Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area

Toric Topology

By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.

https://iopscience.iop.org/article/10.1088/0031-9112/10/5/018/pdf

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

Dan Reznik found, by computer experimentation, a number of conserved quantities associated with periodic billiard trajectories in ellipses. We prove some of his observations using a non-standard generating function for the billiard ball map. In this way, we also obtain some identities valid for all smooth convex billiard tables.

Dan Reznik’s identities and more

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0419

On the integrability of Birkhoff billiards

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

/pdf/a-complete-bibliography-of-publications-in-the-journal-of-3brkwhz8a7.pdf

A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

Angular Billiard and Algebraic Birkhoff conjecture

Consider a Riemannian metric on two-torus. We prove that the
question of existence of polynomial first integrals leads naturally
to a remarkable system of quasi-linear equations which turns out to
be a Rich system of conservation laws. This reduces the question of
integrability to the question of existence of smooth (quasi-)
periodic solutions for this Rich quasi-linear system.

Rich quasi-linear system for integrable geodesic flows on 2-torus

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards.

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits a $C^1$-smooth first integral which is not singular on $\mathcal A$, then the curve $\gamma$ is an ellipse. } 
The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

Andrey E. Mironov

Papers

Angular Billiard and Algebraic Birkhoff conjecture

Rich quasi-linear system for integrable geodesic flows on 2-torus

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

Angular Billiard and Algebraic Birkhoff conjecture

The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables