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We consider (holomorphic) Lagrangian fibrations X->P^n that satisfy some natural hypotheses. We prove that there are only finitely many such Lagrangian fibrations up to deformation.
Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-K\"ahler manifold. We prove that every codimension $1$ fiber of $\pi$ has multiplicity $1$.
A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work,…
A tensor extension of the Poincar\'e algebra is proposed for the arbitrary dimensions. Casimir operators of the extension are constructed. A possible supersymmetric generalization of this extension is also found in the dimensions $D=2,3,4$.
We provide a sufficient criterion for the recurrence of spatial random graphs on the real line based on the scarceness of long-edges. In particular, this complements earlier recurrence results obtained by Gracar et al. (Electron. J. Probab.…
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by…
Parafermions of order two are shown to be the fundamental tool to construct ternary superspaces related to cubic extensions of the Poincar\'e algebra
This work is, in part, a generalization of the article by A.A. Bruen ,T.C Bruen and J.M.McQuillan on Desargues Theorem in arXiv:2007.09175[mathCO]July 17,2020. We prove the extension of Desargues theorem in all dimensions, using 4 different…
Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not…
We study the structure and dynamics of the infinite sequence of extensions of the Poincar{\'e} algebra whose method of construction was described in a previous paper [1]. We give explicitly the Maurer-Cartan (MC) 1-forms of the extended Lie…
We study O'Grady examples of irreducible symplectic varieties: we establish that both of them can be deformed into lagrangian fibrations. We analyse in detail the topology of the six dimensional example: in particular we compute its Euler…
This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can…
A semi-simple tensor extension of the Poincar\'e algebra is proposed for the arbitrary dimensions $D$. A supersymmetric also semi-simple generalization of this extension is constructed in the D=4 dimensions. This paper is dedicated to the…
We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total…
We propose to use a novel master Lagrangian for performing the bosonization of the $D$-dimensional massive Thirring model in $D=d+1 \ge 2$ dimensions. It is shown that our master Lagrangian is able to relate the previous interpolating…
We prove that every irreducible component of a fibre of a complex Lagrangian fibration is Lagrangian subvariety. Especially, complex Lagrangian fibations are equidimensional.
We establish computational results concerning the Lagrangian capacity, originally defined by Cieliebak-Mohnke. More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. Working…
In our previous article Phys. Rev. Lett. 127 (2021) 271601, we announced a novel 'democratic' Lagrangian formulation of general nonlinear electrodynamics in four dimensions that features electric and magnetic potentials on equal footing.…
Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational…
We characterize regularity of Lagrangian submanifolds in Weinstein Lefschetz fibrations, establishing a conjecture of Giroux and Pardon. Our main result is the Weinstein analogue of a closed symplectic Lefschetz pencil result of Auroux,…