We study the equivariant Gromov-Witten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the Gromov-Witten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix. X and N is equipped with the anti-diagonal...

In this paper we study the Hurwitz scheme in terms of the Sato Grassmannian and the algebro-geometric theory of solitons. We will give a characterization, its equations and a show that there is a group of Virasoro type which uniformizes it.

A system of m nonzero vectors in Zn is called an m-icube if they are pairwise orthogonal and have the same length. The paper describes m-icubes in Z for 2 ≤ m ≤ 4 using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in Z.

Every finite group acts as the full isometry group of some compact hyperbolic 3-manifold. In this paper we study those finite groups which act maximally, that is when the ratio |Isom(M)|/vol(M) is maximal among all such manifolds. In two dimensions maximal symmetry groups are called Hurwitz groups, and arise as quotients of the (2,3,7)–triangle group. Here we study quotients of the minimal co-v...

In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet’s L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we ...

Absolute and Relative Stability. A control system is called absolutely stable if the controlled transfer function Hd(s) from the reference signal yd(s) to the output signal y(s) has all of its poles in the open left half plane. One technique for determining absolute stability of a control system is the Routh-Hurwitz stability analysis technique. This very useful technique is presented in Sectio...

We study smooth maps from an open subset of a real projective space to a sphere such that all germs of straight lines go to germs of circles (or points). We give a construction of such maps based on representations of Clifford algebras. It provides a complete purely geometric description of Clifford algebras representations in terms of circles. We also describe a connection between our problem ...

Assuming that φ : Sym(E) → F is a morphism of vector bundles on a variety X, we compute the class of the locus in X where Ker(φ) contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds’ result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construc...

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M , the Hurwitz constant of M . It measures how well all g...

We study the equivariant Gromov-Witten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the Gromov-Witten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix. X and N is equipped with the anti-diagonal...