arXiv research

Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.

Summarizes connections between Euler characteristic theorems and conjectures.

Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.

This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.

We show that, under some technical conditions, the Strong Slope Conjecture proposed by Kalfagianni and Tran is closed under connect sums and cabling. As an application, we establish the Strong Slope Conjecture for graph knots.

Paper connects AJ conjecture and colored Jones polynomial potential function.

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

Verify conjecture for special Hermitian manifolds.

Proves a connectivity conjecture for free groups, showing homotopy type of spheres.

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.

The AJ conjecture is verified for certain connected sums of torus knots.

The paper confirms Arnold's conjecture about hyperbolic polynomials.

Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we allow coefficients in additive G-categories with (involution).

Study proves a conjecture for certain spin manifolds.

The Strominger conjecture is confirmed for compact Hermitian manifolds in 2D and special higher dimensions.

We extend Garsia's conjecture about surface embeddings.

Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.

The article confirms a complex geometry conjecture for a specific type of manifold.

The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.

The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.

We prove the A-theoretic Farrell-Jones Conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S-arithmetic groups and lattices in almost connected Lie groups.

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical not…

Classifies connected shelves up to order six.

Complete classification of Hermitian manifolds with flat Gauduchon connections.

Proves a conjecture for a specific group using spectral sequences and homology.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of th…

The paper proves a quantum modularity conjecture for 3-manifolds.

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

The study refines known counterexamples in 4D to satisfy certain inequalities.

I explain an open conjecture by Braverman/Milatovic/Shubin (BMS) on the positivity of square integrable solutions $f$ of $(-Δ+1)f\geq 0$ on a geodescially complete Riemannian manifold, and its connection to essential self-adjointness problems of covariant Schrödinger operators. The latter conjecture has remained open f…

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.

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of $n$ or fewer crossings approaches $100$ as $n$ approaches infinity. In this paper, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjectur…

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bund…

In this paper, by use of techniques associated to cobordism theory and Morse theory,we give a simple proof of Poincare conjecture, i.e. Every compact smooth simply connected 3-manifold is homeomorphic to 3-sphere.

The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'',…

Compact method proves Brown-York mass positivity and connects to major conjectures.

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $Γ$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions…

The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.

In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.

We prove the existence of a map of spectra $τ_A \colon kA \to lA$ between connective topological K-theory and connective algebraic L-theory of a complex $C^*$-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural e…