This is Part 1 of a two-volume set. 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. 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. 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.
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