The Mathematics of Complex Systems:
Theory and Applications

Description:

This course explores mathematical models used to understand, model and analyse neural dynamics and brain function. Topics include, but are not restricted This course explores mathematical models used to understand, model and analyse neural dynamics and brain function. Topics include, but are not restricted to, brain network analysis, applications of topology in neuroscience, synchronisation phenomena, criticality in neural networks, and applications of differential equations and dynamical systems to neuroscience. Participants will learn about the latest research and develop skills in modelling and analysing neural systems, both theoretically and numerically.to, brain network analysis, applications of topology in neuroscience, synchronization phenomena, criticality in neural networks, and applications of differential equations and dynamical systems to neuroscience. Participants will learn about the latest research and develop skills in modelling and analysing neural systems, both theoretically and numerically.

Description:

Participants will learn about topological methods for analysing complex data. The course includes, but is not limited to, concepts such as simplicial complexes, persistent homology, computational topology, and high- order interactions along with their applications in understanding the structure and dynamics of complex systems. Practical sessions will involve applying these methods to real-world data, such as epidemics, neuroscience, and finance, to name a few. Students will learn modelling and numerical skills (in Python) in topological data analysis.

Description:

This course introduces mathematical models in epidemiology, focusing on the spread of infectious diseases. It includes, but is not restricted to, classic epidemic models, compartmental models, stochastic processes, parameter estimation, anddisease control strategies for and prevention. Case studies on recent epidemics will provide practical insights, both theoretical and in numerical analysis.

Description:

A Markov Decision Processes (MDP) is a model for an agent making sequential in an environment. The agent’s actions influence the evolution of the environment, and garners rewards. The goal of the agent is to choose actions in a way to maximize overallrewards. Reinforcement Learning is a methodology to solve these problems that approximately mimics how humans learn: the agent chooses actions, observes results, and updates behavior accordingly. Applications of RL include robotics, industrial automation, health care, finance and game playing. This course is an introduction to both the mathematical framework and computational techniques necessary to create MDP models and use RL to find optimal solutions. In the applied section of the course, students will use TorchRL to create and optimize a specific MDP model.