PhD Position
“Algebra meets Analysis and Number Theory”

In our research group at the Faculty of Mathematics of the University of Vienna a new PhD-position is available. Candidates are expected to have a strong back- ground in algebra and algebraic geometry and an interest in applying the respective techniques to the study of algebraic aspects of

Fuchsian Differential Equations.

These are ordinary linear differential equations with polynomial or holomorphic coefficients whose intricacy lies in the singularities of the equation: points where the leading coefficient vanishes to a high order. The main question, studied for two centuries without finding a complete answer yet, is to understand the equations whose solutions are algebraic functions, culminating in the famous Grothendieck- Katz p-curvature conjecture for equations defined over the integers: Can one characterize the existence of a basis of algebraic solutions by the reduction of the equation modulo p, for almost all primes p.

Despite many important results, the general case remains open, as well as many related questions and problems. The beauty of the subject is the variety of methods one can apply: commutative algebra and algebraic geometry, differential Galois theory, complex and functional analysis, analytic number theory, combinatorics.

Candidates are encouraged to contact me at <herwig.hauser@univie.ac.at> for further information.