12 citations · 48 across the 26 of their papers we have counts for
39 papers
Integrability of Cauchy problems for discrete conformal maps and circle patterns
Maxim Arnold, Anton Izosimov
A map from a square lattice to the Riemann sphere is called discrete conformal if the image of every elementary square is a harmonic quadrilateral. We prove that the initial value…
Virasoro extensions for diffeomorphisms with breaks
Anton Izosimov, Boris Khesin, Howard Xiao
We study homeomorphisms of the circle that are smooth diffeomorphisms away from a finite set of points. These "broken diffeomorphisms" do not form a Lie group, but instead natu…
Surface topology and incidence theorems over division rings
Anton Izosimov
Incidence theorems concern configurations of points, lines, and, more generally, higher-dimensional subspaces in projective space. Broadly speaking, such theorems fall into two cla…
The dimer model and dynamical incidence geometry
Anton Izosimov, Pavlo Pylyavskyy
We propose a geometric counterpart of the dimer model on bipartite graphs. A state of our model consists of a choice of a point for each white vertex and hyperplane for each black…
Folding of quadrilaterals, zigzags, and Arnold-Liouville integrability
Anton Izosimov
We put Darboux's porism on folding of quadrilaterals, as well as closely related Bottema's zigzag porism, in the context of Arnold-Liouville integrability.
Pentagram maps over rings, Grassmannians, and skewers
Leaha Hand, Anton Izosimov
The pentagram map is a discrete dynamical system on planar polygons. By definition, the image of a polygon under the pentagram map is the polygon whose vertices are the in…
Integrable systems and cluster algebras
Michael Gekhtman, Anton Izosimov
We review several constructions of integrable systems with an underlying cluster algebra structure, in particular the Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on…
Planar networks and simple Lie groups beyond type A
Anton Izosimov
The general linear group , along with its adjoint simple group , can be described by means of weighted planar networks. In this paper we give a network description f…
Coadjoint orbits of area-preserving diffeomorphisms of non-orientable surfaces
Anton Izosimov, Boris Khesin, Ilia Kirillov
We give a classification of generic coadjoint orbits for the group of area-preserving diffeomorphisms of a closed non-orientable surface. This completes V. Arnold's program of stud…
Change of polytope volumes under Möbius transformations and the circumcenter of mass
Anton Izosimov
The circumcenter of mass of a simplicial polytope is defined as follows: triangulate , assign to each simplex its circumcenter taken with weight equal to the volume of the s…
What is a lattice W-algebra?
Anton Izosimov, Gloria Marí Beffa
We employ the Poisson-Lie group of pseudo-difference operators to define lattice analogs of classical -algebras. We then show that the so-constructed algebras coincide with th…
Geometry of generalized fluid flows
Anton Izosimov, Boris Khesin
The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant -metric on the group of vo…
Long-diagonal pentagram maps
Anton Izosimov, Boris Khesin
The pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the…
Polygon recutting as a cluster integrable system
Anton Izosimov
Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with…
Dimers, networks, and cluster integrable systems
Anton Izosimov
We prove that the class of cluster integrable systems constructed by Goncharov and Kenyon out of the dimer model on a torus coincides with the one defined by Gekhtman, Shapiro, Tab…
Intersecting the sides of a polygon
Anton Izosimov
Consider the map which sends a planar polygon to a new polygon whose vertices are the intersection points of second nearest sides of . This map is the inverse of…
The limit point of the pentagram map and infinitesimal monodromy
Quinton Aboud, Anton Izosimov
The pentagram map takes a planar polygon to a polygon whose vertices are the intersection points of consecutive shortest diagonals of . The orbit of a convex polygon un…
The pentagram map, Poncelet polygons, and commuting difference operators
Anton Izosimov
The pentagram map takes a planar polygon to a polygon whose vertices are the intersection points of consecutive shortest diagonals of . This map is known to interact ni…
Pentagram maps and refactorization in Poisson-Lie groups
Anton Izosimov
The pentagram map was introduced by R. Schwartz in 1992 and is now one of the most renowned discrete integrable systems. In the present paper we prove that this map, as well as all…
Smooth invariants of focus-focus singularities and obstructions to product decomposition
Alexey Bolsinov, Anton Izosimov
We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic -manifolds. We show that, in contrast to elliptic…
Vortex sheets and diffeomorphism groupoids
Anton Izosimov, Boris Khesin
In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-…
Classification of Casimirs in 2D hydrodynamics
Anton Izosimov, Boris Khesin
We describe a complete list of Casimirs for 2D Euler hydrodynamics on a surface without boundary: we define generalized enstrophies which, along with circulations, form a complete…
Pentagrams, inscribed polygons, and Prym varieties
Anton Izosimov
The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\…
Characterization of steady solutions to the 2D Euler equation
Anton Izosimov, Boris Khesin
Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or witho…