Stergios M. Antonakoudis(If you are interested in a draft that is not available online, please send me an e-mail.) Isometric disks are holomorphic (Published at Invent. math.) We show that every isometry from the unit disk into a finite dimensional Teichmueller space is either holomorphic or anti-holomorphic; in particular, its image coincides with the projection of an SL(2,R)-orbit to Teichmueller space. This solves a long-standing problem in Teichmueller theory. Our proof is geometric and applies to a large class of complex domains, including rank-one bounded symmetric domains and more generally, strictly convex bounded domains, as well as Teichmueller spaces. Teichmueller spaces and bounded symmetric domains do not mix isometrically (Published at GAFA.) We show that, in dimensions two or more, there are no holomorphic isometries between Teichmueller spaces and bounded symmetric domains in their intrinsic Kobayashi metrics. The complex geometry of Teichmueller spaces and bounded symmetric domains Paper based on plenary talk presented to the 6th Ahlfors-Bers Colloquium at Yale, 23-26 October 2014. You can watch the talkPapershere. Holomorphic maps between moduli spaces (joint with J. Aramayona and J. Souto) Stronger results obtained based on a previous paper of J. Aramayona and J. Souto, proving that a non-constant holomorphic map between two moduli spaces of Riemann surfaces is a forgetful map so long as the genera of the surfaces are not too different. The following papers are in preparation: Royden's theorem and birational geometry We obtain a generalization of Royden's theorem in birational geometry answering a question of S.-T. Yau. The proof uses ideas from functional analysis due to W. Rudin and V. Markovic. As an application, we show that finite dimensional Teichmueller spaces do not admit holomorphic isometric submersions onto round balls; the obstruction is infinitesimal. The bounded orbits conjecture for complex manifolds (Slides) We prove that every holomorphic map on a finite dimensional Teichmueller space with a recurrent orbit has a fixed point. Our theorem applies to a large class of complex domains which might be non-convex and with non-smooth boundary.Isometric maps between Teichmueller spaces are geometric(in preparation) We prove that every isometry between two finite dimensional Teichmueller spaces is the pull-back map induced from a covering map between two topological surfaces.Universality of Diophantine numbers and absolutely winning sets(in preparation) We prove that the set of Diophantine real numbers is a universal absolutely winning set. More precisely, we show that any absolutely winning set contains a k-quasi-symmetric copy of the Diophantine numbers for any k>1.Quasi-convexity and Betti numbers of infinite cyclic covers(in preparation) This papers gives an example of a finitely presented group for which the first Betti numbers of its infinite cyclic covers, considered as a function of the primitive classes in the first integral cohomology of the group, do not exhibit a combination of a convex and periodic behavior. This answers a question of C.T. McMullen, who proved that this function is always convex for fundamental groups of compact, oriented 3-manifolds whose boundary is a union of torii.A short proof of Kra's theta conjecture We give a short proof of Kra's theta conjecture; the conjecture was proved in '89 by C.T. McMullen. An inequality from conformal geometry Richard's E. Schwartz 'A Conformal Averaging Process of the Circle' describes a process that transforms any cyclic polygon to one that is regular. The idea behind the proof is to show that unless the polygon is regular this process decreases the conformal energy of the system. In this note we give a simple proof of the technical inequality needed to prove that claim. Criteria of separatedness and properness This note gives criteria of separatedness and properness for morphisms of schemes of finite type over algebraically closed fields by considering maps from affine curves (as opposed to DVRs). It was written in '09, a few year laters I found out that George Kempf's 'Algebraic Varieties' contains a more concise proof of this result. BackNotes