The Degree-Three Bounded Cohomology of Complex Lie Groups of Classical Type

We establish Monod's isomorphism conjecture in degree-three bounded cohomology for every complex simple Lie group of classical type. Our main ingredient is a bounded-cohomological stability theorem with an optimal range in degree three that we bootstrap from previous stability results by the author and Hartnick. The bootstrapping procedure relies on the occurrence in our setting of a variant of the recently observed phenomenon of secondary stability in the sense of Galatius--Kupers--Randal-Williams.Comment: Added norm computations and relevant references, and modified introduction and general structure for better readability. 33 pages. Comments welcom

A Quillen stability criterion for bounded cohomology

We provide a version of Quillen's homological stability criterion for continuous bounded cohomology. This criterion is exploited in the companion paper (arXiv:2201.03879) in order to derive new bounded cohomological stability results for various families of classical groups.Comment: This is an extended and revised version of parts of (arXiv:2201.03879), which has been split into two parts for better legibilit

Algorithm-assisted discovery of an intrinsic order among mathematical constants

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of $\zeta(3)$. Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.Comment: 21 pages, 6 figures, and 1 table; with 9 appendix sections totaling 12 pages, 1 figure, and 4 table

Evaluation of the relationship between effervescent paracetamol and blood pressure: clinical trial

Background: Paracetamol's solubility is achieved by adding to the excipient sodium salts, either as bicarbonate, carbonate or citrate. As the relationship between salt and hypertension is well known, due to the sodium content it has raised a hypothesis that may interfere with the control of that risk factor. Therefore, the objective of this study is to evaluate the effect on blood pressure of effervescent paracetamol compared to non-effervescent, in hypertensive patients. Methods/Design: This is the protocol of a phase IV multicenter clinical trial, randomized, controlled, crossover, open, which will compare the effect of two different formulations of paracetamol (effervescent or non-effervescent) in the blood pressure of hypertensive patients, with a seven weeks follow up. 49 controlled hypertensive patients will be included (clinical BP lower than 150 and 95 mmHg, and lower than 135 mmHg and 85 mmHg in patients with diabetes or a history of cardiovascular event, and daytime ambulatory measurements lower than 140 and 90 mmHg) and mild to moderate pain (Visual Analog Scale between 1 and 4). The study was approved by the ethics committee of the Fundació Jordi Gol i Gurina and following standards of good clinical practice. The primary endpoint will be the variations in systolic BP in 24 h Ambulatory Blood Pressure Monitoring, considering significant differences 2 or more mmHg among those treated with non-effervescent and effervescent formulations. Intention-to-treat and per-protocol analysis will be held. Discussion: Despite the broad recommendation not to use effervescent drugs in patients with hypertension, there are relatively little studies that show exac tly this pressor effect due to sodium in salt that gives the effervescence of the product. This is the first clinical trial designed to study the effect of effervescence compared to the non-effervescent, in well-controlled hypertensive patients with mild to moderate pain, performed in routine clinical practic

On Bounded-Cohomological Stability for Classical Groups

Bounded cohomology, introduced independently and in different contexts by Johnson, Trauber, and Gromov in the late seventies and early eighties, is a very rich invariant of groups that can detect some of their coarse geometric features. Due to a lack of general computational tools, bounded cohomology remains in general obscure as of today. In the late nineties, Burger and Monod introduced the notion of continuous bounded cohomology, a suitable generalization for topological groups. Apart from having several interesting applications, this theory has remarkably shed light on the bounded cohomology of higher-rank lattices, which is determined to some extent by the continuous bounded cohomology of their ambient semisimple Lie groups. The relationship can be fully exploited in degree two: Being isomorphic to their second continuous cohomology, the second continuous bounded cohomology of any connected semisimple Lie group with finite center (or more generally, of any connected, simply connected, semisimple algebraic group over a local field) is completely understood. This fact and a few examples of groups in this class for which the isomorphism holds in degrees three or four, gave rise to the so-called isomorphism conjecture. Usually attributed to Dupont and Monod, it states that the isomorphism known in degree two should also hold in every degree. The ultimate goal of this thesis is to add further evidence to this conjectural picture: We prove the isomorphism conjecture in degree three for the family of complex symplectic groups. This will be obtained as corollary of the bounded-cohomological stability along said family. One says that continuous bounded cohomology is stable along an infinite nested sequence of topological groups if it is eventually constant in every degree. We develop here a machinery that gives bounded-cohomological stability along any sequence of locally compact, second-countable groups, provided that there exists a sequence of complexes on which the respective groups act. It is based on an original idea of Quillen in the setting of group homology, and on an ad hoc treatment in continuous bounded cohomology by Monod for the families of general and special linear groups. Our method improves Monod's stability range in degree three for special linear groups over non-Archimedean fields. Upon constructing a family of complexes that serves as an input to the aforementioned machinery, the so-called symplectic Stiefel complexes, we then prove stability of continuous bounded cohomology along the families of real and complex symplectic groups. While the stability range produced is insufficient to prove the isomorphism conjecture in degree three, we complete its proof in the complex case via a bootstrapping procedure. Based on a computation by Bucher-Burger-Iozzi, we moreover determine the Gromov norm of a generating class of the third continuous bounded cohomology. Inspired on the situation in continuous cohomology, continuous bounded cohomology is expected to be stable along all families of classical split groups over local fields (indexed by the rank). We conclude by explaining how our methods should extend to other families

Stabilization of Bounded Cohomology for Classical Groups

We show that bounded cohomology stabilizes along sequences of classical real and complex Lie groups, and along sequences of lattices in them. Our method is based on a stability criterion which adapts Quillen's method to the setting of bounded cohomology. This criterion is then applied to a family of measured complexes, the so-called Stiefel complexes, associated to any vector space endowed with a non-degenerate sesquilinear form.Comment: 48 pages. This article supersedes arXiv:1902.0138

Evaluation of the relationship between effervescent paracetamol and blood pressure: clinical trial

Background: Paracetamol's solubility is achieved by adding to the excipient sodium salts, either as bicarbonate, carbonate or citrate. As the relationship between salt and hypertension is well known, due to the sodium content it has raised a hypothesis that may interfere with the control of that risk factor. Therefore, the objective of this study is to evaluate the effect on blood pressure of effervescent paracetamol compared to non-effervescent, in hypertensive patients. Methods/Design: This is the protocol of a phase IV multicenter clinical trial, randomized, controlled, crossover, open, which will compare the effect of two different formulations of paracetamol (effervescent or non-effervescent) in the blood pressure of hypertensive patients, with a seven weeks follow up. 49 controlled hypertensive patients will be included (clinical BP lower than 150 and 95 mmHg, and lower than 135 mmHg and 85 mmHg in patients with diabetes or a history of cardiovascular event, and daytime ambulatory measurements lower than 140 and 90 mmHg) and mild to moderate pain (Visual Analog Scale between 1 and 4). The study was approved by the ethics committee of the Fundació Jordi Gol i Gurina and following standards of good clinical practice. The primary endpoint will be the variations in systolic BP in 24 h Ambulatory Blood Pressure Monitoring, considering significant differences 2 or more mmHg among those treated with non-effervescent and effervescent formulations. Intention-to-treat and per-protocol analysis will be held. Discussion: Despite the broad recommendation not to use effervescent drugs in patients with hypertension, there are relatively little studies that show exac tly this pressor effect due to sodium in salt that gives the effervescence of the product. This is the first clinical trial designed to study the effect of effervescence compared to the non-effervescent, in well-controlled hypertensive patients with mild to moderate pain, performed in routine clinical practic