My own research in mathematical cosmology is concerned with the development of new geometric extensions of general relativity, discovering their cosmological implications by employing global geometric and topological methods, and suggestions of new theoretical predictions for comparing them with future observations. Some highlights of my research are:

**Inflation, multiverse, and the cosmological constant**

I have established the basic property of modified gravity that higher-order curvature lagrangian theories of gravitation in any spacetime dimension are conformally related to general relativity plus a self-interacting scalar field with a particular potential. By the same method of proof, one may show that the same is true in situations involving a string theory lagrangian, a scalar-tensor one, or any other wavemap-type lagrangian theory. This result has been used in a very essential way by more than 600 research papers during the past 30 years because it opens the way to treat such theories as general relativity plus additional self-interacting scalar fields.

This led to the realization that inflation and other properties may be best understood via a transition to the so-called conformal frame of the theory and analyzing properties of the associated scalar field potential. Inflation becomes possible when the spacetime dimension is twice that of the leading degree of the gravity lagrangian. It has also become clear that this conformal potential describes accurately in a generic way the inflationary profile seen in the WMAP and Planck satellites results.

Very recently I have introduced a class of theories, no-scale gravity, where the above restriction on the spacetime dimension is lifted, and the conformal potential is such that the predicted form of the tensor to scalar ratio depends on a freely adjustable parameter. This is an exact prediction which provides a direct way to falsify such theories with more precise future measurements currently under way. Another ingredient of this work is an unconventional solution of the cosmological constant problem, which uses conformal structure, inflation, and the no-scale nature of the theory. This idea may be extended in the future in a great variety of currently popular themes.

* References:* J. D. Barrow and S. Cotsakis, Phys. Lett. B214 (1988) 515; S. Cotsakis, PhD Thesis, University of Sussex, 1990; J. D. Barrow and S. Cotsakis, Eur. Phys. J. C80 (2020) 839; arXiv: 1907.02928.

**Singularities and completeness**

I have been working on criteria for singularity formation in general relativity and related theories for many years. I have provided conditions for global hyperbolicity of spacetimes under generic regularity conditions on the metric. This has been further used to show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable.

Further, I showed that for generic sliced spacetimes global hyperbolicity is equivalent to space completeness under the assumption that the lapse, shift and spatial metric are uniformly bounded. This led to the conclusion that simple sliced spaces are timelike and null geodesically complete if and only if space is a complete Riemannian manifold, and also to a classification of singularities in various cosmological situations.

* References:* Y. Choquet-Bruhat and S. Cotsakis, J. Geom. Phys. 43 (2002) 345-350; S. Cotsakis, Gen. Rel. Grav. 36 (2004) 1183-1188; S. Cotsakis and I. Klaoudatou, J. Geom. Phys. 55

**(2005) 306–315.**

**The ambient cosmological metric**

I have also been interested in a conformal geometry approach to singularities in general relativity for many years. Ambient cosmology-a new approach, within string cosmology, to the issues of spacetime singularities and cosmic censorship in general relativity, is based on the idea that standard 4-dimensional spacetime is the conformal infinity of an `ambient metric’ for the 5-dimensional Einstein equations with fluid sources.

In this approach, the existence of spacetime singularities in four dimensions is constrained by asymptotic properties of the ambient 5-metric, while the non-degeneracy of the latter crucially depends on cosmic censorship holding on the boundary. Both problems find novel solutions in this framework.

* References:* I. Antoniadis and S. Cotsakis, Eur. Phys. J. C. 75:35 (2015) 1-12; I. Antoniadis and S. Cotsakis, Mod. Phys. Lett. A, 30 (2015) 1550161.

**Cosmological dynamical systems**

Perhaps the most distinctive feature of mathematical cosmology is the search for qualitative, long-term behaviors of universes as solutions to dynamical systems arising when we consider evolving spacetimes in the intricate web of possible geometric theories of gravity coupled to matter fields. This is one of the most popular approaches to the field.

Recently, I have been working on the problem of spontaneous order in cosmology through the exciting mechanism of chaotic synchronization. This has led to new advances in the problem of describing the generic cosmological singularity, and may have implications for the BKL conjecture.

* References: *S. Cotsakis, in: Cosmological Crossroads, LNP592 (Springer, 2002), pp. 59–94, arXiv:gr-qc/0201067; S. Cotsakis, Grav. Cosm. 19 (2013) 240; arXiv:1301.4778; S. Cotsakis, J. Miritzis, K. Tzanni, Int. J. Mod. Phys. A34 (2019) 1950092; arXiv:1905.11049; S. Cotsakis, arXiv: 2009.01732