Award Abstract # 0714086
Algebraic stacks and their applications

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: REGENTS OF THE UNIVERSITY OF CALIFORNIA, THE
Initial Amendment Date: January 22, 2007
Latest Amendment Date: January 22, 2007
Award Number: 0714086
Award Instrument: Standard Grant
Program Manager: Tie Luo
tluo@nsf.gov
 (703)292-8448
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 31, 2006
End Date: May 31, 2010 (Estimated)
Total Intended Award Amount: $96,852.00
Total Awarded Amount to Date: $96,852.00
Funds Obligated to Date: FY 2006 = $96,852.00
History of Investigator:
  • Martin Olsson (Principal Investigator)
    molsson@math.berkeley.edu
Recipient Sponsored Research Office: University of California-Berkeley
1608 4TH ST STE 201
BERKELEY
CA  US  94710-1749
(510)643-3891
Sponsor Congressional District: 12
Primary Place of Performance: University of California-Berkeley
1608 4TH ST STE 201
BERKELEY
CA  US  94710-1749
Primary Place of Performance
Congressional District:
12
Unique Entity Identifier (UEI): GS3YEVSS12N6
Parent UEI:
NSF Program(s): ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source: app-0106 
Program Reference Code(s): 0000, OTHR
Program Element Code(s): 126400
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

The research project is focused on a broad range of questions concerning
algebraic stacks and their applications. The project is divided into two
parts. The first part concerns foundational problems which have arisen
in recent important applications of stacks such as the geometric
Langland's program, the theory of moduli of stable maps, and the
construction of moduli spaces for higher dimensional varieties. The
second part of the program concerns applications of the theory. In
particular new applications of the relationship between log geometry in
the sense of Fontaine and Illusie and stacks discovered by the PI in
earlier work, generalizations of the theory of twisted stable maps, as
well as applications to the construction and study of moduli spaces for
varieties of general type, abelian varieties, and vector bundles on
curves.

The notion of stack is a tool used to deal with internal symmetries of
mathematical objects, as well as actions of groups. For example, when
trying to classify geometric objects one is naturally forced to deal with
the symmetries of the objects in question. In recent years, the theory of
stacks has come to play an important role in almost every part of
algebraic geometry, arithmetic geometry, and mathematical physics and a
great number of exciting new applications of stacks have been found. This
is not surprising considering the importance of symmetries in mathematics
and other fields of science. This great interest in stacks has brought to
light a number of important problems about stacks. The research project
aims to broaden our understanding of both the foundational aspects of the
theory of stacks and as well as the many applications.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Yves Laszlo and Martin Olsson "The six operations for sheaves on Artin stacks II: Adic coefficients" Publ. Math. IHES , v.107 , 2008
D. Abramovich, A. Vistoli, and M. Olsson "Tame stacks in positive characteristic" Annales de l'Institut Fourier , v.58 , 2008 , p.1057
Martin C. Olsson "The Picard group of M_{1,1}" Algebra and Number Theory , 2010
M. Lieblich and M. Olsson "Generators and relations for the \'etale fundamental group" Pure and Applied Math Quarterly , 2010
Yves Laszlo and Martin Olsson "Perverse sheaves on Artin stacks" Math. Zeit. , 2009
Yves Laszlo and Martin Olsson "The six operations for sheaves on Artin stacks I: finite coefficients" Publ. Math. IHES , v.107 , 2008 , p.109

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