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Explore the Tate locus and its conjectures in this advanced mathematics lecture from the Institute for Advanced Study's Workshop on Special Cycles and Related Topics. Delve into the degeneracy locus of l-adic local systems on geometrically connected varieties over fields, examining its role as the étale counterpart to the Hodge locus of variation of Hodge structures. Learn about the significant progress made in understanding Hodge loci through o-minimality techniques while discovering the current challenges and open questions surrounding Tate loci. Review the main conjectures about this mathematical object when working over number fields, and examine the current state of what can be proven rigorously. Gain insights into potential applications of these theoretical results and, time permitting, explore sketches of proof techniques. The presentation draws from collaborative research with Jakob Stix and Akio Tamagawa, offering a comprehensive overview of this specialized area at the intersection of algebraic geometry and arithmetic geometry.
