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The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

Let M be a 5-dimensional Riemannian manifold with SecM ∈ [0, 1] and Σ be a locally conformally flat closed hypersurface in M with mean curvature function H. We prove that there exists ε0 > 0 such that (1) ... provided |H| ≤ ε0, where χ(Σ) is the Euler number of Σ. In particular, if Σ is a locally conformally flat minimal hypersphere in M, then V ol(Σ) ≥ 8π²/3, which partially answers a question proposed by Mazet and Rosenberg. Moreover, we show that if M is (some special but large class) rotationally symmetric, then the inequality (1) holds for all H. [ABSTRACT FROM AUTHOR]

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties. [ABSTRACT FROM AUTHOR]

Let p > 3 be a prime, and let a be a rational p-adic integer. Let {En(x)} denote the Euler polynomials given by 2ext/et+1 =Σ∞n=0 En(x) tn/n!. In this paper we show that p-1Σk=0 (a k) (-1 -a k) ≡ (-1)(a)p+(a-(a)p)(p+a-(a)p)Ep-3(-a) (mod p3), p-1Σk=0 (a k) (-2)k ≡ (-1)(a)p-(a-(a)p)Ep-2(-a) (mod p2) for a≢0 (mod p), where (a)p ∊ {0,1,...,p - 1} satisfying a ≡ (a)p (mod p). Taking a = -1/3,-1/4,-1/6 in the first congreuence, we solve conjectures of Z. W. Sun. We also establish a congruence for Σk=0 p-1 K (a k) (-1 -a k) modulo p3. [ABSTRACT FROM AUTHOR]

We consider the real number σ with continued fraction expansion [a0, a1, a2,...]=[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16,...], where ai is the largest power of 2 dividing i+1. We show that the irrationality measure of σ² is at least 8/3. We also show that certain partial quotients of σ² grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson. [ABSTRACT FROM AUTHOR]

We investigate Foulkes characters for a wide class of reflection groups which contains all finite Coxeter groups. In addition to new results, our general approach unifies, explains, and extends previously known (type A) results due to Foulkes, Kerber-Thürlings, Diaconis-Fulman, and Isaacs. [ABSTRACT FROM AUTHOR]

Recently we determined explicitly a Picard modular variety of general type. On the regular locus of this variety there are holomorphic three forms which have been constructed as Borcherds products. Resolutions of quotients of this variety, such that the zero divisors are in the branch locus, are candidates for Calabi-Yau manifolds. Here we treat one distinguished example for this. In fact we shall recover a known variety given by the equations X0X1X2 = X3X4X5, X³ 0 + X³ 1 + X³ 2 = X³ 3 + X³ 4 + X³ 5. as a Picard modular variety. This variety has a projective small resolution which is a rigid Calabi-Yau manifold (h12 = 0) with Euler number 72. [ABSTRACT FROM AUTHOR]

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them are its explicit expression, the connection with the Euler numbers, and its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality (with respect to an L2-inner product), it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed. [ABSTRACT FROM AUTHOR]

Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight s(m, 1)? modules and established a main exponential term in their asymptotic expansions. By different methods, we improve upon the Kac-Wakimoto asymptotics for these characters, obtaining an asymptotic expansion with an arbitrarily large number of terms beyond the main term. More specifically, it is well known that in the case of holomorphic modular forms, asymptotic information may be obtained using modular transformation properties. However, here this is not the case due to the analytic nature of the Kac-Wakimoto series as discovered recently by the first author and Ono. We first "complete" these series by adding to them certain integrals, obtaining functions that exhibit suitable modular transformation laws, at the expense of the completed objects being nonholomorphic. We then exploit this mock-modular behavior of the Kac-Wakimoto series to obtain our asymptotic expansion. In particular, we show that beyond the main term, the asymptotic behavior is dictated by the nonholomorphic part of the completed Kac-Wakimoto characters, which is a priori invisible. Euler numbers (equivalently, zeta-values) appear as coefficients. [ABSTRACT FROM AUTHOR]

We define a finitely generated Q-algebra … with a module structure over the universal enveloping algebra of a Lie algebra. Identities of Euler numbers are investigated using the algebraic structures of …. [ABSTRACT FROM AUTHOR]