Explore the intersection of topology, convexity, and neural networks in this 45-minute lecture by Chad Giusti, presented as part of the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into topics such as receptive fields, neuron convexity, and cofiring events, while examining the Nobel Prize-winning research on brain topology. Investigate the concept of inverse questions and geometric features in neural networks, and learn about open and dark convex structures. Discover the nerve theorem and its applications to feedforward neural networks, and consider the potential implications of convex codes in understanding brain function. Gain insights into cutting-edge research that bridges mathematical concepts with neuroscience, offering a unique perspective on the complexities of neural information processing.