arXiv research

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.

We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of $L^2$-Betti numbers of closed manifolds.

Study properties of group rings of three-manifold groups.

Study confirms uniqueness of minimal sphere in Atiyah-Hitchin manifold.

Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

Formulates a strategy to prove Atiyah-Floer conjecture.

New insights into Coxeter groups' $L^2$-cohomology via weighted analysis.

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

Atiyah's conjecture concerning configurations of N points in the Euclidean three-space is verified for the following nonplanar configurations: The first m points lie on a line L and the remaining n=N-m (>2) points are the vertices of a regular n-gon whose plane is perpendicular to L and whose centroid is on L.

Study of vector bundles over classifying spaces for infinite discrete groups.

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they are linearly independent over $\mathbb{C}$, no matter which configuration $\mathbf{x}$ one s…

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Z…

Compact theorem for $SO(3)$ anti-self-dual equations on cylindrical manifolds.

The paper compares and proves equivalence of invariants in algebraic K-theory.

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds fo…

Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.

The Strong Slope Conjecture is proven for specific knot types.

The Thurston norm is derived from polytopes and applied to group cohomology.

Paper withdrawn because of a gap in the proof of Proposition 3 of Thomas Schick: "Integrality of L2-Betti numbers", Math. Ann. 317, 727-750 (arXiv.org/abs/math.gt/0001101). Most results of the withdrawn paper were based on this proposition.

The paper solves conjectures related to strong openness and optimal $L^2$ extension.

The study confirms a conjecture for a specific type of knot.

The strong slope conjecture helps identify torus knots.

Verifies a conjecture for the figure eight knot.

An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for …

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X . Along a solution of the flow, we show the curvature $iΛF(A_t)$ approaches in $L^2$ an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sha…

Heegaard Floer homology duality applied to 4D mapping tori.

Verify knot polynomial conjectures for specific knots.

Penrose's conjecture links black holes and gravitational singularities.

A group action on a moduli space is shown to be faithful.

Floer theory uses categories to construct 3-manifold invariants.

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relat…

Study Mazur doubles of knots and their relation to the Slope Conjecture.

Log-concave coefficient sequences for two-bridge knots proved.

Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.

Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.

Jones slopes characterize adequate knots under the Strong Slope conjecture.

A minimal hypersurface in a sphere is uniquely determined.

Paper solves a 4D topology problem using Pin(2)-equivariant Mahowald invariants.

Algorithm decides if knots satisfy a conjecture, linking Jones polynomial to surface complexity.

Proves conjectures for pretzel knots using polynomial degrees.

Proposes a weaker version of Strong Cosmic Censorship with curvature bounds.

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…