The main topics of this thesis areNoether's problem and the existence of generic polynomials. These problems areboth related to the inverse Galois problem, which asks the question whetherevery finite group is isomorphic to the Galois group of a Galois extension overthe rational numbers. We solved Noether's problem and found generic polynomialsfor the subgroups of Sn for n < 4 and the quaternion group of order 8.Moreover, we established generic polyomials for the cyclic and dihedral groupsof odd order and discussed their existence for some other groups such asp-groups and Frobenius groups. We also worked out a counterexample forNoether's problem, namely the cyclic group of order 8.
on a real semisimple Lie group. This approach leads to both an inversion theorem
and a Paley-Wiener theorem. No specific expertise in the theory of
real semisimple Lie groups will be assumed.
Everything will be explained in the context of the simplest example SL(2,R).
Automated Geometry Theorem Proving(AGTP) is an algebraic approach to (dis)proving geometric statements in such away that it can be (mostly) fully automated. The main idea is to translatethe given geometric properties to polynomials, where the variables are thecoordinates of the given points, and look at the corresponding variety. One way(a first approach) to see if a certain property holds, is to see if itscorresponding polynomial is contained in the vanishing ideal of the initialvariety. A more complete approach is Wu's Method, which relies on the theorembehind ascending chains and pseudo-remainders.
To try out how far we can automatethis approach, I have implemented the concept of AGTP in Python.
the genus of the curve, and which satisfies some additional conditions. This extends the method of Chabauty-Coleman by making certain aspects of Kim's non-abelian Chabauty program explicit for such curves using p-adic heights. I will then present an application of this technique to the split (and non-split) Cartan modular curve of level 13; this is joint work with J. Balakrishnan, N. Dogra, J. Tuitman and J. Vonk and completes the classification of non-CM elliptic curves over the rationals with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.
Abstract: Predicting the amount of gas or oil extracted from a subsurface
reservoir depends on the soil properties such as porosity and
permeability. These properties, however, are highly uncertain due to the
lack of measurements. Therefore decreasing these uncertainties is of a
Mathematically speaking, permeability can be represented by a random
process, which in turn leads to a random partial differential equation.
The solution of such a partial differential equation, for example
pressure, is only partially observed and, moreover, contaminated with
measurement errors. Therefore, instead of a well-posed forward problem
of finding pressure from certain permeability, we are faced with an
ill-posed inverse problem of finding uncertain random process from a few
pressure measurements. We develop a Bayesian method for inverse problems,
that is both general and computationally affordable.
It is an open problem whether there is a bivariate polynomial f(x,y) with rational coefficients which induces an injective function from the set of pairs of rational numbers into the rational numbers. In the direction of a positive answer, we will prove that there is an affine curve C defined over the rational numbers with a dense set of rational points, and regular polynomial function on CxC which defines an injective function on rational points.
two conical singularities of a flat surface. The proof of this formula involves
both ingredients from representation theory via the Bloch-Okounkov formalism
and from intersection theory on moduli spaces.