Projects
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 ...
Hensel minimality KU Leuven
Studying rational points on algebraic varieties is one of the most fundamental topics in algebraic geometry and number theory. The proposed project fits into this research topic. For a rational number q=a/b with a and b coprime, we define the height of q to be the maximum of |a| and |b|. In some sense, this measures the complexity of the number q. For a tuple (q_1, ..., q_n) of rational numbers, the height is defined to be the maximum of the ...
Curves as covers of the projective line KU Leuven
Most of current public-key cryptography is considered insecure against attacks from sufficiently powerful quantum computers. Post-quantum cryptography studies methods to secure information resistant against such attacks. One proposal is isogeny-based cryptography, which bases its security on computational hardness assumptions related to maps between elliptic curves. We analyze the security of isogeny-based cryptographic schemes, in particular ...
The role of statistical learning in the development of mathematics KU Leuven
Mathematics is a fundamental tool in life, however there is a lack of a clear scientific understanding about how math learning is precisely achieved. The central premise underlying this project is that mathematics is fundamentally a language that uses meaningful symbols which must be organized according to a finite set of rules in order to exchange ideas, concepts and theories among people. One of the most impactful insights that emerged from ...
Absolute sets of rigid local systems KU Leuven
A set of local systems or vector bundles with an integral connection on a smooth complex algebraic variety is said to be absolute if it satisfies certain compatibility conditions of arithmetic-type with the Riemann-Hilbert correspondence. Absolute sets of rank-one local systems admit particularly nice descriptions, by work of Simpson in the projective case and Budur-Wang in general. For higher rank, there are some conjectures of André-Oort ...
Computational aspects of p-adic cohomology. KU Leuven
Motivic integration and motivic Haar measure, zeta functions, and p-adic groups with the Howe - Moore property. KU Leuven
Igusa type zeta functions and the monodromy conjecture KU Leuven
For a polynomial f over the integers, there are important results and open questions relating its arithmetic and geometric properties. The famous monodromy conjecture is such an open problem. Here the arithmetic part consists of numbers of solutions of congruences, that is, solutions of the equation f=0 over the integers, modulo some given number. Technically, one gathers this information in a generating series, called the Igusa zeta ...
Machine Learning for Genomic Data Fusion KU Leuven
It has been shown that while a single genomic data source might not be sufficiently informative, fusing several complementary genomic data sources delivers more accurate predictions. In this regard, genomic data fusion has garnered much interest across biological research communities. Consequently, finding efficient and effective techniques for fusing heterogeneous biological data sources has gained growing attention over the past few ...