of the Dissertation Applications of the Seiberg-Witten equations to the Differential Geometry of non-compact Kähler manifolds by Ilya Elson Doctor of Philosophy in Mathematics Stony Brook University 2014 Soon after the introduction of the Seiberg-Witten equations, and their magnificent application to the differential topology of 4manifolds, LeBrun [LeB95a] used these equations to study differen...

Various real world phenomena can be modeled by a notion called complex network. Much effort has been devoted into understanding and manipulating this notion. Recent research hints that complex networks have an underlying hyperbolic geometry that gives them navigability, a highly desirable property observed in many complex networks. In this internship, a parameter called δ-hyperbolicity, which i...

We investigate the SL2(R) invariant geodesic curves with the associated invariant distance function in parabolic geometry. Parabolic geometry naturally occurs in the study of SL2(R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2dimensional Minkowski space-time) geometries. Initially we attempt to use standard methods of finding geodesics...