Real Places and Torus Bundles

If M is a hyperbolic once-punctured torus bundle over S ¹, then the trace field of M has no real places.     

Bott's formula and enumerative geometry

We outline a strategy for computing intersection numbers on
smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.

     

Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

GENERALIZED SIMPLE NONCOMMUTATIVE TORI

The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are trivial, and C*(kZ × kZ) ×a3 Z ×a4 ... ×an Z is a completely irrational noncommutative torus Ap of rank n. It is shown in this paper that Tkp is strongly Morita equivalent to Ap, and that Tkp (?) Mp∞ is isomorphic to Ap (?) Mk(C) (?) Mp∞ if and only if the set of prime factors of k is a subset of the set of prime factors of p.     

Fillable contact structures on torus bundles over circles

We will construct an example of a strongly symplectically fillable contact structure on a torus bundle over the circle with parabolic monodromy.

     

Formal Markoff maps are positive

This note defines a family of Laurent polynomials indexed in which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each polynomial and prove all the coefficients are positive integers. We also conjecture a generalization of that positivity result.      

The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichlet?s class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.     

Lie Tori of Type BC 2 and Structurable Quasitori

Structurable quasitori are introduced and shown to give all Lie tori of type BC ₂ via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t).