| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 18, 2011 |
| Latest Amendment Date: | June 3, 2013 |
| Award Number: | 1101343 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Andrew Pollington
adpollin@nsf.gov (703)292-4878 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2011 |
| End Date: | August 31, 2015 (Estimated) |
| Total Intended Award Amount: | $360,000.00 |
| Total Awarded Amount to Date: | $360,000.00 |
| Funds Obligated to Date: |
FY 2012 = $120,000.00 FY 2013 = $120,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
9500 GILMAN DR LA JOLLA CA US 92093-0021 (858)534-4896 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
9500 GILMAN DR LA JOLLA CA US 92093-0021 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
01001213DB NSF RESEARCH & RELATED ACTIVIT 01001314DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The PI proposes to develop new explicit relationships between different cohomology theories on algebraic varieties, generalize existing relationships between Betti and de Rham cohomology (Hodge theory), and between etale and de Rham cohomology (p-adic Hodge theory). One particulra project is a description of the etale fundamental groups of nonarchimedean analytic spaces; this will lead to better understanding of period domains and period mappings in p-adic Hodge theory. Other potential results include links between de Rham cohomology and algebraic K-theory, explicit descriptions of etale cohomology suitable for machine computations, and variants of de Rham cohomology giving rise to spectral interpretations of L-functions by analogy with crystalline cohomology.
On the technical side, this project is potentially quite transformative, bringing together areas of research that have historically proceeded in parallel but lacking a coherent synthesis. In the long run, these methods may lead to new techniques for attacking classical question of number theory, such as the distribution and aggregate properties of prime numbers. As evidenced by the PI's prior investigations, they are also likely to lead to improvements in computational and numerical methods in number theory, which may have impacts in areas of application of number theory to computer science (notably information security).
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Number theory is one of the most ancient branches of mathematics, as it includes questions about integer solutions of algebraic equations that date back millennia. In modern times, it is often studied in conjunction with algebraic geometry, the study of solutions of algebraic equations from a geometric point of view, thus forming the hybrid field of arithmetic algebraic geometry.
Much of arithmetic algebraic geometry in general, and the PI's work in particular, involves close analogies between centuries-old methods of calculus and number-theoretic variants introduced in the 20th century. These mostly involve use of p-adic number systems, in which one picks a prime number (called p) and declares two integers to be close together if their difference is divisible by a large power of the chosen prime.
In this project, the PI has focused on p-adic Hodge theory, which amounts to a p-adic analogue of the theorem of Stokes from multivariable calculus. Since its introduction in the 1980s, p-adic Hodge theory has rapidly been adopted as a key ingredient in much recent work in number theory, including the work on modularity of Galois representations triggered by the resolution of the Fermat problem by Wiles.
While numerous results have been obtained by the PI, the most pivotal is a series of foundational results adapting p-adic Hodge theory to a new geometric framework. This involves certain exotic new objects known as "perfectoid spaces", for which much of the basic theory needed to be established from scratch. It is anticipated that the foundations established in this work will lead to an improved understanding of the local Langlands correspondence, one of the key organizing principles in arithmetic algebraic geometry (sometimes likened to the standard model of particle physics).
In addition, the PI has supervised several PhD theses and delivered instructional lectures suitable for graduate students at a variety of venues around the world.
Last Modified: 09/20/2015
Modified by: Kiran Kedlaya
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