3-d Calabi--Yau categories for Teichm\"uller theory

For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic differentials on a genus $g$ surface with simple zeros and $n$ simple poles. For a generic point in the moduli space the corresponding quantum/refined Donaldson--Thomas invariants are computed in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, these counts satisfy wall-crossing formulas.Comment: v2: re-written for streamlined exposition, results unchange

Flags and tangles

We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.Comment: v2: added discussion of dualities, more detailed proofs, typos correcte

Semistability, modular lattices, and iterated logarithms

We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.Comment: v2: new introduction, typos correcte

Dynamical systems and categories

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.Comment: 35 page

Refined combinatorial torsion

Wir untersuchen eine Variante der Reidemeister- und Whitehead-Torsion von CW-Komplexen und glatten Mannigfaltigkeiten von V. Turaev. Die notwendigen algebraischen Hilfsmittel werden dabei in Analogie zu Konstruktionen von Knudsen-Mumford und Deligne in der algebraischen Geometrie entwickelt, wobei das Konzept einer monoidalen Kategorie bzw. einer 2-Gruppe eine zentrale Rolle spielt. Dies liefert einen konzeptuellen Zugang zu diversen Vorzeichenregeln.We investigate the foundations of combinatorial torsion for CW-complexes and smooth manifolds, specifically, the refined variant of V. Turaev. The necessary algebraic tools are developed in analogy to constructions by Knudsen-Mumford and Deligne in algebraic geometry, using the concept of a monoidal category, in particular 2-groups. This provides a more conceptual approach to various sign issues

Perverse schobers, stability conditions and quadratic differentials

There are multiple classes of triangulated categories arising from marked surfaces whose spaces of stability conditions are described by moduli spaces of quadratic differentials on the surfaces. We unify the approaches for describing their spaces of stability conditions and apply this to new classes of examples. This generalizes the results of Bridgeland--Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential). The novel examples include the derived categories of relative graded Brauer graph algebras. The main computational tool are perverse schobers, which allow us to relate hearts of $t$-structures to mixed-angulations of the surface and tilts of the former with flips of the latter. This is complemented by another approach based on deforming Fukaya $A_\infty$-categories of surfaces and transfers of stability conditions.Comment: 75 pages, comments welcome

On pseudo-Anosov autoequivalences

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.Comment: 35 page

Nutrients / Growth, Feeding Tolerance and Metabolism in Extreme Preterm Infants under an Exclusive Human Milk Diet

Background: For preterm infants, human milk (HM) has to be fortified to cover their enhanced nutritional requirements and establish adequate growth. Most HM fortifiers are based on bovine protein sources (BMF). An HM fortifier based on human protein sources (HMF) has become available in the last few years. The aim of this study is to investigate the impact of an HMF versus BMF on growth in extremely low birth weight (ELBW, <1000 g) infants. Methods: This was a retrospective, controlled, multicenter cohort study in infants with a birthweight below 1000 g. The HMF group received an exclusive HM diet up to 32+0 weeks of gestation and was changed to BMF afterwards. The BMF group received HM+BMF from fortifier introduction up to 37+0 weeks. Results: 192 extremely low birth weight (ELBW)-infants were included (HMF n = 96, BMF n = 96) in the study. After the introduction of fortification, growth velocity up to 32+0 weeks was significantly lower in the HMF group (16.5 g/kg/day) in comparison to the BMF group (18.9 g/kg/day, p = 0.009) whereas all other growth parameters did not differ from birth up to 37+0 weeks. Necrotizing enterocolitis (NEC) incidence was 10% in the HMF and 8% in the BMF group. Conclusion: Results from this study do not support the superiority of HFM over BMF in ELBW infants.(VLID)491874