Inspired by mirror symmetry, we investigate some differential geometric aspects of the space Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory Weil-Petersson geometry stringy Kahler moduli space. A few basic examples are studied. In particular, identify our metric with Bergman Siegel modular variety in case self-product an elliptic curve.

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

In this paper, we continue our study of the Weil-Petersson geometry as in the previous paper [10], in which we have proved the boundedness of the Weil-Petersson volume, among the other results. The main results of this paper are that the volume and the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space are rational numbers. In particular, the Ricci curvature define...

In this paper, we continue our study of the Weil-Petersson geometry as in the previous paper [10], in which we have proved the boundedness of the Weil-Petersson volume, among the other results. The main results of this paper are that the volume and the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space are rational numbers. In particular, the Ricci curvature define...

In this paper we study the systole function along WeilPetersson geodesics. We show that the square root of the systole function is uniform Lipschitz on the Teichmüller space endowed with the Weil-Petersson metric. The inradius of a metric space is the largest radius of metric balls contained in the interior of the metric space. As applications of the result above, we study the growth of the Wei...

We develop methods that construct an optimal set of vectors with a specified inner product structure, from a given set of vectors in a complex Hilbert space. The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. Four special cases are considered. In the first, the constructed vectors are orthonormal. In the s...

We extend the reach of functional encryption schemes that are provably secure under simpleassumptions against unbounded collusion to include function-hiding inner product schemes.Our scheme is a private key functional encryption scheme, where ciphertexts correspond tovectors ~x, secret keys correspond to vectors ~y, and a decryptor learns 〈~x, ~y〉. Our schemeemploys asymmetr...

same initial elements. For the case of m= 12 this results in a reduction by a factor of about 80 in the number ofcases examined. Samples at m = 13 and 14 indicate that this pruning method will yield an additional improvement factor of about 2.5 to 3 for each increase in m. Using a characteristic vector representation, the test for a repeated spacing is improved by a factor of m2/8 over the stra...

We consider cubical complexes which are uniformized by an ordered product of regular trees. For these we define the notion of being Ramanujan, generalizing the one-dimensional definition introduced by Lubotzky, Phillips, and Sarnak [15]. As in [12], we also allow local systems. We discuss the significance of this property, and then we construct explicit arithmetic examples using quaternion alge...