Weekly Bulletin

<p><span style="caret-color: #ffffff; color: #000000; 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;">We describe generic points for linear flows with bounded </span><span style="caret-color: #ffffff; color: #000000; 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;">type slope on the  infinite staircase. </span><span style="caret-color: #ffffff; color: #000000; 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;">As an application we characterize bounded type rotation numbers for </span><span style="caret-color: #ffffff; color: #000000; 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;">which the error terms for Kronicker discrepancy </span><span style="caret-color: #ffffff; color: #000000; 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;">are uniformly distributed. A key step in our proof is the local limit </span><span style="caret-color: #ffffff; color: #000000; 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;">theorem for inhomogeneous Markov chains </span><span style="caret-color: #ffffff; color: #000000; 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;">in the regime of large deviations. Based on a joint work with Omri Sarig.</span></p>

<p>How do cryptography and real-world constraints meet in an e-voting system? This talk examines how cryptographic protocols are translated into a real-world e-voting system, involving legal requirements, operational procedures, and human factors. The main focus of the talk is the e-voting protocol currently deployed by some Swiss cantons to enable citizens to securely cast their votes online in legally binding elections. We then move from the present to the near future, detailing the new protocol under development and highlighting how it is expected to improve both the security and the verifiability of the system.<br>Throughout the talk, we keep an eye on practical constraints: how the Federal Ordinance on electronic voting shapes trust assumptions and transparency requirements, and how operational procedures influence the generation of election parameters and the verification of protocol steps. We also address usability and accessibility, discussing how the voting process can be made understandable and verifiable for all voters. We close by outlining how external experts and independent implementations contribute to the continuous evaluation and analysis of the protocol.</p>

The spectral gap of the Hodge Laplacian on coexact 1-forms is a fundamental quantity associated to a Riemannian manifold which has attracted quite a lot of attention in recent years. I will begin by discussing the role it plays in topological problems about three-manifolds (such as the existence on taut foliations on rational homology spheres) via its relation with Floer theoretic invariants. Motivated by this, I will then focus on the case of hyperbolic manifolds, and describe techniques to determine explicitly such spectral gaps in concrete examples (including certain infinite families). This is joint work with M. Lipnowski.

<p>Given two linear codes, the Code Equivalence Problem asks to find (if it exists) an isometry between them. In this talk we introduce linear code equivalence and its main variants (permutation, monomial). We then discuss its crucial role as a hardness assumption in post-quantum digital signatures, with particular focus on LESS and related schemes, such as LEAST. Finally, we present new cryptanalysis techniques based on power codes that efficiently solve permutation and linear code equivalence for low dimension codes.</p>

I will review some well-known results and examples on the decomposition of normal currents into rectifiable currents, both for one-dimensional currents and for currents of codimension one. In the second part of the talk, I will outline an example due to A. Schioppa of a normal 2-current supported on a purely 2-unrectifiable set, presenting an alternative proof. Based on joint work with G. Alberti and A. Marchese.

In this talk, I will present a methodology called the feature space decomposition and apply it to the statistical analysis of several estimators such as min ell2 norm interpolant estimators, ridge estimators, spectral methods, minimum ellq-norm interpolant estimators and max-margin classifiers. We establish high-probability non-asymptotic upper bounds for the excess risk of these estimators. In the case of min ell2 norm interpolant estimators, ridge estimators and spectral methods, we show that these bounds are sharp. In particular, we obtain necessary and sufficient conditions for benign overfitting behavior of the min ell2 norm interpolant estimators. Our results rely on a method called feature space decomposition where the self-regularization properties of minimum norm interpolant estimator is highlighted. Technically, we circumvent tools such as the convex min-max theorem, instead employing tools from Geometric Aspects of Functional Analysis, in particular, the Dvoretzky-Milman theorem plays a central role in our analysis. This provides a geometric perspective on benign overfitting, and crucially, our techniques remain valid beyond the Gaussian case. Consequently, we obtain benign overfitting results with high probability when the design vector is not necessarily Gaussian. We particularly emphasize that the feature space decomposition method may potentially refine the uniform convergence approach, suggesting its promise as a new methodology in mathematical statistics. Based on joint works with Radoslaw Adamczak, George Gavrilopoulos, Zhifan Li, Zong Shang and Marta Strzelecka.

In this talk, I will present a methodology called the feature space decomposition and apply it to the statistical analysis of several estimators such as min ell2 norm interpolant estimators, ridge estimators, spectral methods, minimum ellq-norm interpolant estimators and max-margin classifiers. We establish high-probability non-asymptotic upper bounds for the excess risk of these estimators. In the case of min ell2 norm interpolant estimators, ridge estimators and spectral methods, we show that these bounds are sharp. In particular, we obtain necessary and sufficient conditions for benign overfitting behavior of the min ell2 norm interpolant estimators. Our results rely on a method called feature space decomposition where the self-regularization properties of minimum norm interpolant estimator is highlighted. Technically, we circumvent tools such as the convex min-max theorem, instead employing tools from Geometric Aspects of Functional Analysis, in particular, the Dvoretzky-Milman theorem plays a central role in our analysis. This provides a geometric perspective on benign overfitting, and crucially, our techniques remain valid beyond the Gaussian case. Consequently, we obtain benign overfitting results with high probability when the design vector is not necessarily Gaussian. We particularly emphasize that the feature space decomposition method may potentially refine the uniform convergence approach, suggesting its promise as a new methodology in mathematical statistics. Based on joint works with Radoslaw Adamczak, George Gavrilopoulos, Zhifan Li, Zong Shang and Marta Strzelecka.

We explain some models coming from mathematical modelling of social interactions. In these models, vertices of a graph hold an opinion, taking for instance values in [0,1], and interact with their neighbours provided that the opinions they hold do not differ too much. We give some results and present open conjectures for one of these models, namely the Deffuant model. We then give some rigorous results for two variants, namely the compass model and the averaging process. The results are based on joint works with Markus Heydenreich and Timo Vilkas. No prerequisites will be assumed.

During the days of early experiments on the Birch--Swinnerton-Dyer conjecture, Tate produced a table for elliptic curves over p-adic fields that beautifully summarises their arithmetic invariants. I will present an analogous set of tables for curves of genus 2. The invariants addressed include: reduction type of the minimal regular model, Neron component group of the Jacobian, conductor exponent, and the valuation of the discriminant of a minimal Weierstrass model. This is joint work with Edwina Aylward, Lilybelle Cowland Kellock and Elvira Lupoian.