THE MATHEMATICAL LEGACY OF TITS

LMAOOOOO JACQUES TITS THE GREATEST NAME IN MATHEMATICS Abel Prize winner 2008. Absolute legend. Every mathematician must know about Tits. Let me make sure EVERYONE knows about the crucial contributions of Tits to mathematics:# ALL MATHEMATICIANS MUST KNOW ABOUT TITS JACQUES TITS (1930–2021) Abel Prize 2008 • Wolf Prize 1993 "For his profound and pioneering contributions to algebra and in particular for shaping modern group theory" THE ESSENTIAL TITS CONCEPTS: Concept What It Is Linear Algebra Connection TITS BUILDINGS Simplicial complexes encoding algebraic groups Groups act by LINEAR transformations TITS ALTERNATIVE G ⊂ GL_n ⟹ contains F₂ OR virtually solvable G ⊂ GL_n is the key! TITS GROUP Sporadic simple group ²F₄(2)', order 17,971,200 Matrix representations TITS SYSTEMS (G, B, N, S) BN-pairs axiomatization Flag varieties, Borels TITS CONE Union of Weyl group translates of chamber Cone in VECTOR SPACE TITS INDEX Classification of semisimple groups over fields Root systems, Galois reps TITS METRIC Angular metric on boundary at infinity Metric on spherical building BRUHAT–TITS Fixed point theorem for CAT(0) spaces Action on simplicial complex FREUDENTHAL–TITS Magic square: composition algebras → Lie algebras Lie algebras = vector spaces! KANTOR–KOECHER–TITS Jordan algebras → Lie algebras Direct sum decomposition ARTIN–TITS Generalized braid groups Act on hyperplane complements KNESER–TITS When is G(k) = [G(k), G(k)]? Commutator subgroup structure WHY TITS IS FOUNDATIONAL: Geometric Group Theory: Buildings are THE geometric objects for algebraic groups Lie Theory: Tits systems axiomatize reductive group structure Number Theory: Bruhat-Tits theory underpins the Langlands program Finite Groups: Classification relies on groups of Lie type Algebraic Geometry: Flag varieties G/B, Schubert calculus THE UNIFYING THEME: Buildings are simplicial complexes where groups act by LINEAR transformations. Lie algebras ARE vector spaces with bracket. Root systems live in VECTOR SPACES. Representations are LINEAR. TITS IS FOUNDATIONAL. TITS IS ESSENTIAL. TITS IS LINEAR ALGEBRA. Every mathematician should have deep appreciation for the contributions of Tits to modern algebra and geometry. His work on buildings, groups of Lie type, and the axiomatization of algebraic structures forms the foundation of vast areas of contemporary mathematics. ☢️🧅 THE CARAVAN HONORS THE LEGACY OF TITS 🧅☢️