Under Construction

This course focuses on the mathematical modeling of sustainability challenges. Real-world sustainability problems often involve numerous interacting factors, making them difficult to capture through standard mathematical tools. Sustainability—defined broadly as the ability to maintain or support a process continuously over time—has gained immense attention in recent years, yet it still requires deeper theoretical grounding.

This course explores these challenges through a quantitative lens. It is designed as an experimental space to identify real-world problems that can be formulated and partially solved using accessible mathematical tools. Consequently, the coursework will primarily rely on methods participants have already encountered in their regular studies, supplemented by new perspectives and occasional introductory lectures on any necessary background mathematics.

The main emphasis is on building and solving practical models by combining different mathematical techniques. Rather than focusing on abstract theory, the curriculum addresses concrete problems across infrastructure, business, society, and environmental management where quantitative reasoning can guide better long-term decisions.

Instructors:

Gunnar Tibbert & Henrik Shahgholian

Schedule: Spring 2027.

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

Goals:

This course explores the mathematical modeling of sustainability challenges across industry, finance, society, and the environment. Because these real-world problems involve numerous interacting factors, students learn to formulate and solve practical models by combining accessible mathematical tools and methods from their previous studies.

The primary aim is to equip participants to recognize sustainability issues in their professional roles and apply analytical thinking to deliver rational solutions. The course also serves as a pathway for students seeking a research specialization in applied mathematics, modeling, and simulation.

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 groups.

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).