Related papers: On the second rigidity theorem of Huneke and Wiega…

In 1964, Paul Erd\H{o}s published a paper settling a question about function spaces that he had seen in a problem book. Erd\H{o}s proved that the answer was yes if and only if the continuum hypothesis was false: an innocent-looking question…

We analyze the behavior of polarized complex variations of Hodge structure on the punctured unit disk. For integral variations of Hodge structure, this analysis was first carried out by Wilfried Schmid. We get rid of the assumption that the…

In this paper, we provide some remarks on the scalar curvature rigidity theorem of Brendle and Marques in \cite{BrendleMarques}. The main result is that Brendle and Marques' theorem holds on a geodesic ball larger than that specified in…

In this paper a novel calculus system has been established based on the concept of 'werden'. The basis of logic self-contraction of the theories on current calculus was shown. Mistakes and defects in the structure and meaning of the…

We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes…

This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.

We settle a long-standing problem in the theory of Hecke algebras of complex reflection groups by constructing many (graded) integral cellular bases of these algebras. As applications, we explicitly construct the simple modules of Ariki's…

Paper withdrawn. There is a gap in the proof of Proposition 4.3 (its conclusion ``d'o\`{u} la proposition.'' is incorrect). Therefore theorems 1.1 and 3.1, which were the main results of the paper, are not proved.

Building on the nonabelian Hodge theory in positive characteristic developed by Ogus, Vologodsky, and Schepler, we propose a generalization of the decomposition theorem of Deligne and Illusie from the perspective of mixed Hodge modules.…

We prove some rigidity theorems for configurations of closed disks. First, fix two collections $\mathcal{C}$ and $\tilde{\mathcal{C}}$ of closed disks in the Riemann sphere $\hat{\mathbb{C}}$, sharing a contact graph which…

We prove a stronger version of a termination theorem appeared in the paper "On existence of log minimal models II". We essentially just get rid of the redundant assumptions so the proof is almost the same as in there. However, we give a…

Results in the mentioned paper do not seem valid.

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in…

The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…

In 1931, G\"odel presented in K\"onigsberg his famous Incompleteness Theorem, stating that some true mathematical statements are unprovable. Yet, this result gives us no idea about those independent (that is, true and unprovable)…

Goedel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two…

For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This…

This paper has been removed by the author due to a misstatement in Theorem 1 and a gap in its proof. A corrected and largely extended successor (a joint work with Thomas Bauer and Tomasz Szemberg) can be found under math.AG/0312211,

Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of four decades. In this paper we…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…