- 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.
- Steven Dougherty (Scranton University (USA))
- Title: A characterization of finite commutative Frobenius rings and applications to algebraic coding theory.
- Abstract: The fundamental alphabets for codes over rings are nite Frobenius rings,
largely because their generating character produces MacWilliams relations. We prove
that the following are equivalent statements for a nite commutative ring R: (1) R is
Frobenius; (2) |a| |a|=|R| for all ideals a in R; and (3) (a\bot>)\bot = a for all ideals a in R
and give an algorithmic way of producing the generating character.
- 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.
- Julien Molina (Institut Fourier de l'Université Grenoble Alpes)
- Title: On the rank hierarchy of f-polynomial codes.
- Abstract: In this talk, we investigate f-polynomial codes, which constitute a
generalization of cyclic codes. These codes are equipped with the rank
metric, and we study their generalized rank weights, with particular
emphasis on the first and last weights. For the first weight, we study
the two extreme cases, in other words, when it is equal to 1 and when it
reaches the Singleton bound. In this sense, we establish a bound on the
generalizd rank hierarchy that is better than the Singleton one. Then,
for the last weight, we present a closed-form formula for computing it.
And finally, some generalisations of these results are presented (joint work with Grégory Berhuy).
- Louis Rowen (Bar-Ilan University)
- Title: Linear algebra over semilocal rings
- Abstract: This is the continuation of an ongoing project to find a general
algebraic framework for
semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of
negation, and such basic properties such as unique factorization of polynomials, multiplicativity of determinants,
and the characteristic polynomial of a matrix, all fail. (In fact in the max-plus algebra, the sum of two nonzero
elements is never zero!) Consequently 0 is replaced by a distinguished ideal
$\mathcal A_0$ of $\mathcal A,$ and $(\mathcal A,\mathcal A_0)$ is called a pair.
In this talk we discuss results obtained in the last two years on linear algebra over a (not necessarily
distributive) semiring pair, with a range of applications to tropical algebra as well as related
areas such as hyperrings and fuzzy rings. First we present pairs with their morphisms,
called "weak morphisms". We pay special attention to supertropical pairs and hyperpairs.
Then we turn
to matrices and the question of whether the row rank,
column rank, and submatrix rank of a matrix are equal. The
submatrix rank is less than or equal to the row rank and the column
rank in many cases, including "metatangible pairs" with unique negation, but there is a counterexample to equality,
discovered some time ago by the second author, which we discussed in the talk two years ago. There are
situations when equality holds, encompassing results by Akian, Gaubert, Guterman, Izhakian, Knebusch, and Rowen,
including versions of Cramer's rule. We pay special attention to a recent question of Baker and Zhang whether
n+1 vectors of length n need be dependent.
(joint work with Marianne Akian1, Stephane Gaubert).
- 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.
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