The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic.

Part 1: A. Bayer, Wall-crossing implies Brill-Noether applications of stability conditions on surfacesR. J. Berman, Kahler-Einstein metrics, canonical random point processes and birational geometryT. Bridgeland, Hall algebras and Doanldson-Thomas invariantsS. Cantat, The Cremona groupA.-M. Castravet, Mori dream spaces and blow-upsT. de Fernex, The space of arcs of an algebraic varietyS. Donaldson, Stability of algebraic varieties and Kahler geometryL. Ein and R. Lazarsfeld, Syzygies of projective varieties of large degree: Recent progress and open problemsE. Gonzalez, P. Solis, and C. T. Woodward, Stable gauged mapsD. Greb, S. Kebekus, and B. Taji, Uniformisation of higher-dimensional minimal varietiesH. D. Hacon, J. McKernan, and C. Xu, Boundedness of varieties of log general typeD. Halpern-Leistner, $\Theta$-stratifications, $\Theta$-reductive stacks, and applicationsA. Horing and T. Peternell, Bimeromorphic geometry of Kahler threefoldsS. J. Kovacs, Moduli of stable log-varieties-An updateA. Okounkov, Enumerative geometry and geometric representation theoryR. Pandharipande, A calculus for the moduli space of curvesZ. Patakfalvi, Frobenius techniques in birational geometryM. Paun, Singualar Hermitian metrics and positivity of direct images of pluricanonical bundlesM. Popa, Positivity for Hodge modules and geometric applicationsR. P. Thomas, Notes on homological projective dualityY. Toda, Non-commutative deformations and Donaldson-Thomas invariantsV. Tosatti, Nakamaye's theorem on complex manifoldsPart 2: D. Ben-Zvi and D. Nadler, Betti geometric LanglandsB. Bhatt, Specializing varieties and their cohomology from characteristic 0 to characteristic $p$T. D. Browning, How often does the Hasse principle hold?L. Caporaso, Tropical methods in the moduli theory of algebraic curvesR. Cavalieri, P. Johnson, H. Markwig, and D. Ranganathan, A graphical interface for the Gromov-witten theory of curvesH. Esnault, Some fundamental groups in arithmetic geometryL. Fargues, From local class field to the curve and vice versaM. Gross and B. Siebert, Intrinsic mirror symmetry and punctured Gromov-Witten invariantsE. Katz, J. Rabinoff, and D. Zureick-Brown, Diophantine and tropical geometry, and uniformity of rational points on curvesK. S. Kedlaya and J. Pottharst, On categories of $(\varphi,\Gamma)$-modulesM. Kim, Principal bundles and reciprocity laws in number theoryB. Klingler, E. Ullmo, and A. Yafaev, Bi-algebraic geometry and the Andre-Ooert conjectureM. Lieblich, Moduli of sheaves: A modern primerJ. Nicaise, Geometric invariants for non-archimedean semialgebraic setsT. Pantev and G. Vezzosi, Symplectic and Poisson derived geometry and deformation quantizationA. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomologyT. Saito, On the proper push-forward of the characteristic cycle of a constructible sheafT. Szamuely and G. Zabradi, The $p$-adic Hodge decomposition according to BeilinsonA. Tamagawa, Specialization of $\ell$-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varietiesO. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields.