We prove several rigidity theorems for elliptic genera associated to Toeplitz operators on odd dimensional spin manifolds. We also prove an equivariant odd index theorem for Dirac operators with involution parity and the Atiyah-Hirzebruch vanishing theorems for odd dimensional spin manifolds.

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 1388] constructed a monoid structure on the set of all subsets of I using unipotent -linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.

The global existence and uniqueness of classical solutions to the Boltzmann equation without angular cut-off is proved under the condition that the initial data is in some Sobolev space. In addition, the solutions thus obtained are shown to be positive and C in all variables. One of the key observations is the introduction of a new important norm related to the singularity in the cross section of the collision operator. This norm captures the essential property of the singularity and yields precisely the dissipation description of the linearized collision operator through the celebrated H-theorem.

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