Abstract: Arc-analytic geometry is a fairly new branch of real analytic and algebraic geometry, inspired by John Nash's questions about the role of arcs in geometry. It studies the so-called arc-analytic functions and arc symmetric sets. In the algebraic setting, it provides the most satisfactory real counterpart of Zariski topology over an algebraically closed field. In the analytic setting, in contrast, it consists at present mostly of tough open problems. I will give a brief history of the subject with an overview of classical results from the late 80's and the 90's, and discuss both the recent progress and questions. The talk should be accessible to senior math undergrads.

Animal populations often self-organize into territorial structure from movements and interactions of individual animals. Spatial memory is one of cognitive processes that may affect the movement and navigation of the animals. We will review several mathematical approaches of animal spatial movements: (i) reaction-diffusion-advection model with time-delayed memory-based movement; and (ii) reaction-diffusion-advection model with a non-local advection term driven by a cognitive map representing memory of past animal locations embedded in the environment. The well-posedness of models and bifurcation of spatiotemporal patterns will be discussed.