Related papers: Some finiteness results for algebraic groups and u…

In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.

Given a group satisfying sufficient finiteness properties, we discuss a group algebra criterion for vanishing of all its cohomology groups with unitary coefficients in a certain degree.

In recent years, there has been considerable success in computing Ext-groups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories. In this paper, we…

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic…

We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

In this note we discuss some examples of non torsion and non algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro.

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite…

Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more…

It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

We construct an analogue of Neumann's affiliated algebras for sofic group algebras over arbitrary fields. Consequently, we settle Kaplansky's direct finiteness conjecture for sofic groups.

Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite…

We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and…

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…