Categorical Kahler Geometry and Applications
Categorical Kahler Geometry and Applications
批准号:
2001319
负责人:
Ludmil Katzarkov
金额:
$22.5万
依托单位:
依托单位国家:
美国
项目类别:
Continuing Grant
财政年份:
2020
资助国家:
美国
项目状态:
已结题
起止时间:
2020-05-01 至 2024-04-30
中文摘要
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英文摘要
Birational geometry is a classical mathematical discipline whose roots go back to ancient Greece. Nevertheless, it still offers many difficult unsolved questions. The core part of this project is to tackle these questions with cutting-edge modern methods coming from the homological mirror symmetry program. Homological mirror symmetry is a deep geometric duality that originates in quantum field theory and has been used in studying novel phenomena and proving unexpected results in symplectic geometry suggested by algebraic geometry. This project aims to use homological mirror symmetry to introduce new applications of symplectic topology to algebraic geometry and to answer classical open questions in birational geometry. A postdoctoral fellow will be involved in various aspects of this research project.This project will use an approach based on categorical Kähler geometry. The most notable application of this approach is toward proving the non-rationality of generic, four-dimensional cubics, which is arguably the central problem in algebraic geometry. More specifically, a detailed study of the singularities of quantum D-modules produces a completely new type of birational invariant. This new invariant is a canonical decomposition of the cohomology of a four-dimensional cubic based on simultaneous use of both (algebraic and symplectic) sides of homological mirror symmetry. The example of four-dimensional cubics is only the tip of the iceberg. There are many other potential applications of this approach, for example to uniformization problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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会议论文
FRG: Collaborative Research: New Birational Invariants
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批准号:2245171
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项目类别:Continuing Grant
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资助金额:$50.0万
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财政年份:2023
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负责人:Ludmil Katzarkov
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依托单位:
Conference on Homological Mirror Symmetry
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批准号:2001614
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项目类别:Standard Grant
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资助金额:$2.0万
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财政年份:2020
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负责人:Ludmil Katzarkov
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依托单位:
Homological Mirror Symmetry Conference Miami 2015
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批准号:1502578
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项目类别:Standard Grant
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资助金额:$3.0万
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财政年份:2015
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负责人:Ludmil Katzarkov
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依托单位:
Homological Mirror Symmetry and Categorical Linear Systems
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批准号:1502162
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项目类别:Continuing Grant
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资助金额:$17.0万
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财政年份:2015
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负责人:Ludmil Katzarkov
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依托单位:
Homological Mirror Symmetry MIAMI, Jan 27- Feb 1, 2014
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批准号:1404779
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项目类别:Standard Grant
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资助金额:$3.5万
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财政年份:2014
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负责人:Ludmil Katzarkov
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依托单位:
Homological Mirror Symmetry Conference Miami
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批准号:1303069
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项目类别:Standard Grant
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资助金额:$3.65万
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财政年份:2013
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负责人:Ludmil Katzarkov
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依托单位:
FRG: Collaborative Research: Wall-crossings in Geometry and Physics
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批准号:1265230
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项目类别:Standard Grant
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资助金额:$20.02万
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财政年份:2013
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负责人:Ludmil Katzarkov
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依托单位:
Spectra, gaps, degenerations and cycles
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批准号:1201475
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项目类别:Continuing Grant
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资助金额:$24.3万
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财政年份:2012
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负责人:Ludmil Katzarkov
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依托单位:
Pan American Advanced Studies Institute on Wall Crossing, Stability Hodge Structures and TQFT- Natal, Brazil
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批准号:1242272
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项目类别:Standard Grant
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资助金额:$10.0万
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财政年份:2012
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负责人:Ludmil Katzarkov
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依托单位:
Geometry and Physics Miami - Brazil - Mexico - Conference
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批准号:1201544
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项目类别:Standard Grant
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资助金额:$3.0万
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财政年份:2012
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负责人:Ludmil Katzarkov
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依托单位:
FRG: Collaborative Research: Mirror Symmetry & Tropical Geometry
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批准号:0854977
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项目类别:Standard Grant
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资助金额:$64.29万
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财政年份:2009
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负责人:Ludmil Katzarkov
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依托单位:
Generalized Homological Mirror Symmetry
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批准号:0901330
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项目类别:Standard Grant
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资助金额:$16.24万
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财政年份:2009
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负责人:Ludmil Katzarkov
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依托单位:
FRG Collaborative Research: Homological Mirror Symmetry and its applications
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批准号:0652633
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项目类别:Continuing Grant
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资助金额:$30.0万
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财政年份:2007
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负责人:Ludmil Katzarkov
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依托单位:
Homological Mirror Symmetry and applications
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批准号:0600800
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项目类别:Continuing Grant
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资助金额:$0.0万
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财政年份:2006
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负责人:Ludmil Katzarkov
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依托单位:
Geometry and Arithmetics
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批准号:0535363
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项目类别:Standard Grant
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资助金额:$1.2万
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财政年份:2005
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负责人:Ludmil Katzarkov
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依托单位:
Conference: Algebraic, Symplectic Geometry and Physics; December 15-20, 2004; Coral Gables, FL
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批准号:0506503
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项目类别:Standard Grant
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资助金额:$0.9万
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财政年份:2004
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负责人:Ludmil Katzarkov
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依托单位:
CAREER: Non-abelian Hodge Theory and Monodromy Actions
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批准号:9875383
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项目类别:Standard Grant
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资助金额:$20.0万
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财政年份:1999
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负责人:Ludmil Katzarkov
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依托单位:
Motives, Polylogs, and Non-Abelian Hodge Theory
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批准号:9801647
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项目类别:Standard Grant
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资助金额:$1.1万
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财政年份:1998
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负责人:Ludmil Katzarkov
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依托单位:
Non-Abelian Hodge Theory and Applications
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批准号:9700605
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项目类别:Standard Grant
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资助金额:$9.15万
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财政年份:1997
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负责人:Ludmil Katzarkov
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依托单位:
国内基金
海外基金
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资助金额:3.0万元
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曲率几乎非负的紧致Kahler流形的几何与拓扑
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批准号:11601044
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度量几何及其在Kahler几何中的应用
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批准号:11501501
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资助金额:18.0万元
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Kahler 曲面中特殊曲面的研究
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批准号:11471014
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资助金额:65.0万元
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Kahler几何与辛拓扑中若干问题的研究
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Kahler曲面上极值度量若干问题研究
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资助金额:60.0万元
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