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Miquel dynamics is a discrete-time dynamical system on the space of square-grid circle patterns. For biperiodic circle patterns with both periods equal to two, we show that the dynamics corresponds to translation on an elliptic curve, thus providing the first integrability result for this dynamics. The main tool is a geometric interpretation of the addition law on the normalization of binodal quartic curves.
We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartzs definition of the 2D pentagram map. We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps
In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this
The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We d
We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs $fcolon (X,x_0)to (X,x_0)$, where $X$ is a complex surface having $x_0$ as a normal singularity. We prove that as long as $x_0$ is not a cusp
In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a diff