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Convexity of multiplicities of filtrations on local rings

https://doi.org/10.1112/S0010437X23007972

Blum, Harold; Liu, Yuchen; Qi, Lu (April 2024, Compositio Mathematica)

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.
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ACC for local volumes and boundedness of singularities

https://doi.org/10.1090/jag/799

Han, Jingjun; Liu, Yuchen; Qi, Lu (July 2023, Journal of Algebraic Geometry)

The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities x ∈ ( X , Δ ) x\in (X,\Delta ) satisfies the ACC if the coefficients of Δ \Delta belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of δ \delta -plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
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Mitogenomics reveals phylogenetic relationships of Arcoida (Mollusca, Bivalvia) and multiple independent expansions and contractions in mitochondrial genome size

https://doi.org/10.1016/j.ympev.2020.106857

Kong, Lingfeng; Li, Yuanning; Kocot, Kevin M.; Yang, Yi; Qi, Lu; Li, Qi; Halanych, Kenneth M. (September 2020, Molecular Phylogenetics and Evolution)
null (Ed.)
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Dipole–phonon quantum logic with alkaline-earth monoxide and monosulfide cations

https://doi.org/10.1039/D0CP04574H

Mills, Michael; Wu, Hao; Reed, Evan C.; Qi, Lu; Brown, Kenneth R.; Schneider, Christian; Heaven, Michael C.; Campbell, Wesley C.; Hudson, Eric R. (November 2020, Physical Chemistry Chemical Physics)
null (Ed.)
Dipole–phonon quantum logic (DPQL) leverages the interaction between polar molecular ions and the motional modes of a trapped-ion Coulomb crystal to provide a potentially scalable route to quantum information science. Here, we study a class of candidate molecular ions for DPQL, the cationic alkaline-earth monoxides and monosulfides, which possess suitable structure for DPQL and can be produced in existing atomic ion experiments with little additional complexity. We present calculations of DPQL operations for one of these molecules, CaO + , and discuss progress towards experimental realization. We also further develop the theory of DPQL to include state preparation and measurement and entanglement of multiple molecular ions.
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