N. Kalinin

activity

20132026

most citedSelf-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

24 citations · 93 across the 20 of their papers we have counts for

collaborators

22 papers

math.AG2026

Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans

Nikita Kalinin

Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \] and prove the two…

math.CO2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Turgay Akyar, Artem Beliakov, Konstantin Delchev +6

Place one grain at every nonsink vertex of the wired $n \times n$ square, and let $L (n)$ be the order of this operation in the sandpile group. Thus $L (n)$ is the least positive $q$ …

math.AP2026

Many-point tropical relaxation and the Monge--Ampère equation

Nikita Kalinin, Ernesto Lupercio, Higinio Serrano +1

In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Ampère Dirichlet problem on bounded convex plan…

math.NT2026

The divisor function for matrices

Tim Browning, Nikita P. Kalinin, Alina Ostafe +2

We introduce a matrix divisor function $τ_{n} (T, M)$ , counting factorisations $A B = M$ for $n \times n$ integer matrices $A, B$ of height at most $T$ . For a fixed non-singular $M$ , or for…

math.GR2023★ 2 cited

Sandpile group of infinite graphs

Nikita Kalinin, Vladislav Khramov

For a finite connected graph $G$ and a non-empty subset $S$ of its vertices thought of sinks, the so-called critical group (or sandpile group) $C (G, S)$ has been studied for a long…

math.CO2022★ 4 cited

Sandpile solitons in higher dimensions

Nikita Kalinin

Let $p \in Z^{n}$ be a primitive vector and $Ψ : Z^{n} \to Z, z \to min (p \cdot z, 0)$ . The theory of {\it husking} allows us to prove that there exists a pointwise m…

cond-mat.stat-mech2022

On the origin of the hierarchy of the sciences

Nikita Kalinin

We propose a simple "evolutionary" sandpile model exhibiting self-organised criticality and exactly $1/ f$ -noise (i.e. the critical exponent is equal to $- 1$ ) and observe emergent p…

math.AG2022★ 4 cited

Shrinking dynamic on multidimensional tropical series

Nikita Kalinin, Mikhail Shkolnikov

Let $Ω \subset R^{n}$ be a compact convex domain. An $Ω$ -tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ…

math.CO2021

Equilateral convex triangulations of $R P^{2}$ with three conical points of equal defect

Mikhail Chernavskikh, Altan Erdnigor, Nikita Kalinin +1

Consider triangulations of $R P^{2}$ whose all vertices have valency six except three vertices of valency $4$ . In this chapter we prove that the number $f (n)$ of such triangu…

math.CO2019★ 2 cited

Some statistics about Tropical Sandpile Model

Nikita Kalinin, Yulieth Prieto

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit b…

nlin.AO2018★ 24 cited

Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

Nikita Kalinin, Aldo Guzmán-Sáenz, Yulieth Prieto +3

Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each ot…

cs.GT2018

New mechanism for repeated posted price auction with a strategic buyer without discounting

Nikita Kalinin

On ad exchange platforms the place for advertisement is sold through different kinds of auctions. However, it is not uncommon the situation where the seller repeatedly encounters o…

math.CO2017★ 10 cited

Sandpile solitons via smoothing of superharmonic functions

Nikita Kalinin, Mikhail Shkolnikov

Let $F : Z^{2} \to Z$ be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that unde…

math.NT2017

Tropical formulae for summation over a part of SL(2, Z)

Nikita Kalinin, Mikhail Shkolnikov

Let $f (a, b, c, d) = a^{2} + b^{2} + c^{2} + d^{2} - (a + c)^{2} + (b + d)^{2}$ , let $(a, b, c, d)$ stand for $a, b, c, d \in Z_{\geq 0}$ such that $a d - b c = 1$ . Define \begin{equation} \label{…

math.CO2017★ 9 cited

Introduction to tropical series and wave dynamic on them

Nikita Kalinin, Mikhail Shkolnikov

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dy…

math.CO2015★ 19 cited

Tropical curves in sandpiles

Nikita Kalinin, Mikhail Shkolnikov

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation $ψ$ of the maximal stable state $μ \equiv 3$ is obtained by adding extra…

math.CO2015★ 4 cited

Sandpiles on the heptagonal tiling

Nikita Kalinin, Mikhail Shkolnikov

We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations…

math.AG2015★ 3 cited

A guide to tropical modifications

Nikita Kalinin

This paper surveys {\it tropical modifications}, a notion that has become part of the folklore of tropical geometry. Tropical modifications are used in tropical intersection theory…

math.CO2015

Tropical curves in sandpile models

Nikita Kalinin, Mikhail Shkolnikov

A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one g…

math.AT2013

The Alexander polynomial as an intersection of two cycles in a symmetric power

Nikita Kalinin

We consider a braid $β$ which acts on a punctured plane. Then we construct a local system on this plane and find a homology cicle $D$ in its symmetric power such that $D \cdot β (D)$ …

math.AG2013★ 8 cited

Tropical approach to Nagata's conjecture in positive characteristic

Nikita Kalinin

Suppose that there exists a hypersurface with the Newton polytope $Δ$ , which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of $Δ$ to ea…

math.AG2013★ 4 cited

The Newton polygon of a planar singular curve and its subdivision

Nikita Kalinin

Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F (x, y) = 0$ . Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $Δ$ o…

Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Many-point tropical relaxation and the Monge--Ampère equation

The divisor function for matrices

Sandpile group of infinite graphs

Sandpile solitons in higher dimensions

On the origin of the hierarchy of the sciences

Shrinking dynamic on multidimensional tropical series

Equilateral convex triangulations of RP2 with three conical points of equal defect

Some statistics about Tropical Sandpile Model

Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

New mechanism for repeated posted price auction with a strategic buyer without discounting

Sandpile solitons via smoothing of superharmonic functions

Tropical formulae for summation over a part of SL(2, Z)

Introduction to tropical series and wave dynamic on them

Tropical curves in sandpiles

Sandpiles on the heptagonal tiling

A guide to tropical modifications

Tropical curves in sandpile models

The Alexander polynomial as an intersection of two cycles in a symmetric power

Tropical approach to Nagata's conjecture in positive characteristic

The Newton polygon of a planar singular curve and its subdivision

Equilateral convex triangulations of $R P^{2}$ with three conical points of equal defect