Projects
Definability and Minimality in structures over p-adic fields and polynomial rings. KU Leuven
A language is a collection of symbols for functions, relations and constants, like the ring language (+,-,·,0,1). It can be used to describe properties and subsets of elements of a ‘universe’ like the real or p-adic numbers. The precise interpretation of formulas and sentences may be quite different if the universe changes. For instance, in R the formula (there exists y)(x = y^2) means that x is positive, while in Qp it merely means that x is ...
Motivic integration and motivic Haar measure, zeta functions, and p-adic groups with the Howe - Moore property. KU Leuven
p-adic and motivic integrals, non-archimedean geometry, and diophantine applications KU Leuven
A p-adic variant of the polynomial method Ghent University
The polynomial method in Galois geometry refers to a method of obtaining geometrical information on a geometric structure by studying the algebraic properties of a polynomial associated to the geometric structure. However, this method has its restrictions, of which one very particular restriction is due to the fact that the polynomial is defined over a finite field, which has a non-zero characteristic. We present a new approach by associating ...
Cohomology jump loci, Bernstein-Sato ideals, and the Monodromy Conjecture. KU Leuven
This project is about solutions of systems of polynomial equations and the effect that singularities have on them. The long-term goal is to understand why the numbers of solutions of polynomials modulo prime powers seem to be predicted by the complexity of their singularities. The Monodromy Conjecture is a concrete statement of this phenomenon. The plan is to approach this conjecture via local p-adic differential equations. This approach will ...
New methods in field arithmetic and quadratic form theory. University of Antwerp
Berkovich curves, semistable reduction and wild ramification KU Leuven
Around 1990, Vladimir Berkovich constructed powerful foundations for analytic geometry over non-archimedean fields, such as p-adic fields and fields of Laurent series. This theory has found a broad range of applications in number theory, algebraic geometry and dynamical systems. The aim of the project is to construct generalizations of Michael Temkin's work on norms on pluricanonical forms and its applications to wildly ramified covers of ...
What does covolume tell us about the structure of a lattice? KU Leuven
A Lie group is a mathematical object which consists of the symmetries of a given smooth geometric structure. For example, there is a Lie group consisting of all the isometries of our ambient 3-dimensional space. A lattice in a Lie group G is a discrete subgroup L of G which approximates G up to a set of finite volume. The quotient space G/L then has finite volume, and this volume is called the covolume of L. In the above example, all the ...