Related papers: On the Nochka-Chen-Ru-Wong Proof of Cartan's Conje…

We consider weighted Radon transforms $R_W$ along hyperplanes in $R^3$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel $\mathrm{Ker}R_W$ in the space of infinitely smooth compactly…

In this paper we verify a conjecture by Kozlov (Discrete Comput Geom 18 (1997) 421--431), which describes the convex hull of the set of face vectors of $r$-colorable complexes on $n$ vertices. As part of the proof we derive a generalization…

By finding orthogonal representation for a family of simple connected called $\delta$-graphs it is possible to show that $\delta$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form…

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially…

We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of "typical" covering maps, and use graph Galois theory to "symmetrize" the conjecture. The conjecture is then…

In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total…

In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various…

The paper constructs new Hecke endomorphism algebras with a stratified structure. A novel feature of the proof is to approach difficult Ext^1 vanishing conditions by building entire exact category structures in which the analogous vanishing…

In previous work, the authors established a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of Seshadri constants. We extend our theorem to weighted sums involving closed subschemes in…

We prove a variant of the Arithmetic Fundamental Lemma conjecture of Wei Zhang for n=2. More precisely, we consider the deformation lengths of certain quasi-homomorphisms of quasi-canonical lifts in the sense of Gross. We prove the…

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for…

We consider the weighted Radon transforms $R_W$ along hyperplanes in $R^d, \, d \geq 3$, with strictly positive weights $W = W (x, \theta), \, x \in R^d, \, \theta \in S^{d-1}$. We construct an example of such a transform with non-trivial…

A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of…

In the works of A. Ach\'ucarro and P. K. Townsend and also by E. Witten, a duality between three-dimensional Chern-Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous…

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central…

In this paper we give a new proof of the Quantum Unique Ergodicity conjecture for holomorphic integral weight modular forms on the upper half plane. The proof requires only partial results towards the Ramanujan conjecture and the shifted…

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type…

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some…

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $\sigma_2(\kappa(X))=f(X, \nu(X)).$ In this note, we give a simpler proof of this result, and…

Following the overall strategy of the paper ``Convex hypersurfaces in contact topology" by Ko Honda and Yang Huang on contact convexity in high dimensions, we present a simplified proof of their main result.