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1. The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

   Gong, Sherry; Wu, Jianchao; Yu, Guoliang (August 2024, Geometric and functional analysis)

   We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite-dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov’s theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes’ theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms. 

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2. Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

   https://doi.org/10.1093/imrn/rnz305  

   Benson, Brian; Ralli, Peter; Tetali, Prasad (December 2019, International Mathematics Research Notices)

   Abstract We study the volume growth of metric balls as a function of the radius in discrete spaces and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the 1st author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, $$\lambda _2$$ of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the 2nd eigenvalue (i.e. the 1st nonzero eigenvalue). We also describe a method for proving Buser’s Inequality in graphs, particularly under a lower bound assumption on curvature. 

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3. Asymptotic additivity of the Turaev–Viro invariants for a family of 3-manifolds

   https://doi.org/10.1112/jlms.12655  

   Kumar S., Melby J. (October 2022, Journal of the London Mathematical Society)

   In this paper, we show that the Turaev–Viro invariant volume conjecture posed by Chen and Yang is preserved under gluings of toroidal boundary components for a family of 3-manifolds. In particular, we show that the asymptotics of the Turaev–Viro invariants are additive under certain gluings of elementary pieces arising from a construction of hyperbolic cusped 3-manifolds due to Agol. The gluings of the elementary pieces are known to be additive with respect to the simplicial volume. This allows us to construct families of manifolds which have an arbitrary number of hyperbolic pieces and satisfy an extended version of the Turaev–Viro invariant volume conjecture. 

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4. Neural FIM for learning Fisher information metrics from point cloud data.

   Fasina, O.; Huguet, G.; Tong, A.; Zhang, Y.; Wolf, G.; Nickel, M.; Adelstein, I.; Krishnaswamy, S (January 2003, Proceedings of the 40th International Conference on Machine Learning)

   Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM’s utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells). 

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5. Existentially closed W*-probability spaces

   https://doi.org/10.1007/s00209-022-03038-z  

   Goldbring, Isaac; Houdayer, Cyril (August 2022, Mathematische Zeitschrift)

   Abstract We study several model-theoretic aspects of W $$^*$$ ∗ -probability spaces, that is, $$\sigma $$ σ -finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W $$^*$$ ∗ -spaces and prove several structural results about such spaces, including that they are type III $$_1$$ 1 factors that tensorially absorb the Araki–Woods factor $$R_\infty $$ R ∞ . We also study the existentially closed objects in the restricted class of W $$^*$$ ∗ -probability spaces with Kirchberg’s QWEP property, proving that $$R_\infty $$ R ∞ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III $$_1$$ 1 factors forms a $$\forall _2$$ ∀ 2 -axiomatizable class. We show that for $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , the class of III $$_\lambda $$ λ factors is not $$\forall _2$$ ∀ 2 -axiomatizable but is $$\forall _3$$ ∀ 3 -axiomatizable; this latter result uses a version of Keisler’s Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III $$_\lambda $$ λ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) , there is a family of pairwise non-elementarily equivalent III $$_\lambda $$ λ factors of size continuum. While we cannot prove the same result for III $$_1$$ 1 factors, we show that there are at least three pairwise non-elementarily equivalent III $$_1$$ 1 factors by showing that the class of full factors is preserved under elementary equivalence. 

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