A Mathematical Journey Through Networks, Chemistry, fixation of popularity, and Machine Learning. Meditation on "what is the use?"

What is the use of learning something when its connection to our future plans is nowhere in sight? Students ask this question, but I used to regularly ask this as well. ""What is the use of taking this course?"" ""I am super curious about this specific topic but I don't see how it will help me in my career so let me move on"" etc. In this public lecture, I will tell a story about what can happen when we follow curiosity before we know where it will lead, echoing John Lennon's reminder, that life unfolds while we are busy making plans. Our story begins with the minimum spanning tree. Given a network of possible connections, each with a cost, how can we connect everything as cheaply as possible? This deceptively simple object also lies behind single-linkage clustering, a foundational method for finding structure in unlabeled data. But what does this tree look like when the network (number of data points) is enormous and random? Numerical experiments in statistical physics suggested a remarkable answer: across many different models, its large-scale shape should be universal. Turning that prediction into rigorous mathematics led through places that initially seemed far removed from machine learning. We will encounter models of particles merging in colloidal chemistry, a beautiful random process called the multiplicative coalescent, Erdős's leader problem - a model for the fixation of popularity in political group formation, and network models in which a small amount of choice ca