Related papers: On triangular billiards
In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and…
We show that for almost every $(P,\lambda)$ where $P$ is a convex polygon and $\lambda\in(0,1)$, the corresponding outer billiard about $P$ with contraction $\lambda$ is asymptotically periodic, i.e., has a finite number of periodic orbits…
F. Campana had asked whether a certain threefold is rational. In arXiv:1310.3569v1 [mathAG], this variety was shown to be birational to a specific conic bundle and then to be unirational. We prove that this conic bundle is rational.
We exhibit an infinite family of rational homology balls which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson's diagonalisation theorem, and use this…
In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the…
We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal…
Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi--Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an \'etale quotient of…
We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of…
In this work we completely describe the dynamics of triangle tiling billiards. In the first part of this work, we propose a geometric approach of dynamics by introducing natural foliations associated to it. In the second part, we exploit…
A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of $\pi$. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this…
We prove a Rellich-Vekua type theorem for Schr\"{o}dinger operators with exponentially decreasing potentials on a class of lattices including square, triangular, hexagonal lattices and their ladders. We also discuss the unique continuation…
In this paper, we introduce the zero divisor graph of a multiplicative lattice. We provide a counter example to Beck's conjecture for multiplicative lattices. Further, we prove that Beck's conjecture is true for reduced multiplicative…
A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type…
A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed…
We prove that arboreal Galois extensions of number fields are never abelian for post-critically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of…
This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties…
We show that there are only two equable triangles having vertices on the Eisenstein lattice, up to Euclidean motions.
We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the…
We prove the Parshin's conjecture on the rational triviality of the higher algebraic $K$-theory of smooth projective varieties over finite fields. This is known to imply the Beilinson-Soul\'e conjecture for the fields of positive…
We give a new proof for the directional billiard complexity in the cube, which was conjectured in \cite{Ra} and proven in \cite{Ar.Ma.Sh.Ta}. Our technique gives us a similar theorem for some rational polyhedra.