Weekly Bulletin

We examine a class of semi-Riemannian manifolds that undergo smooth metric signature change—from Riemannian to Lorentzian—across a hypersurface with a transverse radical. This class includes physically mo- tivated cosmological models such as the Hartle-Hawking “no-boundary” proposal, in which the universe transitions smoothly from a Euclidean to a Lorentzian phase. We show that these manifolds admit isometric embeddings into higher-dimensional pseudo-Euclidean spaces and, in particular, prove the existence of global isometric embeddings of the canonical model into both Minkowski and Misner spaces. This framework provides a mathematical setting for studying smooth signature change and its role in higher-dimensional and cosmological models.

Consider a connected surface of finite area without boundary, properly embedded in Euclidean space. By an inequality of Leon Simon from 1993, such a surface must be compact, provided its mean curvature has bounded Lebesgue 2-norm. In 2008, Simon’s inequality was improved by Peter Topping who showed that the diameter of such a surface is in fact cotrolled by the Lebesgue 1-norm of its mean curvature. Over the past decade, Topping’s diameter bound has inspired various generalizations in differential geometry and geometric measure theory. In this talk, I will give an overview of these developments and present an application to Plateau’s problem.

A very interesting limit case of the fractional Laplacian in R^d is given by \(Lu(x) := \int_{B1 (x)} \frac{u(x)−u(y)}{|y−x|^d} dy \) which serves as a principal example of a zeroth-order integro-differential op- erator. This operator arises naturally as the leading term of the logarithmic Laplacian which has been studied in recent years. In contrast with the frac- tional Laplacian, the scaling properties in this scenario are very delicate; in particular, the dilation of the kernel leads to a non-integrable tail, which rep- resents a challenge for the regularity theory of solutions of equations governed by L. In this talk, I will present interior continuity estimates for solutions to a family of operators comparable to the one above, obtained in collaboration with Alberto Saldaña and Sven Jarosh.

<div dir="ltr"><span data-olk-copy-source="MessageBody">A key aspect of Wiles’ proof of Fermat’s Last Theorem is the modularity theorem. The proof crucially relies on the deep connection between Galois representations and </span>L<span aria-hidden="true">L</span>-functions of elliptic curves. In recent years, mathematicians have continued to work on the modularity theorem, seeking various generalizations and proposing many related conjectures. In this talk, I will present an algorithm to compute the local Galois representation of an arbitrary curve via its semistable reduction.</div>

<p><span style="color: #333333;">Let Γ be a geometrically finite subgroup of G := SO(d + 1, 1) for d ≥ 1. By foundational works of Elstrodt, Patterson, Sullivan, and Lax–Phillips, we know that if the critical exponent of Γ is greater than d/2, then the Laplace–Beltrami operator on the L² space of the associated hyperbolic manifold has a spectral gap. In a representation theoretic language, this means that there exists a gap below the critical exponent for which the corresponding spherical complementary series do not occur in L²(G/Γ). It is further known that if the critical exponent is greater than d - 1, then there exists such a gap which also applies for the non-spherical complementary series, called strong spectral gap. This begs the question whether the constraint on the critical exponent can be improved to the optimal one, d/2. Inspired by prior works on the relationship between spectral gap and dynamics, in a joint work with Dubi Kelmer and Osama Khalil, we use exponential mixing of the frame flow to answer the above question in the positive.</span></p>

The classical Peierls argument establishes that percolation on a graph G has a non-trivial (uniformly) percolating phase if G has “not too many small cutsets”. Severo, Tassion, and I have recently proved the converse. Our argument is inspired by an idea from computer science and fits on one page. Our new approach also resolves a conjecture of Babson of Benjamini from 1999 and provides a much simpler proof of the celebrated result of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that percolation on any transitive graph with superlinear growth undergoes a non-trivial phase transition.

We prove that in any dimension n there exists an origin-symmetric ellipsoid {\mathcal{E}} \subset {\mathbb{R}}^n of volume c n^2 that contains no points of {\mathbb{Z}}^n other than the origin, where c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in {\mathbb{R}}^n whose density is at least cn^2 \cdot 2^{-n}. Previously known constructions of sphere packings in {\mathbb{R}}^n had densities of the order of magnitude of n \cdot 2^{-n}, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.

Differential expression analyses for single-cell RNA sequencing typically use empirical Bayes methods such as DESeq2, edgeR, limma, and MAST. These approaches perform univariate statistical testing by modeling gene expression with generalized linear models and borrow strength across genes to stabilize variance estimates. In this talk, I will introduce a framework that borrows strength also for the estimation of the gene expression itself by predicting a gene of interest from the other genes. Our R package, conformeR, combines counterfactual prediction with conformal prediction to leverage the multivariate structure of the data and increase statistical power. This is joint work with Justine Leclerc.

<p style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; caret-color: #000000; color: #000000;">In 2009, Camillo De Lellis and László Székelyhidi Jr. proved a non-uniqueness result for weak solutions to the incompressible Euler equation, through a convex integration method coming from geometry (Nash 54', Gromov 86'). A consequence of their proof is the following surprising result : the set of weak solutions of the equation is dense in $L^{\infty}_t L^2_x$ for the weak topology of this space. By generalizing the geometric framework of their approach, I will explain how to show that the set of weak solutions is path-connected for the strong topology, and I will give the main ideas of the proof of De Lellis and Székelyhidi. </p>

The calibration of mean estimates, which requires that predictions are, on average, equal to the observed responses, is a critical property for reliable decision-making, particularly in actuarial and financial applications. In this presentation, first, we review classic approaches for validating the mean-calibration and introduce the Likelihood Ratio Test (LRT) within the Exponential Dispersion Family (EDF). Next, we investigate the framework of universal inference to test the mean-calibration. We develop a sub-sampled split Likelihood Ratio Test (sub-sampled split LRT) within the EDF that provides finite-sample guarantees with universally valid critical values. We investigate type I error, power and e-power of our sub-sampled split LRT, compare the sub-sampled split LRT to the classic LRT, study the effect of sub-sampling of training and test sets on the split LRT, and propose novel test statistics based on the sub-sampled split LRT to enhance performance the test. In our numerical experiments, we demonstrate that the sub-sampled split LRT and our modifications are appealing alternatives to the classic LRT and achieve high power in detecting miscalibration, offering a practical and powerful toolkit for validating the calibration of mean estimates.

Let K be a local field with residue characteristic p>0 and X a smooth projective curve over K of genus g>=2. The general problem I want to address in this talk is the explicit computation of the semistable reduction of X. If the gonality n of X is strictly less than p, then there is a well-understood method using the theory of admissible reduction. Recently, a lot of effort has been spent on the case n=p, in particular n=p=2 (hyperelliptic curves with residue characteristic p=2). In my talk I will address the first case where n>p, namely smooth plane quartics (non-hyperelliptic curves of genus g=3) and residue characteristic p=2. This is joint work with Kletus Stern and Max Schwegele.

I will talk about how to compute the contributions from each strata in the fiber product for the Torelli pullback of the product loci calculations. I will explain some examples, the algorithm in general, and then we can have a discussion about implementation strategies.