Final Report Summary - BLACKHOLEHOLOGRAM (Black holes, superstrings and holography) The main project objectives stemmed from puzzles in the field of gravitation and high energy physics that concerned black holes, i.e. astrophysical objects which are very massive and have a horizon, a surface surrounding them from behind which no signal, including light, can reach an outside observer. The specific outstanding problem which was addressed was that of black hole entropy. The issue of black holes' entropy concerned the property that, in any physical process, the surface area of the horizon always increased, which was reminiscent of the entropy of a system, a measure of the number of degrees of microscopic states a system could be in for a given, macroscopic, thermodynamic state. This suggested that a black hole was really made up of many such states and the horizon area was simply a measure of the number of such microstates. In a class of examples, one could count the microstates as being the number of field-theoretic excitations of objects, or branes, which made up the black hole and, in many cases, the microstates could be understood as the degrees of freedom of a boundary conformal field theory. The project principal objectives could be divided into the following three groups: 1. finding exact microscopic degeneracy formulas. The specific objective was to compute the exact microscopic degeneracies of black holes using the ideas of string theory. In situations with N=4 supersymmetry there was a lot of control over the analytic aspects of the theory even at strong coupling, therefore the idea was to exploit this control to make predictions for the gravitational theories. 2. interpretation of the microscopic formulas in gravity. The partial objective was to recast the above formulas in terms of a gravitational functional integral which had a semi-classical limit. This method would help recast the exact results as an extension of general relativity. This in turn should be useful for resolving puzzles in quantum gravity. 3. exploring connections with mathematics. The relation with mathematics arose from the fact that the generating functions for the black hole degeneracies were typically modular forms and had a specific transformation under the group Sl(2,\IZ). Modular forms had applications in many areas of mathematics and physics; hence the goal was to develop the connections in a detailed manner. All three objectives were simultaneously pursued during the two years of the project lifetime. Progress was achieved in using the microscopic exact counting formulas from string theory to make rigorous statements about quantum gravity. One concrete piece of progress was to solve an old puzzle called the 'entropy enigma' in the context of N=4 supergravity. A project component which was successfully finished during the first year involved the interpretation of sub-leading saddle points in the quantum gravity path integral. In addition, effects of string theory in the decay of black holes were computed for the first time. A very detailed study of various known microscopic counting formulas was also conducted, some new formulas were found and a detailed interpretation of all these formulas in the theory of gravity was given. Furthermore, many new connections between black holes and mock modular forms in number theory were found, resulting to a very big draft paper, of 70 pages, which was undergoing review by the time of the project completion. Apart from the latter, all partial projects were finished and turned into publications in peer reviewed journals. It should be overall stated that very good progress was achieved towards the objectives listed above. New microscopic counting formulas were found, a detailed consistent interpretation of these formulas in the theory of gravity was produced for the first time and new connections were established between black holes and number theory. These results should have a strong positive impact on the field of quantum gravity as well as on mock modular forms.