Essays on Geometry

A. Muhammed Uludağ, a member of Galatasaray University's Mathematics Department in Istanbul, Turkey, is recognized for his enthusiasm about the modular group and has worked across diverse fields in mathematics. He earned his Ph.D. from Institut Joseph Fourier, France, in algebraic geometry, and has been deeply involved in academic research and teaching at Galatasaray University. Uludağ is also active in contributing to academic journals, editing scholarly works, and organizing international mathematical conferences.Ayberk Zeytin, a member of the Department of Mathematics at Galatasaray University, specializes in number theory and ML techniques, and their application in arithmetic, algebra, and geometry. In addition to his editorial work, he has authored numerous articles in respected journals and book chapters in international book series. He has visited several research institutions including MPIM Bonn, IHES. He has co-organized several international meetings and taken part in several research projects.

• - 1. Introduction.- 2. Interview with Athanase Papadopoulos.- 3. A glance at S. Novikov’s theory of multivalued Morse functions.- 4. A twisted invariant of a compact Riemann surface.- 5. Directional moduli and pseudoconvexity.- 6. Angle Defect for Super Triangles.- 7. Lipschitz and quasiconformal mappings in cartography.- 8. Spherical representations of the group of isometries of semi-homogeneous trees.- 9. Trees of fractions.- 10. Binary quadratic forms: modern developments.- 11. A Note on Reversibility of Unipotent Matrices.- 12. “Le complément supérieur”: On the poetics of mathematics.- 13. Pythagorean Book II of the Elements restored and Pythagorean Incommensurabilities reconstructed.