# Computer Algebra in Scientific Computing

## Invited Talks

University of Montpellier
France

### Tropical geometry of biological systems

Mathematical modelling of biological systems is a daunting challenge. In order to cope realistically with the biochemistry of the cell, systems biology models include hundreds of variables. Ecological modelling also involves many variables representing different interacting organisms, undergoing migration and being spatially distributed. Modelling of infectious diseases and cancer gathers all these intricacies: it has to cope with population dynamics, complex single cell dynamics and spatial heterogeneity.
Denis Noble, a pioneer of multi-cellular modelling of human physiology, advocated the use of middle-out approaches in biological modelling. Middle-out is an alternative to bottom-up that tries to explain everything from first principles and to top-down that uses strongly simplified representations of reality. A middle-out model uses just enough details to render the essence of the overall organisation. Although this is potentially a very powerful principle, the general mathematical methods to put it into practice are still awaited.
Recently, we have used tropical geometry to extract the essence of biological systems and to simplify complex biological models. Tropical methods were applied to systems of differential equations with polynomial or rational right hand side and can be potentially applied also to partial differential equations. Tropical geometry methods exploit a property of biological systems called multiscaleness, summarized by two properties: i) the orders of magnitude of variables and timescales are widely distributed and ii) at a given time, only a small number of variables or components play a driving role, whereas large parts of the system have passive roles.
Tropical geometry is the natural framework for scaling and order of magnitude theory. We are using tropical geometry methods to identify sub-systems that are dominant in certain regions of the phase- and/or parameter-space of dynamical systems. These dominant sub-systems are rather simple, even for large systems. Tropical methods also provide the scaling needed to identify slow/fast decompositions and perform model order reduction in the framework of geometric singular perturbations theory. Interestingly, the dominance relations unravelled by tropical geometry can highlight approximate symmetries that are exploited for the simplification of biological systems.
Several tools for tropical simplification of biological systems are currently being developed and will be made available to the computer algebra and computational biology communities.

Werner Seiler
University of Kassel
Germany

### Singularities of Algebraic Differential Equations

We discuss a framework for defining and detecting singularities of arbitrary fully nonlinear systems of ordinary or partial differential equations with polynomial nonlinearities. It combines concepts from differential topology with methods from differential algebra and algebraic geometry. With its help, we provide for the first time a general definition of singularities of partial differential equations and show that it is at least meaningful in the sense that generic points are regular. Our definition is then extended to the notion of a regularity decomposition of a differential equation at a given order and the existence of such decompositions is proven by presenting an algorithm for their effective determination (with an implementation in Maple). Finally, we rigorously define the notion of a regular differential equation (a fundamental concept in the geometric theory of differential equations). We show that our algorithm automatically extracts one provably regular differential equation from each prime component of a given equations and thus provides an effective answer to an old problem in the geometric theory.