Related papers: Grothendieck-Serre Conjecture I: Appendix

Using the identification of sections of the Galois group of the ground field into the arithmetic fundamental group with neutral fiber functors of the category of finite connections, we define the "packets" in Grothendieck's section…

More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise…

We formulate analogues, for Noetherian local $\mathbb Q$-algebras which are not necessarily regular, of the injectivity part of Gersten's conjecture in algebraic $K$-theory, and prove them in various cases. Our results suggest that the…

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then…

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both…

In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable…

We prove a conjecture by W. Bergweiler and A. Eremenko on the traces of elements of modular group in this paper

We briefly review an open conjecture about Higgs bundles that are semistable with after pulling back to any curve, and prove it in the rank 2 case. We also prove a set of inequalities holding for H-nef Higgs bundles that generalize some of…

We give the details of the construction of a map to restate a conjectural expression about adjoint group action on generic representations in L-packets. We give an application of the construction to give another proof of the classification…

We prove Gieseker conjecture for an homogeneous space $X$, saying that if $X$ has no non-trivial tame coverings then it has no non-trivial regular singular $\mathscr{O}_X$-coherent $\mathscr{D}_{X/k}$-modules. In order to do so we prove a…

We give a new independent proof of a generalised version of the theorem by Rashevskii, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83 -- 94] and from which the classical Chow-Rashevskii Theorem follows as a corollary. The proof…

Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal…

We prove the Gersten conjecture for $p$-adic \'etale Tate twists for a smooth scheme $X$ in mixed characteristic in the Nisnevich topology. Our main observation is that, while $p$-adic \'etale Tate twists are not $\mathbb A^1$-invariant,…

The purpose of this article is to prove that Gersten's conjecture for a commutative regular local ring is true. As its applications, we will prove the vanishing conjecture for certain Chow groups, generator conjecture for certain $K$-groups…

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,

Erd\H{o}s, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k\geq 2$: Every graph on $n\geq k-1$ vertices with at least $(k-1)(n-k+2)+{k-2\choose 2}$ edges contains a subgraph with minimum degree at least…

Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).

The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group $G$ and any set $A\subseteq G$ with $|A|\geq 2$, $G$ is the product of at most $N\frac{\log|G|}{\log|A|}$ conjugates of $A$, for some…

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the…