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In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. 1. The gluing problem for the spectral invariants Since their inception, the eta invariant and the ζ-determinant of Dirac type op- erators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invari- ants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that (2, p. 89) "no precise pasting formulas are obtained but only adiabatic ones." In (23, 24), we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in (25, 24). We begin with describing the gluing problem for compact manifolds. Let D be a Dirac type operator acting onC∞(M, S )w hereM is a closed compact Riemannian manifold of arbitrary dimension and S is a Clifford bundle over M.L etY be an embedded hypersurface in M and let M = M− ∪ M+ be decomposition of M into manifolds with boundary such that ∂M− = ∂M+ = Y. We assume that all geometric structures are of product type over a tubular neighborhood N = (−1, 1) × Y of Y where the Dirac operator takes the product form D = G(∂u + DY ), where G is a unitary operator on S0 := S|Y and DY is a Dirac type operator over Y satisfying G2 = −Id and DY G = −GDY . Recall that the eta function of D and the zeta function of D2 are defined through the heat operator e−tD 2 via

On gluing formulas for the spectral invariants of Dirac type operators