Research by Mattias Jonsson

While my thesis work was in higher-dimensional complex dynamics, my current research takes place in two very different areas of mathematics: Singularity Theory and Mathematical Finance.

Here you can find my publications and preprints.
Singularity Theory is about taking a complicated, or singular, object and try to understand just how singular it is and how to simplify the singularity. Examples of singularities are given by the cusp y2=x3 (left) and the Whitney umbrella x2=y2z (right).
My approach (largely joint with Charles Favre at CNRS, Paris) allows for a treatment of a large variety of objects (curves, ideals, graded systems of ideals, currents and dynamical systems) in a unified way. We use an object that we call the valuative tree to efficiently encode singularities (in two dimensions).
This tree is an intriguing object with a rich self-similar (or fractal) structure, but can still be quite concretely described as a (very large) collection of line segments welded together in such a way that no loops appear.

Here is a pdf file with a more technical description of my general research in singularity theory. You may also read more on my current research on singularities in algebraic geometry, complex dynamics and in pluripotential theory. (It may help to read the three papers in this order.)

In Mathematical Finance, my focus is on pricing, hedging, and utility maximization in incomplete markets. Utility maximization means finding optimal ways of investing in financial assets (stocks, bonds, etc). For this to make sense one has to specify what ``optimal'' means and specify a (probabilistic) model for the asset prices. Pricing and hedging refers to derivative securities, such as call options, whose values (payoffs) depend on the value of an underlying financial asset (e.g. stock). A typical problem is then to find theoretical prices of these derivative securities, as well as ways to hedge them, i.e. replicate their payoffs (a liability) by trading in the underlying asset.

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