Finding all steady states with tropical geometry

Abstract: The point of departure of this talk is the following seemingly simple question: Suppose we are given a reaction network with (generalized) mass action kinetics, and a choice of rate constants and total amounts – how do we find numerical approximations of *all* the positive steady states?

One possible method is to use numerical algebraic geometry to completely solve the steady state equations over the complex numbers, and then use interval arithmetic methods to filter out the real positive solutions. This approach has the advantage of finding all positive steady states with probability 1 (including unstable ones that would be hard to find with traditional ODE methods), and additionally makes it possible to *certify* in the end that all steady states have been found.

In this talk, I will outline the basic ideas behind this type of numerical algebraic geometry solvers, and then describe recent joint work in progress with Feliu, Helminck, Ren, Schröter and Telek, where we use tropical geometry and matroid theory to determine the so-called *steady state degree* of a network, which is needed to ensure efficiency and certifiability.

algebraic geometrydynamical systemsprobability

Audience: researchers in the topic

( video )

---

Seminar on the Mathematics of Reaction Networks

Series comments: Subscription link: list.ku.dk/postorius/lists/morn.list.ku.dk/

This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs twice a month, typically on the 2nd and 4th Thursday of the month, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom! Many of the talks are recorded; to see the recording, from Past Talks, open details of the listed talk for a video link.

The organizers.

---