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We present the proof of Berger and Turaev of Herman's positive entropy conjecture. In every neighbourhood of identity in the set of smooth symplectic diffeomorphisms of the 2-dimensional disc, there exists a diffeomorphism with positive…

We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices, one automatically takes care…

Given a cooriented contact manifold $(M,\xi)$, it is possible to define a notion of positivity on the group $\mathrm{Diff}(M)$ of diffeomorphisms of $M$, by looking at paths of diffeomorphisms that are positively transverse to the contact…

We obtain some Liouville type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary and partially verifies Wang's conjecture (J. Geom. Anal. 31 (2021)). For…

In this paper we establish the convergence case of Khintchine's theorem for affine hyperplanes in function field of positive characteristic. Along with that, we also prove a quantitative version of the same. The main technique used in the…

Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these…

We prove that if the stable foliation and the unstable foliation of an Anosov diffeomorphism on a connected compact manifold are $C^3$, then the diffeomorphism has fixed points. This is a partial positive answer to a Smale conjecture for…

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fr\'{e}chet spaces. We prove a version of the Mountain Pass Theorem which is a key ingredient in the proof of the main theorem.

We prove the A-theoretic Isomorphism Conjecture with coefficients and finite wreath products for solvable groups.

If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.

We introduce a (linear) positive and asymptotic preserving method or solving the one-group radiation transport equation. The approximation in space is discretization agnostic: the space approximation can be done with continuous or…

We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply…

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet $m$-Laplacian problem with different weights and a diffusion term inside the divergence of the form $\big(a(|x|)+g(u)\big)^{-\gamma}$, with…

In this work we proivied a new simpler proof of the global diffeomorphism theorem from [9] which we further apply to consider unique solvability of some abstract semilinear equations. Applications to the second order Dirichlet problem…

We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC…

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane.

We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for…

We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic…

This paper solves in a positive manner a conjecture stated in 2000 by R. G\'omez-Re\~nasco and J. L\'opez-G\'omez regarding the multiplicity of positive solutions of a paradigmatic class of superlinear indefinite boundary value problems.