.

We will use Representation Theory to calculate systematically and efficiently the topological invariants of compact Lie groups and homogeneous spaces. Most of the talk is covered by our second paper on ArXiv with John Jones and Adam Thomas, who are both at Warwick. The paper is a part of the ongoing project to study the topological invariants of the four exceptional Rosenfeld projective planes.

I will discuss the regularity of solutions to a class of semilinear free boundary problems in which admissible functions have a topological constraint, or spanning condition, on their 1-level set. This constraint forces the 1-level set, which is a free boundary, to behave like a surface with singularities, attached to a fixed boundary frame, in the spirit of the set-theoretic Plateau problem. Two such free boundary problems that have been well-studied are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn approximation of the set-theoretic Plateau problem. We establish optimal Lipschitz regularity for solutions, and analytic regularity for the free boundaries away from a codimension two singular set. We further characterize the singularity models for these problems as conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems. This is joint work with Mike Novack and Daniel Restrepo.

This talk is about the Manin-Mumford conjecture, which was proven by M. Raynaud in 1983. The required theory will be presented, including abelian varieties. The first and simplest version of the theorem is as follows: Given a nonzero \(f\in\mathbb{C}[X,Y]\), there exist only finitely many pairs \((q_1,q_2)\in\mathbb{Q}\) with \(f(\mathrm{e}^{2\pi\mathbf{i}q_1},\mathrm{e}^{2\pi\mathbf{i}q_2})=0\), unless there is a very special reason: \(f\) contains a factor of the form \(X^nY^m-\mathrm{e}^{2\pi\mathbf{i}q}\) or \(X^n-Y^m\mathrm{e}^{2\pi\mathbf{i}q}\).

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.

In the derivation of the kinetic equation from the cubic NLS, a key feature is the invariance of the Schrödinger equation under the action of U(1), which allows the quasi-resonances of the equation to drive the effective dynamics of the correlations. In this talk, I will give an example of equation that does not enjoy such type of invariance and show that the exact resonances always take precedence over quasi-resonances. As a result, the effective dynamics is not of kinetic type but still nonlinear and non-trivial. I will present the problem, the ideas behind the derivation of the effective dynamics and some elements of the proof. This is based on a soon-to-appear work in collaboration with de Suzzoni (Ecole Polytechnique) and Touati (CNRS and Université de Bordeaux).

We study the situation of optimizing artificial neural networks (ANNs) with the rectified linear unit activation via gradient flow (GF), the continuous-time analogue of gradient descent. Under suitable regularity assumptions on the target function and the input data of the considered supervised learning problem, we prove that every non-divergent GF trajectory converges with a polynomial rate of convergence to a critical point. The proof relies on a generalized Kurdyka-Lojasiewicz gradient inequality for the risk function. Furthermore, in a simplified shallow ANN training situation, we show that the GF with suitable random initialization converges with high probability to a good critical point with a loss value very close to the global optimum of the loss.