This course is about mathematical modeling of complex systems and sustainability. In mathematics, the word complex system refers to phenomena that are difficult to model through mathematical tools, due to many governing parameters. Examples are abundant, with Meteorological Phenomena being one of the earliest examples of complex systems. In this course, we will address several complex systems from diverse areas that occur in nature, biology, socio-economics, and so on. An important class of problems includes those that are related to sustainability, that is extremely hard to model because of their complexity. The word sustainability (in short) refers to the ability to maintain or support a process continuously over time. The topic has gained an immense momentum in the last few years, and is now one of the hottest topics that still needs to find theoretical ground.

This course is an attempt to explore the above topics with mathematical tools, and raise the awareness on sustainability, and possible modelling of such problems. The course is at an experimental level for finding real-world problems that can be modelled and partially solved with the simplest mathematical tools. Therefore, the mathematical part of the course will mainly comprise tools and methodologies that the participants have already acquired in their regular courses. However, some new thinking and tools will also be introduced in the course. If needed, there will also be a few lectures covering the basics of mathematics needed for the course.

The focus of the course will be on modelling and solving simple problems where different mathematical tools are to be combined.

Most of the course will address various problems which occur in complex systems and sustainability and where applications in infrastructure, business, society and other institutional activities become important.

Instructors:

Henrik Shahgholian (Examiner) henriksh@kth.se Sonja Radosavljevic (TA) sonjara@kth.se

Schedule: Spring 2023.

Lectures/Exercise sessions: Tuesdays 13:15--16:00, room 3424, Lindstedsvägen 25, in Math building.

Jan-March: Lecturing, April: consultation and advice for projects, May: presentations

Goals:

The aim of the course is to introduce the participants with various problems in industry, the financial sector, society, and environmental issues that need to be studied from a sustainability perspective. Since these problems are very complex and their mathematical modelling requires a broad knowledge, we introduce sustainability as a learning goal through complex systems. After completing the course, students must be able to be aware of sustainability problems in their professional roles and be able (through analytical and mathematical thinking) to offer rational solutions to such problems.
The course also provides an opportunity for a student to choose a research specialization in applied mathematics with modelling and simulation of mathematical models in complex systems and sustainability.

Syllabus

1: Climate modelling

Components of a climate model (Atmosphere, Ocean, Sea ice, Land surface) are introduced, as well as their mathematical counterparts (velocity, pressure, temperature, specific humidity, density). Continuity equation, conservation of mass, and first law of thermodynamics, are to follow such models, as well as basic equations. Here we use partial differential equations, at its simplest level, along with Lorenz equation.

2: Ecosystem

Starting with the basic concepts of ecology and its branches, we shall introduce simplest model of population dynamics in ecology and link to simple ecosystems. Interaction between species, such as pre-predatory models with our without environmental factor shall be discussed, using Lotka–Volterra equations, that is a system of differential equation. By introducing laws in population ecology we shall model various interactions (cooperation, competition, interacting species, limiting factors) that may affect the modelling. We shall also take a look at meta-ecosystems, which concerns interaction between various ecosystems.

3: Pollution and air quality modelling

We discuss pollutants in general, and their impact on all life on Earth. We shall also have a look at existing statistics/data on how the environment and people are affected by various pollutants. The focus shall stay on primary human-created pollutions (trash, runoff produced by factories). We shall study the atmospheric transport, diffusion and chemical reaction of pollutants and also many other hazardous concentration of species. Therefore, we shall introduce the equation that governs the mass transportation and study this theoretically and numerically.

4: Energy transition: costs and challenges.

We discuss global energy crisis (causes, effects, possible solutions) and what is shaping the global future of energy. By introducing various energy modes/sources, and presenting factors related to the energy transition and the costs for a fossil free energy. Our first attempt on mathematical modelling is the utilization of elementary mathematics, to create models that (accurately) simulate the deployment of renewable energy production capacities within the context of the energy transition. In this respect Energy Return on (energy) Investment (EROI) is a fundamental thermodynamic metric applied to power generation, measuring relative inputs and outputs. Our second model concerns decision under uncertainty and the evaluation of investment projects, when a transition in energy mode is to take place. We utilize the so-called optimal switching problem, for modelling optimal strategies for switching between energy modes.

Topics: Students may choose any of the below topics for their projects.

Climate and environment (Global warming and climate changes, carbon circulation, pollution, environment versus economic growth)

Traffic flow (congestion, crowd dynamics, ...).

Complex industrial modelling (recycling/circulation, carbon emission, ....)

Socioeconomics (health, criminology, poverty, collective behaviours, ...)

Biological systems, Life sciences, ecology, over-harvesting (fishing), overexploitation (agriculture)

Sustainable/Green Finance (energy market, ..)

Catastrophe theory (environmental disasters, market crash, war, ...)

Examination:

The examination consists of two parts:

i) A few weekly homework.

ii) A project in a group of two participants.

You will receive suggestions and advices to choose a so-called case study and then present this as a project. A written report and a beamer presentation of approx. 30 min. is required. You will receive exact instructions about this in the course.

Literature:

Book chapters, and articles. These will be addressed during the course. The best way is to search on the net, with correctly chosen keywords. However, during the course the instructor will suggest reading materials, and also help with search. Below is a l list of some standard materials:

Modelling literature:

1) Introduction to the Theory of Complex Systems (Stefan Thurner, Rudolf Hanel, and Peter Klimek) https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198821939.001.0001/oso-9780198821939 (Links to an external site.)

2) Complex Systems And Society (Nicola Bellomo, Giulia Ajmone Marsan, Andrea Tosin) https://link.springer.com/book/10.1007/978-1-4614-7242-1 (Links to an external site.)

3) Mathematics and Climate (Hans Kaper, Hans Engler) https://epubs-siam-org.focus.lib.kth.se/doi/book/10.1137/1.9781611972610Links to an external site.

4) Introduction to climate dynamics and climate modeling (Goose) http://www.climate.be/textbook/ (Links to an external site.)

5) Meta-Ecosystem Dynamics: Understanding Ecosystems Through the Transformation and Movement of Matter. Frederic Guichard, Justin Marleau

6) More literature ref. to follow.

Mathematical tools:

Multivariable Calculus & Linear Algebra, ODE, Simple PDEs (Laplacian, heat equation), Equations of continuity, Fourier Analysis, Control theory, Probability and Stochastic, Statistical methods, Numerical methods (some programming will be needed).