Related papers: Toric symplectic singular spaces I: isolated singu…

Let $D$ be an integral domain and $L$ be a field containing $D$. We study the isolated points of the Zariski space $\mathrm{Zar}(L|D)$, with respect to the constructible topology. In particular, we completely characterize when $L$ (as a…

We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.

In this survey article we introduce the notion of frontals, which provides a class of generalised submanifolds with singularities but with well-defined tangent spaces. We present a review of basic theory and known studies on frontals in…

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that…

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit…

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone…

This survey paper addresses uniqueness questions for symplectic forms on closed manifolds, explains what is known in several examples, and reviews some open problems.

This is a report for my Master's reading project where I review some basic ideas in the theory of prequantizing a symplectic manifold. The classic proof that a symplectic manifold is prequantizable if and only if its symplectic form is…

We give a combinatorial description of the embedded contact complex (ECC) of a certain family of contact toric lens spaces that we call concave lens spaces. We also define a notion of a concave toric domain that generalizes the usual…

We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a…

We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants…

This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of…

To find all two-dimensional equivariant symplectic submanifolds in symplectic toric manifolds, we combine the convex geometry of Delzant polytopes with local equivariant symplectic models and obtain a criterion for determining when a…

We study torus actions on symplectic manifolds with proper moment maps in the case that each reduced space is two-dimensional. We provide a complete set of invariants for such spaces. Our proof uses sheaves of groupoids of Hamiltonian…

Over $\C$, Henry Laufer classified all taut surface singularities. We adapt and extent his transcendental methods to positive characteristic. With this we show that if a normal surface singularity is taut over $\C$, then the normal surface…

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the…

In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of…

In this paper we study whether symplectic toric manifolds are symplectically cohomologically rigid. Here we say that symplectic cohomological rigidity holds for some family of symplectic manifolds if the members of that family can be…

In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that…

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective…