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1. On the v -Picard Group of Stein Spaces

   https://doi.org/10.1093/imrn/rnae185  

   Ertl, Veronika; Gilles, Sally; Nizioł, Wiesława (September 2024, International Mathematics Research Notices)

   Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $$\mathbf{Z}_{p}$$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $$1$$. 

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2. Effective entropy of quantum fields coupled with gravity

   https://doi.org/10.1007/JHEP10(2020)052  

   Dong, Xi; Qi, Xiao-Liang; Shangnan, Zhou; Yang, Zhenbin (October 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe. 

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3. Minimal volume entropy of free-by-cyclic groups and 2-dimensional right-angled Artin groups

   https://doi.org/10.1007/s00208-021-02211-9  

   Bregman, Corey; Clay, Matt (April 2021, Mathematische Annalen)

   null (Ed.)

   Let G be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to G has minimal volume entropy equal to 0. In the nonvanishing case, we provide a positive lower bound to the minimal volume entropy of an aspherical simplicial complex of minimal dimension for these two classes of groups. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2-dimensional groups with uniform uniform exponential growth. This criterion is shown by analyzing the fiber π1-growth collapse and non-collapsing assumptions of Babenko-Sabourau. 

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4. De Sitter microstates from T$$ \overline{T} $$ + Λ2 and the Hawking-Page transition

   https://doi.org/10.1007/JHEP07(2022)140  

   Coleman, Evan; Mazenc, Edward A.; Shyam, Vasudev; Silverstein, Eva; Soni, Ronak M.; Torroba, Gonzalo; Yang, Sungyeon (July 2022, Journal of High Energy Physics)

   A bstract We obtain microstates accounting for the Gibbons-Hawking entropy in dS 3 , along with a subleading logarithmic correction, from the solvable T $$ \overline{T} $$ T ¯ + Λ 2 deformation of a seed CFT with sparse light spectrum. The microstates arise as the dressed CFT states near dimension ∆ = c/ 6, associated with the Hawking-Page transition; they dominate the real spectrum of the deformed theory. We exhibit an analogue of the Hawking-Page transition in de Sitter. Appropriate generalizations of the T $$ \overline{T} $$ T ¯ + Λ 2 deformation are required to treat model-dependent local bulk physics (subleading at large central charge) and higher dimensions. These results add considerably to the already strong motivation for the continued pursuit of such generalizations along with a more complete characterization of T $$ \overline{T} $$ T ¯ type theories, building from existing results in these directions. 

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5. Quantile Stein Variational Gradient Descent for Batch Bayesian Optimization

   Gong, Chengyue; Peng, Jian; Liu, Qiang (January 2019, international conference on machine learning)

   Batch Bayesian optimization has been shown to be an efficient and successful approach for black-box function optimization, especially when the evaluation of cost function is highly expensive but can be efficiently parallelized. In this paper, we introduce a novel variational framework for batch query optimization, based on the argument that the query batch should be selected to have both high diversity and good worst case performance. This motivates us to introduce a variational objective that combines a quantile-based risk measure (for worst case performance) and entropy regularization (for enforcing diversity). We derive a gradient-based particle-based algorithm for solving our quantile-based variational objective, which generalizes Stein variational gradient descent (SVGD). We evaluate our method on a number of real-world applications and show that it consistently outperforms other recent state-of-the-art batch Bayesian optimization methods. Extensive experimental results indicate that our method achieves better or comparable performance, compared to the existing methods. 

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