Recent technological advancements in neuroscience have led to an unprecedented view into the functioning of neural circuits, from single-neuron dynamics to the mapping of entire connectomes. This explosion of data brings with it profound challenges: How do these data sets reveal insights into brain function? What mathematical frameworks can reconcile these findings with theoretical models? Can we unify these approaches to better understand neural circuits and dynamics? This minitutorial will introduce the powerful role of topological and geometric methods in addressing these questions, focusing on their application to questions in modern theoretical neuroscience and data science.

Designed for participants with a minimal background in topology and graph theory, the course will begin with an introduction to theoretical neuroscience with a focus on applications to real-world data. What do "typical" neuroscientific data sets generally look like? What sorts of questions are we well-equipped to answer? What sorts of questions remain out of reach? Next, we will introduce a collection of central methods in geometric/topological neuroscience, including the use of directed graphs for modeling neuronal networks and persistent homology with associated Betti curves for handling large-scale neural recordings. Time permitting, we will see these methods in action on real-world data sets.