Related papers: Some remarks concerning Voevodsky's nilpotence con…
We introduce and investigate classes of normed or quasinormed distribution spaces of generalized smoothness that can be obtained by various interpolation methods applied to classical Sobolev, Nikolskii-Besov, and Triebel-Lizorkin spaces. An…
We develop a generalized Floquet-Bloch theory for discrete torsion-free nilpotent groups by exploiting their Malcev completions. Our main result is a branching formula that relates finite-dimensional representations of a discrete nilpotent…
This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and…
Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$…
The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation in a family of canonically po-larised…
Nivat's conjecture is a famous open problem in symbolic dynamics. The existence of nonexpansive lines that when endowed with a given orientation are one-sided nonexpansive directions is at the heart of some advances. In his Ph.D. thesis,…
We develope $\mathbb{C}^{\ast}$-equivariant categorical Donaldson-Thomas theory for local surfaces, i.e. the total spaces of canonical line bundles on smooth projective surfaces. We introduce $\mathbb{C}^{\ast}$-equivariant DT categories…
This master's thesis gives a thorough and pedagogical introduction to the Dijkgraaf-Vafa conjecture which tells us how to calculate the exact effective glueball superpotential in a wide range of $\mathcal{N}=\textrm{1}$ supersymmetric gauge…
We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group: 1. The Strong Novikov Conjecture…
We describe Novikov's "higher signature conjecture," which dates back to the late 1960's, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in…
Over a field of characteristic zero, we establish the homotopy invariance of the Nisnevich cohomology of homotopy invariant presheaves with oriented weak transfers, and the agreement of Zariski and Nisnevich cohomology for such presheaves.…
We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over…
We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived…
We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several…
We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of…
In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.
We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two…
We consider a weak version of the Rost Nilpotence Principle. For characteristic zero fields k, we show that if it holds for all smooth projective schemes over k, then the Rost Nilpotence Principle does also. We also correct the proof of a…
It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…
The smooth second Bogolyubov's theorem is generalized for Lipschitz systems.