| EPSRC Reference: |
EP/W019620/1 |
| Title: |
Characteristic polynomials for symmetric forms |
| Principal Investigator: |
Dotto, Dr E |
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| Department: |
Mathematics |
| Organisation: |
University of Warwick |
| Scheme: |
New Investigator Award |
| Starts: |
15 March 2022 |
Ends: |
14 March 2025 |
Value (£): |
186,588
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
The characteristic polynomial of a matrix A is a fundamental mathematical object which is usually introduced in an undergraduate linear algebra course, defined as the determinant det(A-tI). Its key properties, namely the behavior under direct sums, tensor products, and that for triangular matrices it depends only on the diagonal entries, can be elegantly encoded by a ring homomorphism
K^cyc(R)--->W(R)
from the cyclic K-group of a commutative ring R, whose elements are represented by endomorphisms of finitely generated projective R-modules, to the ring of Witt vectors of R, whose elements are power series on R with constant coefficient 1. Almkvist proves that this map is injective, showing that in a sense the characteristic polynomial is a complete invariant for endomorphisms modulo extensions.
This project will investigate to which extent this construction can be extended to symmetric forms. Kato constructed a complete invariant for the Witt group of symmetric forms over a field of characteristic 2, valued in the two-fold tensor product of the base field over its subfield of squares. We propose to regard this invariant as the rank, or trace, of a symmetric form, and the goal of the project is to lift this rank to a ring of Witt vectors to construct an invariant for the Grothendieck-Witt group.
The techniques we will use for carrying out this program are informed by a version of the cyclotomic trace map of Bökstedt, Hsiang and Madsen for the Grothendieck-Witt spectrum, and will naturally lead us to further investigate the relationship between real topological Hochschild homology, the Witt vectors, and the de Rham-Witt complex.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.warwick.ac.uk |