Today is the first day of Chinese new year 2011. And I also found some good articles and want to share with you:

1, Vector Diffusion Maps and the Connection Laplacian

Abstract. We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of di usion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector elds. VDM provides tools for organizing complex data sets, embedding them in a low dimensional space, and interpolating and regressing vector elds over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on (or near) a low dimensional manifold Md embedded in Rp, we prove the relation between VDM and the connection-Laplacian operator for vector elds over
the manifold.

2, Topological Homotopy Groups

D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.

3, Gromov Witten invariants of exploded manifolds

This paper describes the structure of the moduli space of holomorphic curves and constructs Gromov Witten invariants in the category of exploded manifolds. This includes defining Gromov Witten invariants relative to normal crossing divisors and proving the associated gluing theorem which involves summing relative invariants over a count of tropical curves.

4, Invariant Hilbert schemes

This is a survey article on moduli of affine schemes equipped with an action of a reductive group. The emphasis is on examples and applications to the classification of spherical varieties.

5,  Introduction to representation theory

 These are lecture notes that arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students, and its extended version given by the first author to MIT undergraduate math students in the Fall of 2008. The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers, and contain many problems and exercises. They should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra, and may be used for an undergraduate or introductory graduate course in representation theory.

ps:In the latest version, misprints and errors were corrected and new exercises were added, in particular ones suggested by Darij Grinberg

6,  Is Zero a Natural Number?

 It is argued that zero should be considered as a cardinal number but not an ordinal number. One should make a clear distinction between order types that are labels for well-ordered sets and ordinal numbers that are labels for the elements in these sets.

7, Fuzzy Logic vs Probability

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