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Yayın Gluing formulas for volume forms on representation varieties of surfaces(Springer Nature, 2025-08-06) Erdal, Esma DiricanLet Σg,n be a compact oriented surface of genus g≥4 with n boundary components. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah–Bott–Goldman–Narasimhan symplectic form on the space of representations of π1(Σg,0) in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of Σg,0 in terms of torsions of Σg1,1,Σg2,1, and boundary circle S1, where g=g1+g2 and g1,g2≥2. Then, by using Heusener and Porti’s results on Σg,n, we show that the symplectic volume form on the representation variety of Σg,0 can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces Σg1,1 and Σg2,1.Yayın Complex rays and applications(Işık University Press, 2025-12-01) Hasanoğlu, ElmanComplex rays are a fascinating aspect of modern diffraction theory, typically sought as complex solutions to the eikonal equation. Traditionally, these solutions are obtained by analytically continuing real rays into the complex domain. However, this approach demands the analyticity of initial data, significantly limiting its applicability to many practical problems. Additionally, unlike real rays, complex rays cannot be visualized in space, presenting another drawback. In this paper, we present an alternative interpretation of complex rays, as introduced in [1], and describe a novel approach to two model diffraction problems and Gaussian beams.
