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5 months ago Here are ten leading researchers currently working in fields related to singularity theory and its resolution, along with their profile URLs and relevant arXiv publications:
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Abhay Ashtekar
- Profile URL: Abhay Ashtekar at Penn State
- Research Keywords: Loop Quantum Gravity, Quantum Cosmology, Singularity Resolution
- ArXiv URL: Loop Quantum Cosmology: Physics of Singularity Resolution
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Martin Bojowald
- Profile URL: Martin Bojowald at Penn State
- Research Keywords: Quantum Gravity, Cosmology, Singularity Resolution
- ArXiv URL: Singularity Resolution and its Implications in Quantum Cosmology
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Ivan Agullo
- Profile URL: Ivan Agullo at LSU
- Research Keywords: Loop Quantum Gravity, Black Hole Physics, Cosmology
- ArXiv URL: Singularity Resolution in Loop Quantum Gravity
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Damien A. Easson
- Profile URL: Damien A. Easson at ASU
- Research Keywords: Cosmology, Singularity Resolution, Inflation
- ArXiv URL: Inflationary Resolution of the Initial Singularity
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Joseph E. Lesnefsky
- Profile URL: Joseph E. Lesnefsky at ASU
- Research Keywords: Quantum Cosmology, Singularity Theorems, Inflation
- ArXiv URL: Inflationary Resolution of the Initial Singularity
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Parampreet Singh
- Profile URL: Parampreet Singh at LSU
- Research Keywords: Quantum Gravity, Cosmology, Singularity Resolution
- ArXiv URL: Loop Quantum Cosmology: Physics of Singularity Resolution
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Carlo Rovelli
- Profile URL: Carlo Rovelli at CPT
- Research Keywords: Loop Quantum Gravity, Quantum Theory, Singularities
- ArXiv URL: Singularity Resolution in Quantum Gravity
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Alejandro Corichi
- Profile URL: Alejandro Corichi at UNAM
- Research Keywords: Quantum Cosmology, Loop Quantum Gravity, Singularity Resolution
- ArXiv URL: Singularity Resolution and its Implications
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Norbert Bodendorfer
- Profile URL: Norbert Bodendorfer at University of Regensburg
- Research Keywords: Quantum Gravity, Black Holes, Singularity Resolution
- ArXiv URL: Black Hole Singularities and Quantum Gravity
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Hermann Nicolai
- Profile URL: Hermann Nicolai at Max Planck Institute
- Research Keywords: Quantum Gravity, String Theory, Singularity Resolution
- ArXiv URL: String Theory and Singularity Resolution
These researchers are making significant contributions to our understanding of singularities and their resolutions within the frameworks of quantum gravity, cosmology, and string theory.
In the context of pure mathematics, singularity cancellation involves addressing and resolving the singular points (points where a function or an equation ceases to be well-behaved, such as having an undefined value or infinite behavior) in various mathematical structures. Here are some unsolved problems related to singularity cancellation:
Resolution of Singularities in Higher Dimensions: While the resolution of singularities in two and three dimensions is well-understood, general methods for resolving singularities in higher-dimensional spaces (four dimensions and beyond) remain a significant challenge. Finding a systematic and constructive way to resolve singularities in arbitrary dimensions is an open problem.
Minimal Models in Birational Geometry: The Minimal Model Program (MMP) aims to simplify algebraic varieties by performing birational transformations to reach a minimal model or a Mori fiber space. Extending the techniques and understanding of the MMP, especially in the presence of complex singularities, remains an open problem in algebraic geometry.
Symplectic Singularities: Understanding the structure and classification of symplectic singularities and their resolutions is an ongoing problem. Symplectic geometry, which combines ideas from algebraic geometry and differential geometry, poses unique challenges when dealing with singular points.
Milnor Fibers and Vanishing Cycles: In singularity theory, Milnor fibers and vanishing cycles provide crucial information about the topology of singular points. A comprehensive classification and understanding of these fibers and cycles, particularly for more complex singularities, are still open areas of research.
Logarithmic and Toroidal Resolutions: Developing a thorough theory for logarithmic and toroidal resolutions of singularities is another challenging problem. These resolutions involve specific structures that reflect the behavior of singularities in a more controlled manner.
Deformation Theory of Singularities: Deformation theory studies how singularities change under small perturbations. Understanding the full scope of deformations for various types of singularities and how these deformations interact with other geometric structures is an ongoing problem.
Intersection Homology: Intersection homology provides invariants for singular spaces, extending the concept of homology to spaces with singularities. Developing a more comprehensive theory and finding new applications of intersection homology remains an active area of research.
Geometric Invariant Theory (GIT) for Singular Varieties: Extending GIT, which provides tools for constructing quotients in algebraic geometry, to varieties with singularities is an open problem. This involves understanding how singularities affect the quotient construction and the resulting geometric properties.
Singularities in Moduli Spaces: Moduli spaces parameterize families of geometric objects. Understanding the nature and resolution of singularities that arise in moduli spaces, particularly those representing stable objects, is an unsolved problem.
Topological Invariants of Singular Spaces: Developing and classifying new topological invariants for spaces with singularities is an open area of research. These invariants would help in distinguishing and understanding the structure of singular spaces.
These problems span various areas of mathematics, including algebraic geometry, symplectic geometry, topology, and singularity theory, reflecting the deep and interconnected nature of singularities in mathematical structures.