Euler characteristics of moduli spaces of curves

Let Mgn be the moduli space of n-pointed Riemann surfaces of genus g. Denote by Mgn the Deligne-Mumford compactification of Mgn. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of Mgn for any g and n such that n > 2-2g

Filtered screens and augmented Teichm\"uller space

We study a new bordification of the decorated Teichm\"uller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield short curves having vanishing length in the underlying surface F. As a result, an appropriate quotient of this space of filtered screens on F yields a decorated augmented Teichm\"uller space which is shown to admit a CW decomposition that naturally projects to the augmented Teichm\"uller space by forgetting decorations and whose strata are indexed by a new object termed partially oriented stratum graphs.Comment: Final version to appear in Geometriae Dedicat

Geometric Cross-Modal Comparison of Heterogeneous Sensor Data

In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by detectors at various locations. The information represented by the two modalities is compared using self-similarity matrices (SSMs) corresponding to time-ordered point clouds in feature spaces of each of these data sources; we note that these feature spaces can be of entirely different scale and dimensionality. Several metrics for comparing SSMs are explored, including a cutting-edge time-warping technique that can simultaneously handle local time warping and partial matches, while also controlling for the change in geometry between feature spaces of the two modalities. We note that this technique is quite general, and does not depend on the choice of modalities. In this particular setting, we demonstrate that the cross-modal distance between SSMs corresponding to the same trajectory type is smaller than the cross-modal distance between SSMs corresponding to distinct trajectory types, and we formalize this observation via precision-recall metrics in experiments. Finally, we comment on promising implications of these ideas for future integration into multiple-hypothesis tracking systems.Comment: 10 pages, 13 figures, Proceedings of IEEE Aeroconf 201

Probabilistic Fréchet means for time varying persistence diagrams

© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (D<inf>p</inf>, W<inf>p</inf>), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (D<inf>p</inf>)^N→ℙ(D<inf>p</inf>). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards

An infinite genus mapping class group and stable cohomology

We exhibit a finitely generated group \M whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface \su of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus $g$ with $n$ boundary components, for any $g\geq 0$ and $n>0$. We construct a representation of \M into the restricted symplectic group ${\rm Sp_{res}}({\cal H}_r)$ of the real Hilbert space generated by the homology classes of non-separating circles on \su, which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in H^2(\M,\Z) is the pull-back of the Pressley-Segal class on the restricted linear group ${\rm GL_{res}}({\cal H})$ via the inclusion ${\rm Sp_{res}}({\cal H}_r)\subset {\rm GL_{res}}({\cal H})$.Comment: 14p., 8 figures, to appear in Commun.Math.Phy

Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46234/1/208_2005_Article_BF01446896.pd

From Free Fields to AdS -- Thermal Case

We analyze the reorganization of free field theory correlators to closed string amplitudes investigated in hep-th/0308184 hep-th/0402063 hep-th/0409233 hep-th/0504229 in the case of Euclidean thermal field theory and study how the dual bulk geometry is encoded on them. The expectation value of Polyakov loop, which is an order parameter for confinement-deconfinement transition, is directly reflected on the dual bulk geometry. The dual geometry of confined phase is found to be AdS space periodically identified in Euclidean time direction. The gluing of Schwinger parameters, which is a key step for the reorganization of field theory correlators, works in the same way as in the non-thermal case. In deconfined phase the gluing is made possible only by taking the dual geometry correctly. The dual geometry for deconfined phase does not have a non-contractible circle in the Euclidean time direction.Comment: LaTeX, 1+31 pages, 8 figures v2: refs corrected v3: minor corrections v4: to match with published versio