Related papers: The Scaling Site
In this paper we extend the unramified class field theory for arithmetic surfaces of K. Kato and S. Saito to the relative case. Let X be a regular proper arithmetic surface and let Y be the support of divisor on X. Let CH_0(X,Y) denote the…
We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single…
The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical…
We initiate the theory of a quadratic form $q$ over a semiring $R$. As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear…
The family of complex projective surfaces in projective three space of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system of surfaces of degree $d$ for sufficiently large $d$…
Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed…
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra,…
We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution…
Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of…
A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral…
This is a survey paper on the connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both…
Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an…
The fractionally charged quasiparticles appearing in the 5/2 fractional quantum Hall plateau are predicted to have an extra non-local degree of freedom, known as topological charge. We show how this topological charge can block the…
The metric $D_\alpha (q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^\alpha)^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$…
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on…
The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are…
We study canonical central extensions of the general linear group of the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. By these central extensions and adelic transition matrices of a rank $n$…
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a…
I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a…
We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a choice…