Related papers: Monotone Lagrangians in flag varieties
Lagrangian multiforms provide a variational framework for describing integrable hierarchies. This thesis presents two approaches for systematically constructing Lagrangian one-forms, which cover the case of finite-dimensional integrable…
For certain smooth unimodal families with negative Schwarzian derivative, we construct a set of Collet-Eckmann and subexponentially recurrent parameters $\Omega$, whose complement set has sufficiently fast decaying density, on which…
We introduce a class of generalized Lipkin-Meshkov-Glick (gLMG) models with su$(m)$ interactions of Haldane-Shastry type. We have computed the partition function of these models in closed form by exactly evaluating the partition function of…
For a compact locally symmetric space X, we establish a version of the Selberg trace fromula for a non-unitary representation of the fundamental group of X. On the spectral side appears the spectrum of the "flat Laplacian", acting in the…
This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic…
The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero…
We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the…
We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and…
We prove the Fredholmness of Dirac operators of monopoles with Dirac-type singularities on oriented complete Riemannian $3$-folds, and we also calculate the $L^2$-indices of them.
We study the action functional associated to a smooth Lagrangian function on the cotangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable…
In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…
We found an amazingly simple general "all-band" intensity profile of bandgap (Bragg) solitons for arbitrary parameters of spatially-periodic nonlinear systems, similar to those of multi-frequency stimulated Raman scattering, in particular…
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is…
We present exact expressions, based on the grade restriction rule and window categories, for monodromies associated to certain Calabi-Yau threefold flops. We show a general formula for the monodromy action on the lattice of B-brane charges,…
We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi- Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a…
Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of…
We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study $n$-dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is…
For toric Calabi-Yau threefolds, open Gromov-Witten invariants associated to Riemann surfaces with one boundary component can be written as the product of a disk factor and a closed invariant. Using the Brini-Cavalieri-Ross formalism, these…
The existence and stability of stable bright solitons in one-dimensional (1D) media with a spatially periodical modulated Kerr nonlinearity are demonstrated by means of the linear-stability analysis and in direct numerical simulations. The…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) =…