Some metric properties of spaces of stability conditions

We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped with half the hyperbolic metric. We also make two observations about the way in which the heart changes as we move through the space of stability conditions. Firstly, hearts of stability conditions in the same component of the space of stability conditions are related by finite sequences of tilts. Secondly, if each of a convergent sequence of stability conditions has the same heart then the heart of the limiting stability condition is obtained from this by a right tilt.Comment: 10 page

Contractible stability spaces and faithful braid group actions

We prove that any `finite-type' component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi--Yau-$N$ category $\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$ case. Other classes of triangulated categories with finite-type components in their stability spaces include locally-finite triangulated categories with finite rank Grothendieck group and discrete derived categories of finite global dimension.Comment: Final version, to appear in Geom. Topo

WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

AbstractLet X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.</jats:p

Performing Binary Classification of Contests Profitability for Draftkings

In this Major Qualifying Project, we worked alongside the online daily fantasy sports company DraftKings to build an algorithm that would predict which of the company\u27s contests would be profitable for them. Our goal was to detect contests at risk of not filling to their maximum number of entrants by four hours before the contest closed. We combined categorical and numerical header data provided by DraftKings for hundreds of thousands of contests using modern data science techniques such as ensemble methods. We then utilized parameter estimation techniques to model the time series data of entrants into a given contest. Finally these parameters were fed into a Random Forest algorithm with the header data that provided our final prediction as to whether a contest would fill or not

The astrobiology primer: An outline of general knowledge - Version 1, 2006

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