This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. I show, in particular, that the algebras over the Feynman ...
We study how measures with finite lower density are distributed around $(n-m)$-planes in small balls in $\mathbb{R}^n$. We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.
We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients...
We show that an n-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n-stacks, Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in par...
We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the litera...
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over $\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page of this spectral sequence is an algorithmically computab...
A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of L\'{e}vy-Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possib...
In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the \emph{offspring curve} method, which has been extensively used in the previous works. Our work was inspired by...
We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's mod...
We are concerned with finding explicit generators of the Brauer group of diagonal cubic surfaces in terms of norm residue symbols, which was originally studied by Manin. We introduce the notion of uniform generators and find that the Brauer group of some classes of diagonal cubic surfaces have uniform generators. However, we also prove that the Bra...
