This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ0(s) which generalizes the Artin-Mazur zeta function was given by Lind for Z2-action φ. The n-th order zeta function ζn of φ on Zn×∞, n ≥ 1, is studied first. The trace operator Tn which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally symmetric. The rotational symmetry of Tn induces the reduced trace operator τn and ζn = (det (I − snτn))−1. The zeta function ζ =∞ Qn=1 (det (I − snτn))−1 in the x-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the y-direction and in the coordinates of any unimodular transformation in GL2(Z). Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function ζ0(s). The Taylor series at the origin for these zeta functions are equal with integer coefficients, yielding a family of identities which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions.

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We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension 2m + 1 ≥ 5. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

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