Related papers: On triangular billiards
We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish…
In this paper we show that the billiard ball map of the Liouville billiard tables of classical type on the ellipsoid is non-degenerate at the elliptic fixed point. As a corollary we obtain a spectral rigidity result.
This comment is an analysis of the results presented by Wang et al. in their their 2014 paper on irrational right triangular billiards. They submit numerical evidence that these billiards are a novel kind of nonergodic, incompatible with…
Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has…
Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are called…
We prove that the billiard claimed to be a possible counterexample to the Birkhoff-Poritsky conjecture is actually not a counterexample. We also show that for a billiard in a table obtained by the string construction over any convex…
A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with…
In this paper we show that a certain solvable Lie group constructed in a paper by Benson and Gordon has no lattices. This result answers (in the negative way) a question posed by several authors in the context of symplectic geometry. The…
In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the…
In "Non arithmetic super rigid groups: counter examples to Platonov's conjecture" Bass and Lubotzky gave a counter example to Platonov's conjecture by presenting an example of a linear group with super-rigidity which is not an arithmetic…
We consider the billiard map in a convex polyhedron of $\mathbb{R}^3$, and we prove that it is of zero topological entropy.
It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the…
The conjecture that every modular lattice is integral is disproved.
We prove that a closed, simply connected, positively curved, cohomogeneity-three manifold whose quotient space has no boundary is rationally elliptic, thus providing a generalization of similar results regarding rational ellipticity of…
The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the BGS conjecture. We show that the length spectra of their periodic orbits divides into subspectra…
Athanasiadis and Kalampogia-Evangelinou recently conjectured that the chain polynomial of any geometric lattice has only real zeros. We verify this conjecture for families of geometric lattices including perfect matroid designs, Dowling…
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a…
We prove that certain fibered, $-$amphicheiral knots are rationally slice. Moreover, we show that the concordance invariants $\nu^+$ and $\Upsilon(t)$ from Heegaard Floer homology vanish for a class of knots that includes rationally slice…
A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…
We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in…