| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | March 14, 2017 |
| Latest Amendment Date: | March 14, 2017 |
| Award Number: | 1701635 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Sandra Spiroff
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2016 |
| End Date: | August 31, 2019 (Estimated) |
| Total Intended Award Amount: | $65,730.00 |
| Total Awarded Amount to Date: | $65,730.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
360 HUNTINGTON AVE BOSTON MA US 02115-5005 (617)373-5600 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
Boston MA US 02115-5005 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
|
| Program Reference Code(s): | |
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Research at the interface of physics and mathematics has a long history of being beneficial to both fields. This project concerns string theory, one of the most vibrant areas of research at this interface. As we explore the boundaries of the regime of validity of string theory and address some of its theoretical shortcomings and the conditions of its peaceful co-existence with the theory of gravity, we rely on increasingly sophisticated mathematical models. F-theory is one of the most geometric corners of string theory and relies on elliptic fibrations to geometrically engineer models for particle physics and gravity at very high energy. This project explores the mathematics that governs the physics of F-theory in order to bring new insights to the mathematics of elliptic fibrations. Elliptic fibrations are geometric spaces that locally look like a long-exposure picture of an elliptic curve moving over another space. Elliptic curves are some of the oldest but yet most prominent objects across mathematics. They are studied in algebraic geometry and number theory and play a central role in representation theory, cryptography (where they are used to secure internet's transactions), computer modeling, and theoretical physics. This multi-disciplinary project will explore connections between these mathematical and physical concepts and will also encourage graduate students both in mathematics and physics to study problems at the interface of physics and mathematics. The PI will also reach out to students from underrepresented groups to inspire them to participate in the scientific endeavor.
Geometric engineering is the construction of physical systems by geometric methods. This is a natural approach in the context of string theory, where the geometry of extra dimensions and the dualities between different formulation and regimes of the theory allow for a geometric formulation of many physical constraints. This project focuses on the geometry of elliptic fibrations as seen from the point of view of string theory, with strong interest in their topology, arithmetic properties, and connection to gauge theories, Grand Unified Theories, and topological defects. The research will also address the problem of resolution of singularities of Weierstrass models and its connection to Mori's theory and representation theory and will explore the structure of the singular fibers of elliptic fibrations over higher dimensional spaces. The PI will also study the physics of topological defects such as cosmic string solutions using new models of elliptic fibrations. A central theme of this project is the prominent role of singularities, their resolutions, and their topological invariants.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
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 goal of this project was to study elliptic fibrations in the context of string theory. The project focused on the mathematics of resolutions of singularities of elliptic fibrations. In algebraic geometry, curves are organized by their genus, a curve of genus-zero is a sphere, an elliptic curve is a curve of genus one. They play a key role in many areas of mathematics and physics. When an elliptic curve moves over another object, it defines what is called an elliptic fibration. There is a general model to describe elliptic fibration (the Weirtrass model) but for many interesting cases, the Weierstrass model has singularities that have to be repaired (resolved).
This project produced new results on the resolutions of certain elliptic fibrations that are used in string theory. It explains how their singularities are resolved and sometimes when there are different ways to do the resolution, it also explains how different resolutions are connected to each other by what is called flop transitions. The description of flops transitions of elliptic fibrations relies on some ideas that were first developed in quantum field theory and gauge theory. In this sense, it used ideas from theoretical physics to improve our understanding of elliptic fibrations.
The project also introduced new tools to compute the topological invariants of these elliptic fibrations.
The project resulted in 5 papers and involved the training of three Ph.D. students and two undergraduate students. The results of this project were also presented in seminars and (international) conferences.
Last Modified: 07/02/2021
Modified by: Jonathan M Esole
Please report errors in award information by writing to: awardsearch@nsf.gov.
