Related papers: Polytopes, supersymmetry, and integrable systems

Foundational cases of the generalized Stokes' theorem are visualized using geometric algebra. From considering bivector valued fields, two seldom used instances of the theorem are obtained. Graphical representations are given, showing a…

In these lecture notes, we give a quick account of the theory of Poisson groupoids and Lie bialgebroids. In particular, we discuss the universal lifting theorem and its applications including integration of quasi-Lie bialgebroids,…

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic…

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact…

This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.

We investigate the interplay and connections between symmetry properties of equations, the interpretation of coordinates, the construction of observables, and the existence of physical relativity principles in spacetime theories. Using the…

Twisted K-theory has received much attention recently in both mathematics and physics. We describe some models of twisted K-theory, both topological and geometric. Then we state a theorem which relates representations of loop groups to…

The potential group method is applied to the n-dimensional Coulomb-Rosochatius potential, whose bound states and scattering states are worked out in detail. As far as scattering is concerned, the S-matrix elements are computed by the method…

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields…

We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the…

Diblock copolymers blended with homopolymer may self-assemble into spherical, cylindrical or lamellar aggregates. Transitions between these structures may be driven by varying the homopolymer molecular weight or the molecular weight or…

In these notes, we describe several geometric interpretations of $H^2(X)$ when $X$ is a trisected 4-manifold. The main insight is that, by analogy with Hodge theory and sheaf cohomology in algebraic geometry, classes in $H^2(X)$ can be…

We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class $A_q$ for $q \in (1,\infty)$. We show that the Stokes problem is well-posed for all $q$. In…

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B…

In this paper we prove some orthogonality relations for representations arising from deep level Deligne--Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig (2004), and of Chan and the second author (2019).…

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

Superintegrable systems on a symplectic manifold conventionally are considered. However, their definition implies a rather restrictive condition 2n=k+m where 2n is a dimension of a symplectic manifold, k is a dimension of a pointwise Lie…

In this paper, we show that there is a close relationship between generalized subtangent manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the…

In arXiv:2004.01871 Satake introduced the notions of admissible triplets and good basic invariants for finite complex reflection groups. For irreducible finite Coxeter groups, he showed the existence and the uniqueness of good basic…

Let C be the set of all possible quantum states. We study the convex subsets of C with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of…