Related papers: The Total Rank Conjecture in Characteristic Two

We first formulate a general scheme for the classification of 2-compact groups in terms of maximal torus normalizer pairs. Applying this scheme, we show that all connected and some non-connected 2-compact groups are N-determined. We also…

By a theorem of Roberts, the integral closure of a regular local ring in a finite abelian extension of its fraction field is Cohen-Macaulay, provided that the degree of the extension is coprime to the characteristic of the residue field. We…

The (co)completeness problem for the (projectively) stable module category of an associative ring is studied. (Normal) monomorphisms and (normal) epimorphisms in such a category are characterized. As an application, we give a criterion for…

We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-Huneke's vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot's theorem that…

Over a field of characteristic two, we develop a theory of standard monomials for polynomial rings modulo a Frobenius power of the maximal ideal generated by all variables. As a result, we obtain a filtration by modular GL_n-representations…

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of…

This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically…

We prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli space of strongly parabolic Higgs bundles of rank two or three, with full flags. Although the main theorem is proved only for rank at most three, most of…

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the…

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with…

Over a complete Noetherian local domain of mixed characteristic with perfect residue field, we construct a perfectoid ring which is similar to an explicit representation of a perfect closure in positive characteristic. Then we demonstrate…

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed…

We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our…

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this…

We study Koszul homology over Gorenstein rings. If an ideal is strongly Cohen-Macaulay, the Koszul homology algebra satisfies Poincar\'e duality. We prove a version of this duality which holds for all ideals and allows us to give two…

This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing…

With the long-term goal of proving local structure theorems of algebraic stacks in positive characteristic near points with reductive (but possibly non-linearly reductive) stabilizer, we conjecture that quotient stacks of the form…

In this study, it is proven that the universal equivalence of general linear groups (admitting the inverse-transpose automorphism) of orders greater than $2$, over local, not necessarily commutative rings with $1/2$, is equivalent to the…

The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional…

We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for…