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Details of Grant 

EPSRC Reference: EP/W019620/1
Title: Characteristic polynomials for symmetric forms
Principal Investigator: Dotto, Dr E
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of Warwick
Scheme: New Investigator Award
Starts: 01 February 2022 Ends: 31 January 2025 Value (£): 186,588
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
24 Nov 2021 EPSRC Mathematical Sciences Prioritisation Panel November 2021 Announced
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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Organisation Website: http://www.warwick.ac.uk