International Mathematics Research Notices

doi: 10.1093/imrn/rnp140pmid: N/A

doi: 10.1093/imrn/rnp138pmid: N/A

doi: 10.1093/imrn/rnp139pmid: N/A

Toth, John A.; Wigman, Igor

doi: 10.1093/imrn/rnp052pmid: N/A

We compute the asymptotic expectation of the number of open nodal lines for random waves on smooth planar domains. We find that for both the long energy window [0, ], and the short one [, 1], the expected number of open nodal lines is proportional to , asymptotically as . Our results are consistent with the predictions of Blum, Gnutzmann, and Smilansky [4] in the physics literature.

Fitzpatrick, Sean

doi: 10.1093/imrn/rnp057pmid: N/A

We consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients whose properties we describe.When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only and standard equivariant characteristic forms. This formula generalizes the formula given in [10] for the case of a contact manifold.

Groe-Brauckmann, Karsten; Korevaar, Nicholas J.; Kusner, Robert B.; Ratzkin, Jesse; Sullivan, John M.

doi: 10.1093/imrn/rnp058pmid: N/A

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism.

Gillibert, Jean

doi: 10.1093/imrn/rnp059pmid: N/A

Nous revisitons, dans le langage des log schmas, le problme de prolongement de biextensions de schmas en groupes commutatifs lisses par le groupe multiplicatif tudi par Grothendieck dans [10]. Nous montrons que ce problme admet en gnral une solution dans la catgorie des faisceaux pour la topologie log plate, contrairement ce que l'on peut observer en topologie fppf pour laquelle Grothendieck a dfini des obstructions monodromiques. En particulier, dans le cas d'une varit ablienne et de sa duale, il est possible de prolonger la biextension de Weil sur la totalit des modles de Nron; ceci permet de dfinir un accouplement sur les points qui combine l'accouplement de classes dfini par Mazur et Tate et l'accouplement de monodromie.We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in [10]. We show that this problem admits a solution in the category of sheaves for Kato's log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole Nron model. This allows us to define a pairing on the points that combines the class group pairing defined by Mazur and Tate and Grothendieck's monodromy pairing.

Sasaki, Atsumu

doi: 10.1093/imrn/rnp060pmid: N/A

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with a slice S if D G S is open in D and there exists an antiholomorphic and orbit-preserving diffeomorphism of D such that S id S. In this article, we study linear, strongly visible actions. We prove that irreducible multiplicity-free space V of a connected compact Lie group is strongly visible. Furthermore, we find an explicit description of S and according to Kac's classification. Our result gives an evidence to Kobayashi's conjecture [10, Conjecture 3.2] in the case of irreducible multiplicity-free spaces, asserting that we can take S to have the same dimension as the rank of V.

Dolgachev, Igor V.; Iskovskikh, Vasily A.

doi: 10.1093/imrp/rnp061pmid: N/A

We show that the plane Cremona group over a perfect field k of characteristic p 0 contains an element of prime order 7 not equal to p if and only if there exists a two-dimensional algebraic torus T over k such that T(k) contains an element of order . If p 0 and k does not contain a primitive th root of unity, we show that there are no elements of prime order > 7 in and all elements of order 7 are conjugate.

Mezzadri, Francesco; Mo, Man Yue

doi: 10.1093/imrn/rnp062pmid: N/A

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble ( 2) with weight . We compute the leading-order term of the partition function and the coefficients of its Taylor expansion. Our results are valid in the region . Such a partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian unitary ensemble that was introduced by Berry and Shukla [2]. It can also be interpreted as the moment-generating function of the singular linear statistics .