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Explore advanced topics in algebraic geometry through this comprehensive lecture series delivered during the Junior Trimester Program at the Hausdorff Research Institute for Mathematics. Delve into cutting-edge research areas including modular forms, Shimura varieties, K3 surfaces, birational geometry, and tropical geometry through presentations by leading mathematicians in the field. Learn about CM values of regularized theta lifts, pro-étale sites, arithmetic transfer conjectures, and exotic formal moduli spaces while examining the connections between algebraic geometry and number theory. Study toric varieties via Cox rings, moduli spaces of stable sheaves, non-commutative nodal curves, and Hilbert schemes of points on various algebraic surfaces. Investigate mirror symmetry for orbifold del Pezzo surfaces, semiorthogonal decompositions in birational geometry, and stability conditions with flat connections. Examine the geometry of singularities in the Minimal Model Program, hyperholomorphic sheaves, and generalized deformations of algebraic varieties. Discover applications to rational points on elliptic curves, Beilinson-Flach elements, and p-adic Gross-Zagier formulas on Shimura curves. Analyze Betti numbers of hypersurfaces, defects of linear systems, and the equations defining projective varieties through multiple detailed sessions. Gain insights into the geography and geometry of 3-folds of general type, Q-Gorenstein deformations, and derived categories in global dimension two.
