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Mirzakhani wrote two papers on counting curves of given type on a surface: one for simple curves, and one for arbitrary ones. We give a complete argument deriving Mirzakhanis result for general curves from the one about simple ones. We then sketch an argument to give a new proof of both results -- full details and other related matters will appear in a book we intend to write.
We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda, such as $mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles o
We show that Mirzakhanis curve counting theorem also holds if we replace surfaces by orbifolds.
Let $Sigma$ be a hyperbolic surface. We study the set of curves on $Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $gamma_0$. For example, in the particular case that $Sigma$ is a once-punctured to
We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new funct
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves a