Weekly Bulletin

Given an effective divisor D in projective space P^n (equipped with its usual symplectic form), we obtain a distinguished subset, the skeleton, L, which has many interesting symplectic properties. When D is smooth and of degree >1, L is known to be a barrier, meaning that the Gromov width of P^n-L is strictly smaller than that of P^n. We can ask if a similar result holds if D is not smooth. This talk will give a partial answer to this question. We will see how to obtain an upper bound for the Gromov width of P^n-L for a certain class of non-smooth divisors D using neck-stretching, which is tight in the case of a generic arrangement of at least n+1 hyperplanes.

The failure of De Giorgi - Nash - Moser theory in the vectorial setting opens to the study of partial regularity, namely smoothness of solutions outside a negligible, singular set. This is a classical phenomenon for systems and harmonic maps. I will discuss recent advances on nonlinear integro-differential systems, with particular emphasis on the structure and size of the singular set. From recent, joint work with Giuseppe Mingione (Parma) and Simon Nowak (Bielefeld).

Complete embedded minimal surface with integrable Gauss curvature such as the plane and the catenoid are fundamental objects in geometry. In this talk, I will show that the asymptotic slope of such a surface is bounded from below in an optimal way by a systolic quantity called the neck-size. A consequence of this inequality is a new characterization of the catenoid purely in terms of its extrinsic properties. This result confirms a conjecture of G. Huisken and can be viewed as an analog in extrinsic geometry of the Riemannian Penrose inequality in mathematical relativity. The proof is based on an analysis of so-called minimal capillary surfaces, which are compact minimal surfaces that intersect a given complete embedded minimal surface with integrable Gauss curvature at a constant angle. This is joint work with M. Eichmair.

<p><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">There are intimate relations between the geodesic flow and Brownian motion on hyperbolic surfaces (for instance, Hopf-Tsuji-Sullivan theorem and the expression of harmonic measure). In this talk, I will show how we can use Brownian motion to express the lengths of closed geodesics, length of orthogeodesics, zeta-regularized determinant of the Laplace-Beltrami operator of a hyperbolic surface. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a hyperbolic surface and that of the same surface with an arbitrary number of additional cusps. This is mainly based on a joint work with Yuhao Xue (IHES). </span></span></p>

<p>Incidence geometries are a powerful and general framework to axiomatically describe classical geometric objects, in terms of relations between objects of different types. Important cases of incidence geometries are induced by point -- line geometries, i.e. geometries determined by two sets \({\cal P}\neq\emptyset\) and \(\cal L\) (called respectively points and lines) and an incidence relation such that and for any two given distinct points there is at most a line which is incident with them both. In a point-line geometry, we say that two points \(p\) and \(q\) are collinear (written \(p\perp q\)) if there is a line \(\ell\in{\cal L}\) such that both \(p\) and \(q\) are incident with \(\ell\).</p> <p>In this talk we shall focus mostly on \(\Gamma\)-geometries i.e. point--line geometries where given \((p,\ell)\in{\cal P}\times{\cal L}\) we have either \(\forall q\in\ell: p\perp q\) or \(|\{ q\in\ell: p\perp q\}|\leq 1\). Projective spaces, polar spaces as well as their grassmannians are all \(\Gamma\) geometries.</p> <p>Next, we shall discuss the notion of projective embeddings as providing useful "concrete" models for geometries, by representing their points as varieties embedded in a suitable projective space.</p> <p>By regarding embedded geometries as  projective systems it is possible to construct codes which usually admit large automorphism groups and sport a rich "local" structure which can be used in order to perform efficient decoding.</p> <p>In this talk, I will report on the general constructions and then focus on the case of polar Grassmannians and of the representation of flag geometries of a given geometry.</p> <p>All of these results are joint work with Ilaria Cardinali from University of Siena.</p> <p> </p> <p><strong>References</strong></p> <p>[1] Ilaria Cardinali and Luca Giuzzi, <em>Codes and caps from orthogonal Grassmannians, </em>Finite Fields and their Applications <strong>24</strong> (2013) 148–169.<br>[2] ________, <em>Enumerative Coding for Line Polar Grassmannians with Applications to Codes</em>, Finite Fields and their Applications <strong>46 </strong>(2017) 107–138.<br>[3] ________, <em>Linear codes arising from the point-hyperplane geometry – part I: the Segre </em><em>embedding,</em> Finite Fields and their Applications <strong>111</strong> (2026) 102766.<br>[4] ________, <em>Linear codes arising from the point-hyperplane geometry – part II: the twisted </em><em>embedding,</em> <a href="https://arxiv.org/abs/2507.16694" target="_blank" rel="noopener">arXiv:2507.16694</a> (2025).<br>[5] ________, <em>On minimal codes arising from projective embeddings of point–line geometries</em>, <a href="https://arxiv.org/abs/2511.22747" target="_blank" rel="noopener">arXiv:2511.22747</a> (2025).</p>

A very old and natural question in topology is how much information about a space X is contained in its singular homology H_*(X). It is well known that homology by itself is not a complete invariant: different spaces can have exactly the same homology groups without being homotopy equivalent. The situation becomes much more interesting once additional structure is taken into account. Instead of passing immediately to homology, one can look at the full chain complex C_*(X) together with the cross product, which is closely related to the cup product on cochains. This extra structure encodes subtle interactions between chains that are lost on homology, and it is responsible for familiar phenomena such as Massey products and Steenrod operations. In this talk, we discuss the result that singular cochains, when viewed as an E_infinity coalgebra, actually form a complete invariant of a space: the homotopy type of X can be recovered from this structure in a functorial way. This generalizes classical rational homotopy theory as well as deep results of Mandell, Yuan and Bachmann--Hahn. More generally, we explain how the result is based on a classification of "perfect" E-infinity coalgebras. All of this is joint work with F. Riedel.

<p>The talk focuses on the integration of composite DNA alphabets and rank modulation, two promising paradigms for high-density DNA-based data storage. Composite DNA leverages the inherent redundancy of synthesis by representing positions as mixtures of nucleotides, while rank modulation utilizes the relative ordering of these motifs to provide robustness against signal variations. We first address this combination through a framework of fixed-length permutations subject to ranking errors measured by Kendall’s tau distance. For this setting, we establish the channel capacity and present a construction for sequences of ranked symbols using Tensor Permutation Codes (TPC). We then extend the theory to a more physically faithful model involving variable-length permutations, specifically addressing insertion and deletion (indel) errors occurring at the "tail" of the ranking, the lower-frequency motifs. Within this paradigm, we establish a theoretical equivalence between tail deletion, insertion, and indel codes, and provide optimal constructions for both individual symbols and sequences via Tail Tensor Permutation Codes (TTPC). Together, these results offer a comprehensive theoretical and practical framework for leveraging rank-modulated composite symbols in error-resilient DNA storage systems.</p>

This talk is based on a joint work with Elie Aidékon (Shanghai). Attach to each edge of the complete graph on $n$ vertices, i.i.d. exponential random variables with mean $n$. Aldous (1998) proved that the longest path with average weight below $p$ undergoes a phase transition at $p=\frac{1}{e}$: it is $o(n)$ when $p<\frac{1}{e}$ and of order $n$ if $p>\frac1e$. Later, Ding (2013) revealed a finer phase transition around $\frac{1}{e}$: there exist $c'>c>0$ such that the length of the longest path is of order $\ln^3 n$ if $ p \le \frac{1}{e}+\frac{c}{\ln^2 n}$ and is polynomial if $p\ge \frac{1}{e}+\frac{c'}{\ln^2 n}$. We identify the location of this phase transition and obtain sharp asymptotics of the length near criticality. The proof uses an exploration mechanism mimicking a branching random walk with selection introduced by Brunet and Derrida (1999).

Non-linear statistical inverse problems pose major challenges both for statistical analysis and computation. Likelihood-based estimators typically lead to non-convex and possibly multimodal optimization landscapes, and Markov chain Monte Carlo (MCMC) methods may mix exponentially slowly. We discuss recent progress in devising both statistical and global polynomial-time computational guarantees in such settings. In particular, we will discuss a class of computationally tractable estimators--plug-in and PDE-penalized M-estimators--for inverse problems defined through parametric PDEs. These estimators arise from conditionally convex and, in many PDE examples, nested quadratic optimization formulations which avoid evaluating the forward map G(f) and do not require PDE solvers. For prototypical non-linear inverse problems arising from elliptic PDEs, such as the well-known Darcy model, we prove that these estimators attain the best currently known statistical convergence rates while being globally computable in polynomial time. In the Darcy model, we obtain novel sub-quadratic o(N^2) arithmetic runtime bound for estimating f from N noisy samples. Our analysis is based on new generalized stability estimates, extending classical stability beyond the range of the forward operator, combined with tools from nonparametric M-estimation. Our estimators also provide principled warm-start initializations for polynomial-time Bayesian computation. Based on the preprint https://arxiv.org/abs/2601.09007

We will consider coarse correlated equilibria (CCE) in continuous time mean field games. CCEs are generalizations of Nash equilibria, where a moderator (aka correlation device) privately recommend strategies to the players that are not convenient to unilaterally reject. We will provide a linear programming approach through the notion of relaxed strategies in the same spirit as the works by Kurtz and Stockbridge, which have been recently extended to mean field games in several papers by Bouveret, Dumitrescu, Leutscher and Tankov. Within such a relaxed setting and under some regularity assumptions, we will show existence of an optimal CCE with respect to a fixed objective for the moderator. Finally, we will propose an equivalent Lagrangian formulation and a primal-dual algorithm to compute an optimal CCE numerically. This talk is based on a joint project with F. Cannerozzi and I. Tzouanas.