Lagrangian 3-torus fibrations

We prove that Mark Gross' topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual

Modular Invariant of Quantum Tori II: The Golden Mean

In our first article in this series ("Modular Invariant of Quantum Tori I: Definitions Nonstandard and Standard" arXiv:0909.0143) a modular invariant of quantum tori was defined. In this paper, we consider the case of the quantum torus associated to the golden mean. We show that the modular invariant is approximately 9538.249655644 by producing an explicit formula for it involving weighted versions of the Rogers-Ramanujan functions

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.Comment: 45 page

On Petersson's partition limit formula

For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $\chi=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $\lambda>0$ such that $\chi(\lambda)=\pm 1$. Petersson proved a beautiful limit formula for the ratio of $p_+(n)$ to $p_-(n)$ as $n\to\infty$ expressed in terms of important invariants of the real quadratic field $\mathbb{Q}(\sqrt{p})$. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Ces\`aro theorem. In this paper we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman-Erd\H{o}s

SYZ mirror symmetry for hypertoric varieties

We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution using the wall and chamber structure of the SYZ base.Comment: v_2: 31 pages, 5 figures, minor revision. To appear in Communications in Mathematical Physic

Traces of past activity in the Galactic Centre

The Milky Way centre hosts a supermassive Black Hole (BH) with a mass of ~4*10^6 M_Sun. Sgr A*, its electromagnetic counterpart, currently appears as an extremely weak source with a luminosity L~10^-9 L_Edd. The lowest known Eddington ratio BH. However, it was not always so; traces of "glorious" active periods can be found in the surrounding medium. We review here our current view of the X-ray emission from the Galactic Center (GC) and its environment, and the expected signatures (e.g. X-ray reflection) of a past flare. We discuss the history of Sgr A*'s past activity and its impact on the surrounding medium. The structure of the Central Molecular Zone (CMZ) has not changed significantly since the last active phase of Sgr A*. This relic torus provides us with the opportunity to image the structure of an AGN torus in exquisite detail.Comment: Invited refereed review. Chapter of the book: "Cosmic ray induced phenomenology in star forming environments" (eds. Olaf Reimer and Diego F. Torres

Consistent patterns of common species across tropical tree communities

D.L.M.C. was supported by the London Natural Environmental Research Council Doctoral Training Partnership grant (grant no. NE/L002485/1). This paper developed from analysing data from the African Tropical Rainforest Observatory Network (AfriTRON), curated at ForestPlots.net. AfriTRON has been supported by numerous people and grants since its inception. We sincerely thank the people of the many villages and local communities who welcomed our field teams and without whose support this work would not have been possible. Grants that have funded the AfriTRON network, including data in this paper, are a European Research Council Advanced Grant (T-FORCES; 291585; Tropical Forests in the Changing Earth System), a NERC standard grant (NER/A/S/2000/01002), a Royal Society University Research Fellowship to S.L.L., a NERC New Investigators Grant to S.L.L., a Philip Leverhulme Award to S.L.L., a European Union FP7 grant (GEOCARBON; 283080), Leverhulme Program grant (Valuing the Arc); a NERC Consortium Grant (TROBIT; NE/D005590/), NERC Large Grant (CongoPeat; NE/R016860/1) the Gordon and Betty Moore Foundation the David and Lucile Packard Foundation, the Centre for International Forestry Research (CIFOR), and Gabon’s National Parks Agency (ANPN). This paper was supported by ForestPlots.net approved Research Project 81, ‘Comparative Ecology of African Tropical Forests’. The development of ForestPlots.net and data curation has been funded by several grants, including NE/B503384/1, NE/N012542/1, ERC Advanced Grant 291585—‘T-FORCES’, NE/F005806/1, NERC New Investigators Awards, the Gordon and Betty Moore Foundation, a Royal Society University Research Fellowship and a Leverhulme Trust Research Fellowship. Fieldwork in the Democratic Republic of the Congo (Yangambi and Yoko sites) was funded by the Belgian Science Policy Office BELSPO (SD/AR/01A/COBIMFO, BR/132/A1/AFRIFORD, BR/143/A3/HERBAXYLAREDD, FED-tWIN2019-prf-075/CongoFORCE, EF/211/TREE4FLUX); by the Flemish Interuniversity Council VLIR-UOS (CD2018TEA459A103, FORMONCO II); by L’Académie de recherche et d’enseignement supérieur ARES (AFORCO project) and by the European Union through the FORETS project (Formation, Recherche, Environnement dans la TShopo) supported by the XIth European Development Fund. EMV was supported by fellowship from the CNPq (Grant 308543/2021-1). RAPELD plots in Brazil were supported by the Program for Biodiversity Research (PPBio) and the National Institute for Amazonian Biodiversity (INCT-CENBAM). BGL post-doc grant no. 2019/03379-4, São Paulo Research Foundation (FAPESP). D.A.C. was supported by the CCI Collaborative fund. Plots in Mato Grosso, Brazil, were supported by the National Council for Scientific and Technological Development (CNPq), PELD-TRAN 441244/2016-5 and 441572/2020-0, and Mato Grosso State Research Support Foundation (FAPEMAT)—0346321/2021. We thank E. Chezeaux, R. Condit, W. J. Eggeling, R. M. Ewers, O. J. Hardy, P. Jeanmart, K. L. Khoon, J. L. Lloyd, A. Marjokorpi, W. Marthy, H. Ntahobavuka, D. Paget, J. T. A. Proctor, R. P. Salomão, P. Saner, S. Tan, C. O. Webb, H. Woell and N. Zweifel for contributing forest inventory data. We thank numerous field assistants for their invaluable contributions to the collection of forest inventory data, including A. Nkwasibwe, ITFC field assistant.Peer reviewe