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We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a…
It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow from the corresponding linear estimates.…
We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive $\delta'$ interaction supported by a smooth surface in $\R^3$, either infinite and asymptotically planar, or compact and closed. Its second term is…
The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied for a massless fermion particle under the action of external fields. Using the covariant approach based on general relativity, the Dirac operator stemming from a…
We study the second order invariants of a Lorentzian surface in $\mathbb{R}^{2,2},$ and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate…
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with…
Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an…
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime…
We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic…
Recently, it has been revealed that a variety of novel phenomena emerge in hyperbolic spaces, while non-Hermitian physics has significantly enriched the landscape of condensed matter physics. Building on these developments, we construct a…
The string vertices of closed string field theory are subsets of the moduli spaces of punctured Riemann surfaces that satisfy a geometric version of the Batalin-Vilkovisky master equation. We present a homological proof of existence of…
It is known that, if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported--that is, if the…
The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral…
In this paper we present a theoretical analysis of the effect of local geometrical structure of the Fermi surface on the surface impedance of a metal at the anomalous skin effect. We show that when the Fermi surface includes nearly…
Bi-Hamiltonian structures involving Hamiltonian operators of degree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms of an algebra structure on the space of 1-forms, related to so-called Fermionic Novikov…
In this paper we construct a non-commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non-perturbative quantum gauge theory coupled to a fermionic field on a…
Hyperbolic buildings are central objects in both hyperbolic geometry and geometric group theory, exhibiting a wide range of intriguing characteristics, especially with respect to group actions. In this paper, we develop the theory of…
A practical solution for the mathematical problem of functional calculus with Laplace-Beltrami operator on surfaces with axial symmetry is found. A quantitative analysis of the spectrum is presented.
The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…