Trends in mathematics. Courant-Colloquium, October 2007 by Meyer, Schick, Tschinkel. (eds.)

By Meyer, Schick, Tschinkel. (eds.)

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3 ([1]). The quiver of the m-cluster category C m = C m (A n−1 ) is isomorphic to the quiver (Γ(m, n), τm ) obtained from m-diagonals in an nm + 2-gon. The proof of our result uses Happel’s description of the Auslander-Reiten-quiver of D b (kQ) where Q is of Dynkin type A n−1 and combinatorial analysis of Γ(n, m). For details we refer to [1, Section 5]. The description in type D . We have a similar description of the m-cluster categories of D-type. Instead of working with a polygon (or unpunctured disc) we now have to use a punctured polygon.

This is exactly what we want, a polynomial search bound! The circle method approach described above is also useful for another problem on search bounds for positive definite integral quadratic forms Q: Suppose Rainer Dietmann: Forms in many variables 51 that for an integer n the congruences Q(x 1 , . . , x s ) ≡ n (mod m) are soluble for all positive integers m. Then according to the philosophy of the Local-Global-principle, one should expect also an integer solution x ∈ Zs of the equation Q(x) = n to exist.

If the classical flow is not also ergodic, there is no reason to expect eigenfunctions will not concentrate on the orbit. However, the unstable nature of the geodesic suggests if there is concentration, it will be very weak. We have the following Theorem. Theorem 2 (Main Theorems [7, 6]). Suppose that γ ⊂ M is an unstable closed geodesic, making only transversal reflections with ∂M if γ ∩ ∂M = , and U ⊃ γ is a sufficiently small neighbourhood. 1), M \U |u|2 d x 1 , |λ| → ∞. C log |λ| Remark 1. The conclusion of Theorem 2 applies also to semi-hyperbolic geodesics; that is, those which are unstable in a minimum of one direction in phase space.

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