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We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90%$ accuracy with ${sim}0.5%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95%$. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of fake HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered.
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Classical and exceptional Lie algebras and their representations are among the most important tools in the analysis of symmetry in physical systems. In this letter we show how the computation of tensor products and branching rules of irreducible representations are machine-learnable, and can achieve relative speed-ups of orders of magnitude in comparison to the non-ML algorithms.
The many ways in which machine and deep learning are transforming the analysis and simulation of data in particle physics are reviewed. The main methods based on boosted decision trees and various types of neural networks are introduced, and cutting-edge applications in the experimental and theoretical/phenomenological domains are highlighted. After describing the challenges in the application of these novel analysis techniques, the review concludes by discussing the interactions between physics and machine learning as a two-way street enriching both disciplines and helping to meet the present and future challenges of data-intensive science at the energy and intensity frontiers.
We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.
Let $X$ be a compact Kahler manifold and $Lto X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences ${ |Lambda, krangle }_{k=1}^infty$, where $forall k$ $|Lambda, krangle$ is a holomorphic section of $L^{otimes k}$. The terms in each sequence concentrate on $Lambda$, and a sequence itself has a symbol which is a half-form, $sigma$, on $Lambda$. We prove estimates, as $ktoinfty$, of the norm squares $langle Lambda, k|Lambda, krangle$ in terms of $int_Lambda sigmaoverline{sigma}$. More generally, we show that if $Lambda_1$ and $Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $langleLambda_1, k|Lambda_2, krangle$ have an asymptotic expansion as $ktoinfty$, the leading coefficient being an integral over the intersection $Lambda_1capLambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincare series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.
Sets of zero-dimensional ideals in the polynomial ring $k[x,y]$ that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Grobner cells. Conca-Valla and Constantinescu parametrize such Grobner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of $(x,y)$-primary ideals in Grobner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring $k[[x,y]]$ with a given leading term ideal with respect to a local term ordering.
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