Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. These volumes include 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. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic $p$ and $p$-adic tools, etc. The resulting articles will be important references in these areas for years to come.
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Tommaso de Fernex, University of Utah, Salt Lake City, UT.
Brendan Hassett, Brown University, Providence, RI.
Mircea Mustata, University of Michigan, Ann Arbo, MI.
Martin Olsson, University of California, Berkeley, CA.
Mihnea Popa, Northwestern University, Evanston, IL.
Richard Thomas, Imperial College of London, United Kingdom.
Part 1: A. Bayer, Wall-crossing implies Brill-Noether applications of stability conditions on surfaces
R. J. Berman, Kahler-Einstein metrics, canonical random point processes and birational geometry
T. Bridgeland, Hall algebras and Doanldson-Thomas invariants
S. Cantat, The Cremona group
A.-M. Castravet, Mori dream spaces and blow-ups
T. de Fernex, The space of arcs of an algebraic variety
S. Donaldson, Stability of algebraic varieties and Kahler geometry
L. Ein and R. Lazarsfeld, Syzygies of projective varieties of large degree: Recent progress and open problems
E. Gonzalez, P. Solis, and C. T. Woodward, Stable gauged maps
D. Greb, S. Kebekus, and B. Taji, Uniformisation of higher-dimensional minimal varieties
H. D. Hacon, J. McKernan, and C. Xu, Boundedness of varieties of log general type
D. Halpern-Leistner, $\Theta$-stratifications, $\Theta$-reductive stacks, and applications
A. Horing and T. Peternell, Bimeromorphic geometry of Kahler threefolds
S. J. Kovacs, Moduli of stable log-varieties-An update
A. Okounkov, Enumerative geometry and geometric representation theory
R. Pandharipande, A calculus for the moduli space of curves
Z. Patakfalvi, Frobenius techniques in birational geometry
M. Paun, Singualar Hermitian metrics and positivity of direct images of pluricanonical bundles
M. Popa, Positivity for Hodge modules and geometric applications
R. P. Thomas, Notes on homological projective duality
Y. Toda, Non-commutative deformations and Donaldson-Thomas invariants
V. Tosatti, Nakamaye's theorem on complex manifolds
Part 2: D. Ben-Zvi and D. Nadler, Betti geometric Langlands
B. 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 curves
R. Cavalieri, P. Johnson, H. Markwig, and D. Ranganathan, A graphical interface for the Gromov-witten theory of curves
H. Esnault, Some fundamental groups in arithmetic geometry
L. Fargues, From local class field to the curve and vice versa
M. Gross and B. Siebert, Intrinsic mirror symmetry and punctured Gromov-Witten invariants
E. Katz, J. Rabinoff, and D. Zureick-Brown, Diophantine and tropical geometry, and uniformity of rational points on curves
K. S. Kedlaya and J. Pottharst, On categories of $(\varphi,\Gamma)$-modules
M. Kim, Principal bundles and reciprocity laws in number theory
B. Klingler, E. Ullmo, and A. Yafaev, Bi-algebraic geometry and the Andre-Ooert conjecture
M. Lieblich, Moduli of sheaves: A modern primer
J. Nicaise, Geometric invariants for non-archimedean semialgebraic sets
T. Pantev and G. Vezzosi, Symplectic and Poisson derived geometry and deformation quantization
A. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology
T. Saito, On the proper push-forward of the characteristic cycle of a constructible sheaf
T. Szamuely and G. Zabradi, The $p$-adic Hodge decomposition according to Beilinson
A. Tamagawa, Specialization of $\ell$-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varieties
O. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields.
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