Weekly Bulletin

Model theory is a fast growing branch of mathematical logic with deep interactions with algebra, algebraic geometry, combinatorics, and, more recently, topological dynamics. I will focus on a few interactions with topological dynamics and applications to additive combinatorics. I will discus type-definable components of definable groups, which lead to model-theoretic descriptions of Bohr compactificatios of groups and rings, and also to so-called locally compact models of approximate subgroups and subrings which in turn are crucial to get structural or even classification results about approximate subgroups and subrings. I will discuss my result that each approximate subring has a locally compact model, and mention some structural applications. In contrast to approximate subrings, not every approximate subgroup has a locally compact model. However, Ehud Hrushovski showed that instead it has such a model in a certain generalized sense (with morphisms replaced by quasi-homomorphisms). In order to do that, he introduced and developed local logics and definability patterns. In my recent paper with Anand Pillay, we gave a shorter and simpler construction of a generalized locally compact model, based on topological dynamics methods in a model-theoretic context. I will briefly discuss it, if time permits.

In recent years, DNA-based storage systems have emerged as promising solutions for long-term data archival due to their high data density and durability. This talk explores the intricacies of DNA storage, highlighting its advantages and challenges. DNA-based storage possesses distinct characteristics that distinguish it from conventional storage mediums. Synthesized DNA strands contain inherent redundancy due to replication during synthesis and sequencing processes, yet their unordered arrangement introduces complexity to data retrieval. Additionally, synthesis and sequencing introduce errors such as insertions, deletions, and substitutions, posing significant challenges to data integrity. This talk will begin with an examination of the background of DNA storage and the challenges posed by the DNA storage channel. Throughout the presentation, we will emphasize several open problems related to the channel, underscoring the need for innovative solutions to advance DNA storage technology. To bridge theory with application, we will discuss two recent research projects. The first focuses on the DNA coverage depth problem, which aims to reduce both, cost and latency in DNA storage. We will investigate how error-correcting codes and retrieval algorithms impact the sequencing coverage depth and explore how this coverage can be optimized. Additionally, we will explore a scalable end-to-end DNA storage pipeline, integrating deep neural networks and tensor-product based error-correcting codes, significantly improving speed, accuracy, and code rate. These insights contribute to advancing the efficiency and feasibility of DNA storage solutions, setting the stage for future advancements.

Although contact geometry has its origins in the 19th century, it wasn't until the 1970s that it began to be studied via topological methods. More recently, in 2002, the Giroux correspondence theorem established, in dimension 3, a close relationship between contact manifolds and open book decompositions (i.e fibered links). This means that properties of contact 3-manifolds can be studied by studying properties of mapping classes of surfaces. In this talk I will start by providing an overview to contact geometry in dimension 3, and introducing one of the most relevant properties, the dichotomy between tight and overtwisted contact structures. This can be studied, by a result of Honda-Kazez-Matic, via right-veering diffeomorphisms of surfaces. I will show that this property is not easy to detect in general before presenting a combinatorial way of detecting it.

Code-based cryptosystems are promising candidates for post-quantum cryptography, such as the McEliece and Niederreiter cryptosystems. Their security is based on the difficulty of decoding a general linear code. In this thesis, we study the security of algebraic geometry codes in code-based cryptography. These codes include very well-known families like generalized Reed-Solomon, BCH and Goppa codes. Furthermore, they can be described using certain tools, e.g. the theory of algebraic function fields, which we will focus on.

More information: https://ethz.ch/en/news-and-events/events/details.triangles-quadrilaterals-pentagons.70122.htmlcall_made

Exceptional floods, i.e. flood events with magnitudes or spatial extents occurring only once or twice a century, are rare by definition. Therefore, it is challenging to estimate their frequency, magnitude, and future changes. In this talk, I discuss three methods that enable us to study exceptional extreme events absent in observational records thanks to increasing sample size: stationary and non-stationary stochastic simulation, reanalysis ensemble pooling, and single-model initialized large ensembles. I apply these techniques to (1) study the frequency of widespread floods, (2) quantify future changes in spatial flood extents, (3) estimate the magnitude of floods happening once or twice a century, and (4) shed light on the relationship between future increases in extreme precipitation and flooding. These applications suggest that simulation approaches that substantially increase sample size provide a better picture of flood variability and help to increase our understanding of the characteristics, drivers, and changes of exceptional extreme events.

We study the invariant Gibbs measure of mean-field interacting diffusions and prove optimal global and local rates of convergence to its thermodynamic limit in the full sub-critical regime of temperatures \(T>T_c\) for a large class of potentials. Our proof relies on a non-asymptotic Sanov-type upper bound for the global rate (which is of independent interest itself) combined with an application of Stein's method for the local rate. We also apply these techniques to prove sharp exponential concentration inequalities for i.i.d empirical measures in negative Sobolev norms. This is joint work with Matías G. Delgadino (U. T. Austin).

This work aims to study an extension of the celebrated Sannikov’s Principal-Agent problem to the multi-Agents case. In this framework, the contracts proposed by the Principal consist in a running payment, a retirement time and a final payment at retirement. After discussing how the Principal may derive optimal contracts in the N-Agents case, we explore the corresponding mean field model, with a continuous infinity of Agents. We then prove that the Principal’s problem can be reduced to a mixed control-and-stopping mean field problem, and we derive a semi-explicit solution of the first best contracting problem. This is a joint work with Thibaut Mastrolia and Nizar Touzi.

The Cohen-Lenstra heuristic studies the distribution of the p-part of the class group of quadratic number fields for odd prime p, and Gerth’s conjecture regards the distribution of the 2-part of the class group of quadratic fields. The main difference between these two conjectures is that while the (odd) p-part of class group behaves completely “randomly”, the 2-part of class group does not since the 2-torsion of the class group is controlled by the genus field. In this talk, we will discuss a new conjecture generalizing Cohen-Lenstra and Gerth’s conjectures. The techniques involve Galois cohomology and embedding problems of global fields. If time permits, we will also discuss how to prove a function field analog of this new conjecture, by counting points on the Hurwitz spaces.