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The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the…

We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.

In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the additive energy of multiplicative subgroups and…

Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by the second author. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower…

Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold.…

Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of…

Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint…

The Donald-Flanigan conjecture asserts that for any finite group and for any field, the corresponding group algebra can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group over a field of…

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain "higher Todd genera" are birational…

Base on a conjecture, we prove that for any smooth separated stack of finite type over a number field, its descent obstruction equals its iterated descent obstruction. As a consequence, we show that for any algebraic stack over a number…

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection.…

This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index…

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus,…

We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform…

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink…

We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all…

Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified…

The torsion anomalous conjecture states that for any variety V in an abelian variety there are only finitely many maximal V-torsion anomalous varieties. We prove this conjecture for V of codimension 2 in a product E^N of any elliptic curve…