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Explore the intersection of o-minimality and topology in this advanced mathematical lecture delivered at the Institute for Advanced Study's Workshop on Special Cycles and Related Topics. Delve into how topology has fundamentally shaped the development of o-minimal structures, beginning with the framework of 'tame topology' built upon euclidean order topology. Examine the crucial role of topological objects definable in o-minimal structures, including definable groups with their natural manifold topology, definable complex analytic spaces used in Hodge theory applications, and definable linear orders, function spaces, and metric spaces. Learn about recent advances in understanding the general nature of topological spaces definable in o-minimal structures, including classification and embedding results for one-dimensional spaces. Discover analogues of topological properties adapted for the o-minimal setting such as compactness, separability, and first-countability, along with their various applications. Understand how this research connects to open conjectures in set-theoretic topology while employing methods grounded in elementary o-minimal theory, representing collaborative work with Andújar Guerrero and Walsberg, and relating to independent research by Peterzil and Rosel.
