Award Abstract # 1711178
Manifolds with Special Holonomy and Applications

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ROCHESTER
Initial Amendment Date: July 10, 2017
Latest Amendment Date: August 11, 2021
Award Number: 1711178
Award Instrument: Standard Grant
Program Manager: Christopher Stark
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 15, 2017
End Date: July 31, 2023 (Estimated)
Total Intended Award Amount: $165,166.00
Total Awarded Amount to Date: $198,199.00
Funds Obligated to Date: FY 2017 = $165,166.00
FY 2021 = $33,033.00
History of Investigator:
  • Sema Salur (Principal Investigator)
    sema.salur@rochester.edu
Recipient Sponsored Research Office: University of Rochester
910 GENESEE ST
ROCHESTER
NY  US  14611-3847
(585)275-4031
Sponsor Congressional District: 25
Primary Place of Performance: University of Rochester
Dept. of Mathematics 915 Hylan Hall
Rochester
NY  US  14627-0140
Primary Place of Performance
Congressional District:
25
Unique Entity Identifier (UEI): F27KDXZMF9Y8
Parent UEI:
NSF Program(s): GEOMETRIC ANALYSIS
Primary Program Source: 01001718DB NSF RESEARCH & RELATED ACTIVIT
01002122DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 102Z
Program Element Code(s): 126500
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

The unification of the four fundamental forces of nature--electromagnetism, gravity, the strong and weak nuclear forces--is one of the greatest unsolved mysteries of physics. Over the last few decades, M-theory, a "theory of everything",  has emerged as a candidate for such a unification of these forces.  This project is about manifolds with special holonomy, spaces whose infinitesimal symmetries allow them to play a crucial role in M-theory 'compactifications'---that is, they model the tiny 'curled up' dimensions lurking at every point of spacetime.   In this project, the principal investigator will focus in particular on 6-dimensional Calabi-Yau manifolds (which play the analogous role of the curled-up dimensions of superstring theory) and spaces of dimension 7 and 8 whose symmetries fill out the special holonomy groups known as G2 and Spin(7), respectively.  Despite extensive research on Calabi-Yau manifolds, the geometric properties of G2 and Spin(7) manifolds are not well understood, and the problem of the existence of calibrated (i.e., volume minimizing) submanifolds is still wide open.  One goal of this project is to develop techniques that are robust enough to handle these difficult existence questions. Another goal is to study the deformation spaces of calibrated submanifolds, as understanding these spaces will ultimately be useful for M-theory compactifications.  The PI also believes that manifolds with special holonomy is an excellent topic for graduate research, and intends to continue to supervise PhD students.  She plans to encourage women and members of other under-represented groups to take up graduate study and continue to research careers in differential geometry, through activities that include advising, organizing seminars, special sessions, conference and "Women in Math" workshops.

In this project, the PI plans to continue her work on Ricci-flat manifolds, their calibrated geometries and the compactifications of moduli spaces. In recent joint work with F. Arikan and H. Cho, she showed that every G2 manifold is an almost contact manifold. Studying the relations between contact and G2 structures can be useful to find the existence conditions of a G2 metric  on 7-manifolds (similar to the existence conditions of the Calabi-Yau metric). In another joint work with Cho and A.J. Todd, she investigated the properties of G2 manifolds from a symplectic point of view. Using contact and symplectic structures, the PI plans to construct Lagrangian and Legendrian type submanifolds  of G2 and Spin(7) manifolds. Also, in joint work with C. Robles, she applied the Cartan-Kahler theory to associative and Cayley embeddings into G2 and Spin(7) manifolds, and she plans to use these techniques to construct new examples of G2 and Spin(7) manifolds and study their contact and symplectic structures. In other joint work with D. Joyce, the PI studied deformations of asymptotically cylindrical coassociative submanifolds and their topological quantum field theories, and with Todd she also proved similar results for asymptotically cylindrical special Lagrangian submanifolds. The PI plans to apply these techniques on special Lagrangian moduli spaces inside Calabi-Yau manifolds to obtain a framework for the Floer homology program. Understanding the moduli spaces of these submanifolds will provide a better understanding of the mirror symmetry phenomenon.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Salur, Sema and Yalcinkaya, Eyup "Almost Symplectic Structures on Spin(7)Manifolds" 12th ISAAC Congress, Portugal, Geometries Defined by Differential Forms Proceedings, 2020. , 2020 https://doi.org/ Citation Details

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.

The subject of geometric structures on Riemannian manifolds is of great importance and has been intensively studied in differential geometry. Special holonomy geometries, along with geometric structures such as contact, symplectic, and calibrated, are defined by differential forms and are closely connected. Since Harvey and Lawson's groundbreaking work in 1982, studying the relationship between calibrated geometry and special holonomy has been essential for understanding "Mirror Symmetry," a phenomenon that arises in string theory and theoretical physics. A significant challenge in this field is the absence of a theorem providing necessary and sufficient conditions for a 7-dimensional manifold to have a G_2 metric. In earlier papers, the P.I. proposed a program to explore the connections between (almost) contact structures and G_2 structures, aiming to understand the topological obstructions to the existence of G_2 metrics (they are also Ricci-flat) on such manifolds.

Key outcomes of this research include discovering important relationships between G_2 structures, 2-plane fields, and (almost) contact structures on G_2 manifolds, as well as their Lagrangian-type Harvey Lawson and calibrated associative submanifolds. Additionally, the research has extended mirror dualities between Calabi-Yau manifolds inside the same G_2 manifold through 2-plane fields to similar mirror dualities between Calabi-Yau manifolds inside the same Spin(7) manifold through 4-plane fields.

This research not only advances the field of differential geometry and its applications in low-dimensional topology and physics but also aims to inspire future generations, especially women, to pursue studies in mathematics and science. Through extensive collaborations, mentorship activities, and organizational efforts, the P.I. aims to make a lasting impact on the mathematical and scientific communities by promoting diversity, inclusion, and collaboration in research.

 


Last Modified: 03/10/2024
Modified by: Sema Salur

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