Most algebraic structures can be given a lattice structure. For instance, any R-module defines a lattice where the meet and the join operations are given by the intersection and the sum of modules. Furthermore, any R-module morphism gives rise to a usual lattice morphism between the corresponding lattices. Actually, these two correspondences comprise a functor from the category of

R-modules to the category of complete modular lattices and usual lattice morphisms. However, this last category does not summon some basic algebraic properties that modules have (for example, the first theorem of isomorphism). With this in mind, we consider the category of linear modular lattices and linear morphisms, where we extend the notions of preradicals, and thus, describe the big lattice of lattice preradicals. In the process, we define some isomorphic structures to such lattice of lattice preradicals.

Surprising connections have recently emerged between two very active and previously independent research domains: random permutations and random geometry. This talk will uncover these connections, showing how random geometric objects can be directly used to reconstruct universal limits for random permutations.

We will illustrate this new general theory through concrete examples of Baxter permutations and monotone meanders, helping the audience build intuition. In the last part of the talk, we will explain how similar ideas led us to a new conjecture for the scaling limit of uniform random meanders and share progress on this long-standing open problem.

I will present two results on the formation and the interior structure of black holes in general relativity.
The first result proves that extremal (zero temperature) black holes can form dynamically in gravitational collapse. This provides a definitive disproof of the ”third law of black hole thermodynamics.” This is joint work with Ryan Unger (Princeton).
The second result concerns black hole interiors and the Strong Cosmic Censorship conjecture due to Penrose. This conjecture asserts the deterministic character of general relativity. I will present work in the context of a negative cosmological constant that shows that whether determinism holds or not surprisingly depends on the Diophantine properties of the black hole geometry.

In this work, we present an overview and comparison of spectral bundle methods for solving both primal and dual semidefinite programs (SDPs). In particular, we introduce a new family of spectral bundle methods for solving SDPs in the primal form. The algorithm developments are parallel to those by Helmberg and Rendl, mirroring the elegant duality between primal and dual SDPs. The new family of spectral bundle methods achieves linear convergence rates for primal feasibility, dual feasibility, and duality gap when the algorithm captures the rank of the dual solutions. The original spectral bundle method by Helmberg and Rendl is well-suited for SDPs with low-rank primal solutions, while on the other hand, our new spectral bundle method works well for SDPs with low-rank dual solutions. These theoretical findings are supported by a range of large-scale numerical experiments.