Projective Structure,Symplectic Connection and Quantization

Let X be a connected Riemann surface equipped with a projective structure . Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using , this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using , a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.     

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Orthogonal and symplectic parabolic bundles

We investigate orthogonal and symplectic bundles with parabolic structure, over a curve.     

Quantum phenomena in magnetic nano clusters

One of the fascinating fields of study in magnetism in recent years has been the study of quantum phenomena in nanosystems. While semiconductor structures have provided paradigms of nanosystems from the stand point of electronic phenomena, the synthesis of high nuclearity transition metal complexes have provided examples of nano magnets. The range and diversity of the properties exhibited by these systems rivals its electronic counterparts. Qualitative understanding of these phenomena requires only a knowledge of basic physics, but quantitative study throws up many challenges that are similar to those encountered in the study of correlated electronic systems. In this article, a brief overview of the current trends in this area are highlighted and some of the efforts of our group in developing a quantitative understanding of this field are outlined.     

Synthesis,characterization, and in vitro biological evaluation of highly stable diversely functionalized superparamagnetic iron oxide nanoparticles

In this article, we report the design and synthesis of a series of well-dispersed superparamagnetic iron oxide nanoparticles (SPIONs) using chitosan as a surface modifying agent to develop a potential T ₂ contrast probe for magnetic resonance imaging (MRI). The amine, carboxyl, hydroxyl, and thiol functionalities were introduced on chitosan-coated magnetic probe via simple reactions with small reactive organic molecules to afford a series of biofunctionalized nanoparticles. Physico-chemical characterizations of these functionalized nanoparticles were performed by TEM, XRD, DLS, FTIR, and VSM. The colloidal stability of these functionalized iron oxide nanoparticles was investigated in presence of phosphate buffer saline, high salt concentrations and different cell media for 1 week. MRI analysis of human cervical carcinoma (HeLa) cell lines treated with nanoparticles elucidated that the amine-functionalized nanoparticles exhibited higher amount of signal darkening and lower T ₂ relaxation in comparison to the others. The cellular internalization efficacy of these functionalized SPIONs was also investigated with HeLa cancer cell line by magnetically activated cell sorting (MACS) and fluorescence microscopy and results established selectively higher internalization efficacy of amine-functionalized nanoparticles to cancer cells. These positive attributes demonstrated that these nanoconjugates can be used as a promising platform for further in vitro and in vivo biological evaluations.     

Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise

Financial systems are known to have irregular and erratic fluctuations due to diverse influences and often result in economic crisis and huge financial losses. Recent models of financial systems show that they behave chaotically and have long range memory dependence. Mitigating these undesirable chaotic natures of financial systems by appropriate control policies is important in order to reduce investment risks and improve economic performance. In this paper, a fractional order fuzzy control policy is employed to suppress the chaotic dynamics of a representative chaotic fractional order financial system. An intelligent Regrouping Particle Swarm Optimization (Reg-PSO) is used to design the numeric weights of the control policy and the methodology is demonstrated by credible simulations. The designed fractional fuzzy control policies are shown to work well with respect to conventional fuzzy control policies in the presence of persistent and anti-persistent noise, which can be due to additional extraneous influences on the system.     

Monodromy for principal bundles

Given a strongly semistable principal bundle E_G over a curve, in Biswas et al. (2006) [4], a group-scheme for it was constructed, which was named as the monodromy group-scheme. Here we extend the construction of the monodromy group-scheme to principal bundles over higher dimensional varieties.