Related papers: Equivalent version of Huppert's conjecture for $K_…

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig…

In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Chern--Simons invariants. We also establish a version of a conjecture of Goncharov relating…

The paper provides a version of the rational Hodge conjecture for $\3\dg$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed in \cite{perry2020integral} for admissible subcategories. We obtain examples of…

In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying…

We prove that any link in $S^3$ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either $\mathbb{Z}$ or…

The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.

We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate…

We prove a covariant version of the Stinespring theorem for Hilbert C*-modules.

We give a counterexample to a recently conjectured variant of the Penrose inequality.

We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.

We formulate and prove an analogue of Beurling's theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.

In this paper we prove the equivalence of two conjectures on linear systems through fat points on a generic K3 surface. The first conjecture is exactly as Segre conjecture on the projective plane. Whereas the second characterizes such…

In 2005, A. Knutson--R. Vakil conjectured a puzzle rule for equivariant K-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors' recently…

We give a new criterion for solvability of group equations, providing proofs of various generalizations of the Kervaire-Laudenbach conjecture for Connes-embeddable groups.

We prove Simon's conjecture for 3-manifolds.

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.

We prove the conjectures of Hodge and Tate for any six-dimensional hyper-K\"ahler variety that is deformation equivalent to a generalized Kummer variety.

An equivalent but useful version on the Homological Nerve Theorem is proved.

Let $Hilb^d(A^3)$ be the Hilbert scheme of $d$ points in $A^3$, and let $T_z$ denote the tangent space to a point $z \in Hilb^d(A^3)$. Okounkov and Pandharipande have conjectured that $\dim T_z$ and $d$ have the same parity for every $z$.…

We give an elementary proof of the well-known fact that the third cohomology group H^3(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext^2(G, M) of equivalence classes of crossed module extensions…