We prove an estimate for Donaldson's -operator on a prequantized compact symplectic manifold. This estimate is an ingredient in the recent result of Keller and Lejmi about a symplectic generalization of Donaldson's lower bound for the -norm of the Hermitian scalar curvature.
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New quantum integrals discovered for a spin chain model.
problem Exploring quantum integrals for a spin chain model.
method Using surface defects and observables in 4D super-QCD.
result First construction of quantum integrals and their joint eigenvectors.
We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application…
We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the -Weyl algebra of -operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discus…
Defines a new energy for submanifolds, comparing to Willmore energy.
problem Finding new conformally invariant energies for submanifolds.
method Coupling tractor connection to GJMS operators for higher-dimensional analogues.
result Shows comparison with existing energy in 4D.
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…
We establish a theoretical link between adversarial training and operator norm regularization for deep neural networks. Specifically, we prove that -norm constrained projected gradient ascent based adversarial training with an -norm loss on the logits of clean and perturbed inputs is equivalent to data-…
New algorithms improve robust reinforcement learning under uncertainty.
problem Robust reinforcement learning in MDPs with contamination.
method Non-asymptotic convergence analysis of -learning and actor-critic methods.
result Efficient algorithms learn robust policies with minimal samples.
We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic probl…
Study of surface defects in gauge theories leads to duality and separation of variables.
problem Understanding surface observables and their transitions in gauge theories.
method Utilized Fourier transformations and spectral problems to derive dualities and separation of variables.
result Exact duality between spectral problems of spin chains and Gaudin models.
New optimizers control network width scaling, improving stability and transfer across different model sizes.
problem Designing stable optimizers for networks of varying widths.
method Interpreting optimizers as steepest descent under mean-normalized operator norms, enabling layerwise composability and width-independent bounds.
result New optimizers like row normalization and column normalization provide stable learning-rate transfer across different model widths.
The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gau…