Hermen Jan Hupkes

Assistant Professor
Applied Mathematics
Mathematical Institute
University of Leiden

PhD Positions Available

There are two open PhD Positions within the NWO funded project Preparing MFDEs for the Modelling World. This project is focussed on understanding the impact of spatial and/or temporal discreteness on several fundamental patterns such as waves, spots and spirals that are often encountered in nature.

Postdoc Position Available

There is an open postdoc position within the NWO funded project Preparing MFDEs for the Modelling World. This project is focussed on understanding the role MFDEs can play when studying optimal control problems that involve time delays. The mathematical techniques we intend to develop can be used in a variety of settings, such as the analysis of Nash equilibria in competitive markets and the design of optimal taxation policies to fairly price environmental pollution.

China Scholarship Council (CSC)

The council offers Chinese nationals the opportunity to pursue a PhD degree at a foreign university in order to stimulate the exchange of ideas.

Research Overview

Motivated by the study of physical structures such as crystals, grids of neurons and population patches, an increasing interest has arisen in mathematical modelling techniques that reflect the underlying spatial discreteness. My main research goal is to develop tools that can be used to characterize the differences and similarities between such discrete models and their more traditional continuous counterparts. Although most of my work is analytical, I have also worked on numerical algorithms to help visualize the theoretical concepts. Some of my results also turned out to be applicable in the seemingly unrelated area of economic modelling. As a consequence, my interests have also extended to this field.

Waves and Patterns on Two Dimensional Lattices

Embedding the standard square grid into the plane breaks the natural isotropy of space that is often taken for granted, as two preferred lattice directions are chosen. In particular, patterns that travel in a straight line will experience different effects depending on the orientation of their trajectory versus the grid. In this project we study the existence and stability of travelling waves and explore how the direction dependence comes into play. We use both comparison principles and spectral methods in our stability analysis. This allows us to consider small perturbations in a non-monotonic setting and large perturbations in a monotonic setting.

Papers

Talks

Waves and Patterns on One Dimensional Lattices

Many results concerning the existence and stability of travelling waves in one dimensional lattices are based on comparison principles. This project explores alternative techniques to analyze travelling patterns, by providing an infinite dimensional analytical underpinning to classic geometrical ideas.

Papers

Talks

Numerical Aspects

In this project we explore the role that discretization schemes and numerical algorithms play when studying spatial evolution systems.

Talks

Initial Value Problems involving MFDEs with Economic Applications

A common assumption in economic theory is that agents have perfect foresight. While not entirely realistic, it is of course true that expectations play an important role when making decisions, which can lead to well-known phenomena such as self-fulfilling prophecies. Functional differential equations of mixed type (MFDEs) arise naturally in certain macro-economic agent-based models as they can incorporate such expectations. Models of this type suffer from an incompatibility between the mathematical state space and the (smaller) modelling state space. This project investigates the consequences of this mismatch, focussing on the effects of incomplete information on decision-making.

Papers

Talks

MFDEs: bifurcation theory near equilibria and periodic solutions

Functional differential equations of mixed type (MFDEs) are differential equations with both retarded and advanced arguments. As such, they can be seen as a natural generalization of delay differential equations. In this project we show how a number of classical tools can be adapted for use in the MFDE world.

Papers

Talks

Preprints

Mathematics

Other

Publications

Mathematics

Economics

Other

Thesis

Phone and Email

Phone: +31-71-527-4837
Email: hhupkes_AT_math.leidenuniv.nl

Visiting Address

Room 206a
Snellius Building
Niels Bohrweg 1
2333 CA Leiden
The Netherlands

Postal Address

Mathematical Institute
University of Leiden
P.O. Box 9512
2300 RA Leiden
The Netherlands

Spring 2015

Old Talks

Links