Georgia Institute of Technology College of Sciences

Let g be a positive integer and let Gamma_g be the mapping class group of the genus g closed orientable surface. We show that every finite group is involved in Gamma_g. (Here a group G is said to be involved in a group Gamma if G is isomorphic to a quotient of a subgroup of Gamma of finite index.) This answers a question asked by U. Hamenstadt. The proof uses quantum representations of mapping class groups. (Joint work with A. Reid.)

In this talk, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear problem. We show that the complexity of the problem changes significantly depending on how both P and X are described, that is, on the encoding of the input data. For example, we analyze the case where the complete linear formulation of P is provided, as opposed to the case where P is given by an implicit description (to be defined in the talk). When P has binary vertices only, we provide additional tractability results and linear formulations of polynomial size. Some applications and extensions to integral polytopes will be discussed. Joint work with Shabbir Ahmed, Santanu S. Dey, and Volker Kaibel.

A panel discussion with John Etnyre, Michael Lacey, Wing Suet Li, and Tom Trotter.

Reproducing kernel Hilbert spaces (RKHS) have recently emerged as a powerful mathematical framework for many problems in machine learning, statistics, and their applications. In this talk, we will present a formulation in vector-valued RKHS that provides a unified treatment of several important machine learning approaches. Among these, one is Manifold Regularization, which seeks to exploit the geometry of the input data via unlabeled examples, and one is Multi-view Learning, which attempts to integrate different features and modalities in the input data. Numerical results on several challenging multi-class classification problems demonstrate the competitive practical performance of our approach.