Over a complete Riemannian manifold of finite dimension, Greene and Wu introduced a convolution of f defined aswhere κµ is a radial kernel that integrates to 1, and dΠ is the base measure on TpM. In this paper, we introduce a reformulation of the Greene-Wu (GW) convolution. Using our reformulation, many properties of the GW convolution can be easily derived, including a new formula for how the curvature of the space would affect the curvature of the function through the GW convolution. Also enabled by our new reformulation, an improved method for gradient estimation over Riemannian manifolds is introduced. Theoretically, our gradient estimation method improves the order of estimation error from O (n + 3) 3/2 to O n 3/2 , where n is the dimension of the manifold. Empirically, our method outperforms the best existing method for gradient estimation over Riemannian manifolds, as evidenced by thorough experimental evaluations.

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Let pR, mq be a complete discrete valuation ring with the finite residue field R{m " Fq. Given a monic polynomial P ptq P Rrts whose reduction modulo m gives an irreducible polynomial P ptq P Fqrts, we initiate the investigation of the distribution of cokerpP pAqq, where A P MatnpRq is randomly chosen with respect to the Haar probability measure on the additive group MatnpRq of n ˆn R-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the P -part of a random matrix A P MatnpFqq with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, not dividing q, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |Aut Z pHq|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between MatnpRq and MatnpFqq to translate our problems about a Haar-random matrix in MatnpRq into problems about a random matrix in MatnpFqq with respect to the uniform distribution.Standard notations. We write p to mean an arbitrary prime (number) and q an arbitrary prime power. We do not assume any relations between p and q, unless specified otherwise. We fix arbitrary n P Z ě0 , although we will often use it as index or let it go to infinity. By a ring, we mean a commutative ring with the multiplicative identity 1. By a distribution, we mean a probability measure. Given an ideal I of a ring R and a module M over it, we define M rI 8 s :" tx P M : I N x " 0 for N " 0u and call it the I 8 -torsion or the I-part of M . If M " M rI 8 s, we call M an I 8 -torsion module. For t P R, we say t 8 -torsion or t-part to mean ptq 8 -torsion or ptq-part, and write M rt 8 s :" M rptq 8 s. We write Mat n pRq to mean the set of n ˆn matrices over R, and I n P Mat n pRq means the identity matrix.

Given an elliptic curve E E defined over C \mathbb {C} , let E × E^{\times } be an open subset of E E obtained by removing a point. In this paper, we show that the i i -th Betti number of the unordered configuration space C o n f n ( E × ) \mathrm {Conf}^{n}(E^{\times }) of n n points on E × E^{\times } appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the F q \mathbb {F}_{q} -point counts of C o n f n ( E × ) \mathrm {Conf}^{n}(E^{\times }) , which can be obtained from the zeta function of E E over any fixed finite field F q \mathbb {F}_{q} . We show that the mixed Hodge structure of the i i -th singular cohomology group H i ( C o n f n ( E × ) ) H^{i}(\mathrm {Conf}^{n}(E^{\times })) with complex coefficients is pure of weight w ( i ) w(i) , an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro’s spectral sequence computation that describes the weight filtration of the mixed Hodge structure on H i ( C o n f n ( E × ) ) H^{i}(\mathrm {Conf}^{n}(E^{\times })) .