- Fatma Kader Bingöl (Scuola Normale Superiore di Pisa)
- Title: Classification theorems for involutions by cohomological invariants.
- Abstract: Quadratic forms over a field F are classified up to isometry by dimension, discriminant, and
Clifford invariant when I3F, the third power of the fundamental ideal in the Witt ring of F, vanishes, or
in different terms, when the norm form of every quaternion algebra over F is surjective. Quadratic pairs on
central simple algebras can be seen as twisted forms of quadratic forms, as in the split case, the quadratic pair
is given explicitly by a quadratic form.
The classification theorem has been extended to the setting of quadratic pairs, as well as to involutions, under a
stronger condition that the base field has 2-separable dimension at most 2. A field F has 2-separable
dimension at most 2 if and only if I3L=0 for all finite separable extensions L/F. We show that
the classification theorems hold under the weaker condition that the base field F satisfies I^3F=0, if
the underlying central simple algebra has exponent at most 2. This is joint work with Anne Quéguiner-Mathieu
and Karim Johannes Becher.
- Stephen Scully
(University of Victoria)
- Title: Low-dimensional symmetric bilinear forms in In in characteristic 2.
- Abstract: Following the proof of Milnor's conjecture relating
the graded Witt ring of a field to its mod-2 Milnor K-theory, a major problem in the theory
of symmetric bilinear forms (or quadratic forms in characteristic not 2) is to understand, for
each n, the low-dimensional part of the nth power of the fundamental ideal in the Witt ring of a
field. Using algebraic-geometric methods, Karpenko showed that a non-zero anisotropic form of dimension <2n+1
representing an element of In has dimension 2n+1 - 2i for some
1 ≤ i ≤ n. When i = n, a classical
result of Arason-Pfister says that the form is similar to a Pfister form. When i=n-1, a conjecture of Hoffmann predicts
that the form should be isometric to the tensor product of an (n-2)-fold Pfister form and an Albert form (of dimension
6). Over fields of characteristic not 2, this is wide open for all n ≥ 4. In this talk, I will outline the
general picture, and then discuss the (simpler) case where the characteristic is 2. Here, we can give a direct
and elementary proof of Karpenko's theorem, and, more interestingly, a proof of Hoffmann's conjecture for all n. The methods also
yield additional results on low-dimensional forms in In and forms of height 2 in this setting.
- Thomas Unger (University College Dublin)
- Title: Pfister's local-global principle for Azumaya algebras with involution.
- Abstract: This is joint work with Vincent Astier. We prove Pfister's local-global principle
for hermitian forms over Azumaya algebras with involution over semilocal rings,
and show in particular that the Witt group of nonsingular hermitian forms is
2-primary torsion. Our proof relies on a hermitian version of Sylvester's law of
inertia, which is obtained from an investigation of the connections between a
pairing of hermitian forms extensively studied by Garrel, signatures of hermitian
forms, and positive semidefinite quadratic forms.
- Rasool Hafezi (Nanjing University of Information Science and Technology)
- Title: Representation theory of monomorphism categories
- Abstract: There has been significant recent interest in monomorphism (submodule) categories, as
they provide a powerful framework for addressing open problems in linear algebra using tools and ideas from homological
algebra, combinatorics, and geometry. The modern development of monomorphism categories began with the work of Ringel
and Schmidmeier on invariant subspaces of nilpotent linear operators. In particular, they provided a complete description
of the indecomposable objects in the submodule category of mod-k[x]/(x6) using Auslander–Reiten theory.
As a generalization of submodule categories, X. Luo and P. Zhang introduced the concept of separated monic representations over
an acyclic quiver, with the aim of describing Gorenstein-projective modules. In my talk, I will begin by explaining, based on
[RS1, RS2], how the study of monomorphism categories in the case where the algebra is k[x]/(xn) connects
to basic problems in linear algebra. This is related to the study of pairs X = (U,V), where V is a
finite-dimensional vector space equipped with a nilpotent operator T satisfying Tn = 0, and U
is a subspace of V such that T(U) ⊆ U. In the second part of the talk, I will recall Gabriel’s theorem, which
classifies hereditary finite-dimensional algebras of finite representation type over an algebraically closed field in terms of
ADE Dynkin quivers - a foundational result in the representation theory of algebras. Based on my joint work with Naser Bahlekeh [HB], I will then present a Gabriel-style classification of representation-finite separated monomorphism categories over
G-semisimple algebras.
References.
[HB] R. Hafezi and A. Bahlekeh, G-semisimple algebras. J. Pure Appl. Algebra 228 (2024), no. 12, Paper No. 107738, 34 pp.
[RS1] C. M. Ringel and M. Schmidmeier, Invariant subspaces of nilpotent operators. Level, mean, colevel: the triangle T(n).
Bull. Iranian Math. Soc. 51 (2025), no. 3, Paper No. 37, 179 pp.
[RS2] C. M. Ringel and M. Schmidmeier, Invariant subspaces of nilpotent linear operators. I. J. Reine Angew. Math.
614 (2008), 1–52.
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