How do our brains make sense of a complex and unpredictable world? In this talk, I will discuss a physicist's approach to the neural topography of information processing in the brain. First I will review the brain's architecture, and how neural circuits map out the sensory and cognitive worlds. Then I will describe how highly complex sensory and cognitive tasks are carried out by the cooperative action of many specialized neurons and circuits, each of which has a simple function. I will illustrate my remarks with one sensory example and one cognitive example. For the sensory example, I will consider the sense of smell ("olfaction"), whereby humans and other animals distinguish vast arrays of odor mixtures using very limited neural resources. For the cognitive example, I will consider the "sense of place", that is, how animals mentally represent their physical location. Both examples demonstrate that brains have evolved neural circuits that exploit sophisticated principles of mathematics - principles that scientists have only recently discovered.