EPSRC Reference: 
EP/W019620/1 
Title: 
Characteristic polynomials for symmetric forms 
Principal Investigator: 
Dotto, Dr E 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
New Investigator Award 
Starts: 
15 March 2022 
Ends: 
14 March 2025 
Value (£): 
186,588

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

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(AtI). 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 Kgroup of a commutative ring R, whose elements are represented by endomorphisms of finitely generated projective Rmodules, 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 twofold 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 GrothendieckWitt 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 GrothendieckWitt spectrum, and will naturally lead us to further investigate the relationship between real topological Hochschild homology, the Witt vectors, and the de RhamWitt complex.

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Potential use in nonacademic contexts 
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Summary 

Date Materialised 


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Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 