Weekly Bulletin

https://www.math.uzh.ch/mat074

<p align="justify"> Various algebraic phenomenons in homogeneous dynamics have non algebraic counterparts. For example, the equidistribution of Hecke neighbors can be seen as a non-algebraic counterpart of Duke's Theorem. In this talk, we consider non-algebraic counterparts of the p-Adic Littlewood Conjecture and of Duke's Theorem for subcollections. <BR> One of the non-algebraic counterparts of the p-Adic Littlewood Conjecture involves unboundedness of the A-orbits of arbitrary choices of p-Hecke neighbors of a lattice as p goes to infinity along the primes. <BR> We prove, using expanders, a bootstrap argument and the equidistribution of Hecke neighbors, that the set of exceptions for this conjecture has Hausdorff dimension strictly smaller than 1 in [0,1] (where we assign lattices to points in [0,1]). Moreover, we discuss evidence for the conjecture in some cases using GRH. <BR> <BR> This talk is based on a joint ongoing work with Erez Nesharim from the Technion. </p>

I will prove that the field of meromorphic functions on a Stein compact set of dimension n (for instance on the closed unit ball in C^n) has cohomological dimension n. The main tool is an extension to the Stein setting of Artin's theorem comparing étale and singular cohomology. I will give applications of these results to sums of squares problems in analytic geometry.

The isoperimetric problem is to find the figure in the plane having the greatest area among all those with the same perimeter. This problem and its apparently obvious solution, the circle, date back to at least Ancient Greece but for a rigorous proof we have to wait until the 19th century. We will overview the history of the isoperimetric inequality and sample some of its evolutions in recent times.''

The design of secure post-quantum digital signatures is a particularly important and current topic, especially considering the presence of initiatives such as NIST's call for proposals. While lattice-based designs offer intriguing solutions (some of which are about to be standardised) NIST itself expressed the desire for alternatives, based on different security assumptions. Code-based signatures are historically challenging to design, due to the intrinsic nature of the Hamming metric, and the syndrome decoding problem; however, a recent approach exploiting the notion of code equivalence offers an interesting alternative. In this talk, we briefly summarise the state of the art, introduce the LESS signature scheme, and then present recent developments which greatly contribute to making it one of the most promising code-based signature schemes in literature.

Group actions are fundamental mathematical tools, both for classical cryptography with discrete logarithm and for post-quantum cryptography, such as isogeny-based and code-based ones. They have received a lot of interest from the cryptographic community, who are also attracted by the possibility of defining additional functionalities over standard primitives. However, different families of group actions may differ significantly in their core characteristics, so some works usually focus on specific schemes, usually with abelian acting groups like CSI-FiSh. In this talk, we will see some additional functionalities for general cryptographic group actions, particularly the one arising from isomorphism problems in coding theory used in LESS and MEDS signature schemes, such as a threshold implementation and different commitment design strategies.

We discuss various models of random walks with a reinforced memory originating from the well-known Elephant Random Walk. We concentrate on models with a linear reinforcement mechanism, where the weight of a step is increased by an additive factor if the step is remembered, making it therefore likelier to repeat the step again and again in the future. We will also discuss the counterbalanced versions of these walks.

The polaron model describes an electron interacting with a polarizable crystal which is modelled by a nonrelativistic continuous quantum field. If the interaction between the electron and the field is strong, it is known that the ground state energy is to leading order given by the ground state energy of the semiclassical polaron model, where the field is treated as a classical variable. In this talk, we give a detailed description of the full low-energy spectrum of the (confined) polaron by providing arbitrarily high corrections to the semiclassical energy. More precisely, we present an asymptotic series expansion for every low-energy eigenvalue in inverse powers of the coupling constant. Towards the end of the talk, we will discuss what is known about the low-energy spectrum of the non-confined translation-invariant polaron, in particular, the existence of excited bound states at fixed total momentum. The talk is based on joint works with M. Brooks, K. Mysliwy and R. Seiringer.

Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense between zeros at fixed pre-specified times. Their additive randomization allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomized arcade processes can thus be interpreted as a generalization of anticipative stochastic bridges. The filtrations generated by these processes are utilized to construct a class of martingales which interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally Markov randomized arcade processes, the dynamics of FAMs are informed through Bayesian updating. FAMs can be connected to martingale optimal transport (MOT) by considering optimally coupled target random variables. Moreover, FAMs allow to formulate an information-based martingale optimal transport problem, which enables the introduction of noise in MOT, in a similar fashion to how Schrödinger's problem introduces noise in optimal transport. This information-based transport problem is concerned with selecting an optimal martingale coupling for the target random variables under the influence of the noise that is generated by an arcade process.

Matrix games are the most basic problem in Game Theory, but robustness to small perturbations is not yet fully understood. A perturbed matrix game is one where the entries depend on a parameter which varies smoothly around zero. We introduce two new concepts: (a) value-positivity if, for every sufficiently small error, there is a strategy that guarantees the value of the error-free matrix game; and (b) uniform value-positivity if there exists a fixed strategy that guarantees, for every sufficiently small error, the value of the error-free matrix game. While the first concept captures the dependency of optimal strategies to small perturbations, the second naturally arises where the data is uncertain and a strategy is sought which remains optimal despite that uncertainty. In this paper, we provide explicit polynomial-time algorithms to solve these two problems for any polynomially perturbed matrix game. For (a), we further provide a functional form for the error-dependent optimal strategy. Last, we translate our results into robust solutions for LPs.