Related papers: A counterexample to vanishing conjectures for nega…

In 2008, the author proposed a version of duality theory for (not necessarily, Abelian) complex Lie groups, based on the idea of using the Arens-Michael envelope of topological algebra and having an advantage over existing theories in that…

We propose an Artinian version of Berger's Conjecture for curves, concerning the module of K\"ahler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a…

Walker's Cancellation theorem for abelian groups tells us that if $A$ is finitely generated and $G$ and $H$ are such that $A \oplus G \cong A \oplus H$, then $G \cong H$. Michael Deveau showed that the theorem can be effectivized, but not…

Inspired by Lurie's theory of quasi-unital algebras we prove an analogous result for $\infty$-categories. In particular, we show that the unital structure of an $\infty$-category can be uniquely recovered from the underlying non-unital…

The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets…

We establish a generic vanishing theorem for surfaces in characteristic $p$ that lift to $W_2(k)$ and use it for surface classification of surfaces of general type with Euler characteristic 1 and large Albanese dimension.

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted…

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert…

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result…

We prove that the Fibered Isomorphism Conjecture of T. Farrell and L. Jones holds for various mapping class groups. In many cases, we explicitly calculate the lower algebraic K-groups, showing that they do not always vanish.

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the…

We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles…

We prove the vanishing of bounded cohomology with separable dual coefficients for many groups of interest in geometry, dynamics, and algebra. These include compactly supported structure-preserving diffeomorphism groups of certain manifolds;…

Counterfactual definiteness must be used as at least one of the postulates or axioms that are necessary to derive Bell-type inequalities. It is considered by many to be a postulate that is not only commensurate with classical physics (as…

We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more…

Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse…

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new…

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the…

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…