Related papers: Strong Arnold chord conjecture via normalized capa…

We study the complete diagonal of the Laurent series expansion of a rational function in $n$-complex variables. For a denominator that is nondegenerate for its Newton polyhedron, we prove that the complete diagonal, initially defined in a…

Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily…

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit…

Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $\Theta \in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a…

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one…

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by…

We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants which are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon--Tarsi conjecture on Latin cubes considered…

Let $E$ be a real Banach space with dimension at least 2. In this paper, we prove that if $D\subset E$ is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in $D$ is a cone arc.

We prove that a non-classical flag domain is pseudoconcave if it satisfies a certain condition on the root system. Moreover, we prove that every point in a one-codimensional real boundary orbit of a non-classical period domains is a…

We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise…

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary…

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a…

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…

We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic…

We study concavity properties of positive solutions to the Logarithmic Schr\"odinger equation $-\Delta u=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-\Delta…

In a Riemannian manifold a regular convex domain is said to be $\lambda$-convex if its normal curvature at each point is greater than or equal to $\lambda$. In a Hadamard manifold, the asymptotic behaviour of the quotient…

We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010-12, are now…

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains $\mathbb Y$, any homeomorphism $\varphi…

Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We…

There are two long-standing conjectures in Hamiltonian dynamics concerning Reeb flows on the boundaries of star-shaped domains in $\mathbb{R}^{2n}$ ($n \geq 2$). One conjecture states that such a Reeb flow possesses either $n$ or infinitely…