Laura Cossu (University of Cagliari)
Geometry and Algebra of Spaces of Subrings
We investigate spaces of subrings of a commutative ring under the Zariski and patch topologies. To analyze these structures from topological, geometric, and algebraic perspectives, we introduce patch bundles, patch presheaves, and patch algebras. In particular, we show how subrings in spaces like the Zariski-Riemann space can be recovered as canonical homomorphic images or localizations of patch algebras, and we explore the sheaf-theoretic properties of this construction.
Magdalena Wiertel (Vrije Universiteit Brussel)
Indecomposable set-theoretic solutions of the Yang–Baxter equation associated with some classes of braces
The Yang–Baxter equation is one of the fundamental equations of mathematical physics that appears in various areas of pure mathematics, such as quantum groups and low-dimensional topology. It was proposed by Drinfel'd to study solutions of the set-theoretic version of this equation. We will focus on the class of involutive indecomposable solutions which can be viewed as the building blocks of the theory. A key tool in their study relies on the associated permutation group, that acts transitively on the underlying solution.
During this talk, we will show how the brace structure of the associated permutation group can be used to describe solutions whose permutation groups belong to some specific families. Based on a joint work with Andrew Darlington.
Rahinatou Njah epouse Nchiwo (Aalto University)
Strengthening the Foundations of Quantum-Resistant Encryption: From RLWE to MP-LWE
The rapid development of quantum computing devices poses a serious threat to many widely used encryption and key exchange protocols. Lattice-based cryptography (LBC) is one of the leading candidates for building secure systems that can resist attacks by quantum computers. One practical and efficient method within LBC is the Ring Learning with Errors (RLWE), which is believed to be computationally hard. The hardness of RLWE depends on the choice of a polynomial used to define the underlying algebraic structure. Some choices lead to weaker instances. However, it is unknown which choices lead to provably hard instances.
This motivates the Middle-product Learning with Errors (MP-LWE) whose hardness does not depend on the choice of a particular polynomial. Previous work showed that MP-LWE is at least as hard as RLWE for a large class of polynomials. However, this class represents only a small fraction of all polynomials of a given degree and coefficient bound.
In this work, we improve this reduction to cover a much larger class of polynomials. In our experiments, the improved reduction applies to up to $90\%$ of the irreducible polynomials with a given degree and coefficient bound. This gives us a better understanding of the security of RLWE across a much broader range of polynomial choices.
Eda Yildiz (Yildiz Technical University)
S-Zero-Divisor Graphs: Structural Reduction and Topological Characterizations
Let R be a commutative ring and S \subseteq R a multiplicatively closed subset not containing zero. In this talk, we introduce and investigate a new generalized framework called the S-zero-divisor graph of R, denoted by \Gamma_S(R). The vertex set of this graph consists of elements outside the filter ideal 0_S = \x \in R \mid sx = 0 for some }s \in S} that are annihilated by some element outside of 0_S, with classical adjacency condition xy = 0. We will demonstrate how this definition precisely captures the essential zero-divisor structure of R. By systematically excluding the elements absorbed by the filter ideal 0_S, the graph undergoes a powerful structural reduction. We will establish that \Gamma_S(R) is always an induced subgraph of the classical zero-divisor graph \Gamma(R). Furthermore, we will present complete characterizations of when \Gamma_S(R) realizes an empty, complete, or star graph. Throughout the presentation, it will be shown that these simplified topological classifications are intrinsically linked to the algebraic properties and internal decompositions of the quotient ring R/0_S.
Filippa Lo Biundo (Leeds)
Pansu pullback and spectral complexes on Carnot groups
In sub-Riemannian geometry, Carnot groups play a role analogous to that of Euclidean spaces in the Riemannian setting. Their special structure allows one to define an intrinsic notion of differentiability, namely Pansu differentiability, which in turn gives rise to the Pansu pullback on differential forms. A natural question is whether this pullback commutes with the differentials of the main complexes associated with Carnot groups. In this talk, I will exhibit counterexamples to this commutativity for the de
Rham complex and the Rumin complex, the latter being specifically adapted to the geometric structure of Carnot groups. I will then turn to a recently intro
duced family of complexes, the spectral complexes associated with the de Rham complex, and explain how, in this setting, the Pansu pullback does commute
with the corresponding differentials.
Nesibe Ayhan (University of Graz)
Well-Posedness for the Generalized Camassa-Holm Equations
The classical Camassa-Holm (CH) equation is used to describe dynamics of shallow water waves, and features interesting behavior such as solitons or wave breaking. The study of CH has been extensively investigated in the literature. In this talk, we consider a generalized version of CH where the momentum can be of arbitrarily high order and the nonlinearity can be of any polynomial order. More precisely, the equation reads as
m_t + m_x u^p + b m u^{p-1}u_x = -(g(u))_x + (b+1)u^p u_x, where m = (1-\partial_x^2)^k u,
where p \geq 1, k \geq 1, b is a real parameter, and g(u) is a smooth function. We prove local well-posedness using Kato's semigroup, where nonlinearity is treated directly using commutator estimates and the fractional Leibniz rule without having to use tricky manipulation. Furthermore, in the case where the momentum is conserved, we show that the solution is in fact global.