ECMI Modelling Week

Nowadays, when humanity is faced with various challenges regarding ecology and environmental protection, it is interesting to observe the climate of a region through the prism of mathematics and mathematical models. Mathematical models are used to better understand the nature of the behavior of physical characteristics such as temperature, pressure, wind speed, etc. in certain regions. The goal is to first confirm the trend of climate behavior through mathematics based on real data, and then use simulations to predict future behavior. The models used for these purposes are complex equations that require knowledge and application of various areas of mathematics, and for simulations, programming languages.

In this project the students will decide about the time and size of investments in green energy, when subsidies are in place, but there is uncertainty about their duration. Mathematically speaking, this problem is an optimal stopping problem. Under some assumptions, the problem can be analytically solved. But as we introduce fewer assumptions, the problem becomes harder to solve analytically. Therefore, we need to use numerical methods or machine learning methods.

In the modern world, reliable wireless internet connectivity is a necessity in homes, offices, and public spaces. The strategic placement of Wi-Fi routers directly affects signal strength, coverage, and overall network performance. Optimizing router placement can enhance productivity in office spaces, ensure smooth streaming and gaming experiences in homes, and reduce costs by minimizing the need for additional networking equipment. Industries such as telecommunications, smart home technology providers, and facility management services increasingly seek automated tools that predict and optimize Wi-Fi signal distribution for varied building layouts.

Despite its significance, determining the best locations for Wi-Fi routers is a complex task involving various factors: building geometry, the presence of obstacles (walls, windows, furniture), and user-defined priorities for different areas. By employing mathematical modeling, this project aims to develop a robust solution to automate router placement recommendations using physical principles of wave propagation and mathematical equations that describe signal behavior.

Students participating in this project will:
1. Apply differential equations to model signal propagation in a two-dimensional space.
2. Explore the impact of different geometries and boundary conditions (representing walls, windows, and doors).
3. Model multiple signal sources by introducing source functions for various router locations.
4. Learn numerical techniques for solving PDEs and ODEs, including finite element or finite difference methods.
5. Develop python skills in programming and visualization tools to create user-friendly models.

The ultimate objective is to create a general-purpose tool that simplifies Wi-Fi optimization for users with no mathematical background.

Complex biological organisms interact with their environment through various external stimuli, one of the most important being vision. When light hits the eye, it triggers a cascade of neural activity: the signal travels through the optic nerve to the primary visual cortex (V1), and then through higher-level visual pathways to encode and interpret the incoming information.

This raises an intriguing question: can we decode this visual information using high-temporal-resolution data such as electroencephalography (EEG)?
This problem focuses on the analysis of multi-channel EEG recordings collected from several subjects who were shown diverse natural scene images from various categories. The recordings span multiple sessions, during which images were presented in structured blocks.

The aim is to use generative deep learning architectures to reconstruct the visual stimuli, that is, the images seen by the subjects, based on their recorded neural activity.