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1. Torus actions, maximality, and non-negative curvature

   https://doi.org/10.1515/crelle-2021-0035  

   Escher, Christine; Searle, Catherine (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n -manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we showthe Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action. 

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2. Tests of independence for pairs of paths of non-stationary Gaussian processes

   Ernst, Philip A; Viens, Frederi G; Yan, Shuo (October 2025, arXiv)

   For a pair of paths of continuous-time stochastic processes, the notion of Pearson correlation $$\rho(T)$$, over a time interval $[0,T]$, is an easy extension of the same notion of empirical correlation for two time series of fixed length $$n$$, and indeed they are explicitly tied to each other when observing the stochastic processes at a fixed time frequency (time step $=T/n$). When the processes are self-similar, such as with standard Brownian motion, or fractional Brownian motion (fBm), a simple observation leads to showing that the univariate distribution of $$\rho(T)$$ is constant as $$T$$ changes. Therefore, such highly non-stationary processes cannot be treated the same way that i.i.d. data or stationary time series are treated, where in many cases, $$\rho(T)$$ is known to converge to 0 as $$T$$ tends to infinity if the stationary time series are independent. The statistician G. Udny Yule had noticed this phenomenon empirically in 1926, giving rise to the notion of ``nonsense correlation’’ for the behavior of $$\rho(T)$$ which is frequently far from 0, and ``highly dispersed’’ as a distribution, irrespective of the value of $$T$$. In this paper, using $T=1$ without loss of generality, we show that a notion of convergence does hold, can be analyzed in detail, and even gives rise to procedures for testing the independence of two paths of standard or fractional Brownian motions. This is the convergence to $$\rho = \rho(1)$$ of its discrete-observation version $$\rho_n$$, as the observation frequency $$n$$ tends to infinity. Using explicit computations based on characteristic functions of the trivariate elements that enter the expressions for the ratios defining $$\rho$$ and $$\rho_n$$, we show that $$\rho_n - \rho$$ converges to 0 at the rate $1/n$, and that the fluctuations of $$n(\rho_n - \rho)$$ are asymptotically non-normal. Based on a fully explicit representation, we show that the limit in distribution of $$n(\rho_n-\rho)$$, conditionally on the paths, is a normal distribution with mean and variance being non-linear functionals of the paths. Because of this, the testing procedure which arises from our limit theorem can be understood as a procedure based on ordinary Gaussian quantiles, where the asymptotic mean is non-zero and is explicitly computable from the data, and where the asymptotic covariance is also explicitly computable from the data. Perhaps most notably from an applicability perspective, the rate of scaling in the conditionally normal convergence is $$n^{-1}$$ rather than the usual $$n^{-1/2}$$, opening a path to significantly greater power in testing procedures compared to those based on ordinary central limit theorems. 

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3. On an Equivalence of Divisors on $$\overline {\text {M}}_{0,n}$$ from Gromov-Witten Theory and Conformal Blocks

   https://doi.org/10.1007/s00031-022-09752-6  

   Chen, L.; Gibney, A.; Heller, L.; Kalashnikov, E.; Larson, H.; Xu, W. (January 2022, Transformation Groups)

   Abstract We consider a conjecture that identifies two types of base point free divisors on $$\overline {\text {M}}_{0,n}$$ M ¯ 0 , n . The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on $$\overline {\text {M}}_{0,n}$$ M ¯ 0 , n to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on $$\overline {\text {M}}_{0,n}$$ M ¯ 0 , n for a large class, and we give sufficient conditions for the non-vanishing of these divisors. 

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4. Dimension Reduction for Polynomials over Gaussian Space and Applications

   Ghazi, Badih; Kamath, Pritish; Raghavendra, Prasad (June 2018, 33rd Computational Complexity Conference, CCC 2018)

   We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest. 

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5. Dimension Reduction for Polynomials over Gaussian Space and Applications

   Ghazi, B.; Kamath, P.; Raghavendra, P. (June 2018, Computational Complexity Conference (CCC))

   We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest. 

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