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Explore advanced topics in algebraic groups, arithmetic lattices, and geometric applications through this comprehensive conference featuring over 40 lectures from leading mathematicians. Delve into Zariski dense subgroups and their connections to number theory, covering classification and reduction of toric principal bundles, uniform stability of higher-rank arithmetic groups, and the existence problem of compact quotients of pseudo-Riemannian homogeneous spaces. Examine property (T) for various group structures, including automorphism groups of free groups and fiber products, while investigating sum of squares methods and positivity conditions. Study the Hasse principle for reductive algebraic groups over finitely generated fields and p-adic function fields, alongside linear representations of the Grothendieck-Teichmuller group. Learn about bounded generation of linear groups and its applications to Diophantine approximation through a dedicated mini-course, and gain insights into Bruhat-Tits theory across multiple lectures. Investigate spectral gaps, orbit closures on homogeneous spaces, and their applications to number theory, including thin orbits and character varieties. Explore the systolic geometry of arithmetic locally symmetric spaces, non-commensurable isospectral spaces, and hypergeometric groups with their arithmeticity properties. Discover rigidity phenomena in various contexts, from representation rigidity to profinite rigidity, finite step rigidity, and the non-left-orderability of arithmetic groups of higher real rank. Examine local-global principles for norm one tori, the genus of division algebras over discrete valued fields, and homogeneity results for invariant distributions on p-adic symmetric spaces.
