Related papers: The $\beta$-symplectic critical surfaces in Hermit…
We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the K\"{a}hler cone of the latter. We then propose a version of mirror symmetry…
The paper clarifies some subtle points surrounding the definition of scalar curvature in generalized K$\ddot{a}$hler (GK) geometry. We have solved an open problem in GK geometry of symplectic type posed by R. Goto \cite{Go1} on relating the…
Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1(M; \mathbb{Z})\to\mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a…
Given a function $\mathcal{H} \in C^1(\mathbb{S}^2)$, an $\mathcal{H}$-surface $\Sigma$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_\Sigma$ satisfies $H_\Sigma = \mathcal{H} \circ \eta$, where $\eta$ is the…
This paper contains the motivation for the study of critical surfaces. In previous work the only justification given for the definition of this new class of surfaces is the strength of the results. However, when viewed as the topological…
The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the…
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…
It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature K\"ahler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature…
The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for…
The $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component of a closed oriented surface is a preferred component of the character variety consisting of homomorphisms from the fundamental group of the surface to the projective linear group…
This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant K\"ahlerian structure. These…
In this paper we classify constant angle surfaces in $\H^2\times\R$, where $\H^2$ is the hyperbolic plane.
The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either $1$ or $2$. There are three types of edge, and three corresponding parameters $a$, $b$, $c$. It is proved…
We study the K\"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from…
We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in…
The effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices. In particular, results of Monte Carlo simulations for flat, perfect surfaces are…
Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a…
A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be…
We complete the classification of order $5$ nonsymplectic automorphisms on hyper-K\"ahler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of…
This article provides an overview of the techniques related to classification of spherical and more general objects within triangulated categories, and its relationship with algebraic geometry, representation theory and symplectic geometry.…