Árpád Tóth

The Splitting of Primes in Division Fields of Elliptic Curves

We give a global description of the Frobenius for the division fields of an elliptic curve E that is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in a subfield of such a division field. These subfields include a large class of nonsolvable quintic extensions and our application provides an arithmetic counterpart to Kleins “solution” of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the reduced curve modulo p.

Holomorphic diffeomorphisms of semisimple homogeneous spaces

The density property for a Stein manifold X implies that the group of holomorphic diffeomorphisms of X is infinite-dimensional and, in a certain well-defined sense, as large as possible. We prove that if G is a complex semisimple Lie group of adjoint type and K is a reductive subgroup, then G/K has the density property. This theorem is a non-trivial extension of an earlier result of ours, which handles the case of complex semi-simple Lie groups. We also establish the density property for some other complex homogeneous spaces by ad hoc methods. Finally, we introduce a lifting method that extends many results on complex manifolds with the density property to covering spaces of such manifolds.

3-D Brownian Motion Simulator for High-Sensitivity Nanobiotechnological Applications

A wide variety of nanobiotechnologic applications are being developed for nanoparticle based in vitro diagnostic and imaging systems. Some of these systems make possible highly sensitive detection of molecular biomarkers. Frequently, the very low concentration of the biomarkers makes impossible the classical, partial differential equation-based mathematical simulation of the motion of the nanoparticles involved. We present a three-dimensional Brownian motion simulation tool for the prediction of the movement of nanoparticles in various thermal, viscosity, and geometric settings in a rectangular cuvette. For nonprofit users the server is freely available at the site http://brownian.pitgroup.org.

On the evaluation of Salié sums

The Salie sum S(m,n;c) can be evaluated as the product of a Gauss sum and an exponential sum involving square roots of mn mod c. We give a new proof of this fact that can simultaneously handle a twisted version of these sums that arise in the theory of half-integral weight modular forms.

Mathematical modelling and computer simulation of Brownian motion and hybridisation of nanoparticle–bioprobe–polymer complexes in the low concentration limit

A wide variety of nano-biotechnological applications are being developed for nanoparticles based on in vitro diagnostic and imaging systems. Some of these systems allow highly sensitive detection of molecular biomarkers. Frequently, the very low concentration of the biomarkers makes very difficult the mathematical simulation of the motion of nanoparticles based on classical, partial differential equations. We address the issue of incubation times for low concentration systems using Monte Carlo simulations. We describe a mathematical model and computer simulation of Brownian motion of nanoparticle–bioprobe–polymer contrast agent complexes and their hybridisation to immobilised targets. We present results for the dependence of incubation times on the number of particles available for detection, and the geometric layout of the DNA-detection assay on the chip.