| September 27, 2001 |
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| October 4, 2001 |
Abstract: In this talk I will present some work in progress in vision research. We consider the problem of recognizing what parts of an image are perceived as being in the foreground. We use a variant of the Pao-Geiger-Rubin model, which uses an energy dissipation approach to this problem. The model is surface-based, rather than contour-based. Specifically, the edges in the image are not viewed as isolated contours, but are viewed as bounding a surface. Each local edge has a local hypothesis; for example, a north-south edge might think "the region immediately to the left of me is part of the figure". The model then uses energy dissipation methods to seek assignments of local hypotheses that are mutually agreeable, yielding a segmentation of the image that might be perceived. We test the model on various images to address questions like: does the model "perceive" smaller objects to be in the foreground (the way we do)? convex objects to be in the foreground (the way we do)? how does it perform on optical illusions that viewers report to have two different segmentations? This is joint work with Nava Rubin of the Center for Neural Science, NYU. I thank Anita Disney (CNS, NYU), Davi Geiger (Courant, NYU), Bob Shapley (CNS, NYU), and Dave McLaughlin (Courant, NYU) for useful discussions. Contact person: Bisi Agboola |
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| October 11, 2001 |
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| October 18, 2001 |
Abstract: In this talk I will present links between the control-theoretic concept of controllability and dynamical systems concepts of integrability and ergodicity. The links comes from studying some elements of the theory of control of mixing. This is, in fact, the control theory of measure-preserving systems with specific objectives, typically coming from ergodic theoretic considerations. The mixing objectives such as partition and Kolmogorov-Sinai entropy will be considered and a simple prototypical example of optimal control solved. Some results on controllability of vortex systems will be shown using flat coordinates, showing the use of the concept of integrability in achieving controllability. Finally, a contrast will be made between these results on controllability and a new concept of controllability via ergodicity. A result questioning robustness of the KAM-type results in the context of perturbations with arbitrary time-dependence will be presented using such ergodicity-controllability link. Contact person: Doug Moore |
| October 25, 2001 |
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| November 1, 2001 |
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| November 8, 2001 |
Abstract: In geometric quantization, the quantum Hilbert space depends not only on the symplectic manifold, but on additional data such as the prequantum line bundle and polarization. These Hilbert spaces can be identified if there is a projectively flat connection on the space of polarizations considered. A well-known example is the quantization of the Chern-Simons gauge theory, in which case there is a projectively flat connection on the moduli space of complex structures on a Riemann surface. In this talk, we discuss the projective flatness in geometric quantization, especially when the symplectic manifold is a linear space, and relate it to the Maslov index. Contact person: Xianzhe Dai |
| November 15, 2001 |
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| November 22, 2001 |
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| November 29, 2001 |
Abstract: Quantum computation and information enables more efficient problem solving in four main areas (so far): Experimental number theory (e.g. factoring), quantum physics modelling, combinatorial searching, and communication. Algorithms in these areas are designed for the standard model of quantum computation. There are many other models of quantum computation. These models can be cast in terms of linear representations of monoids presented by generators. What are the relationships between the computational power of these models? For the few examples known, the power is either very weak in a sense to be defined, or equivalent to standard quantum computation. The talk will begin with a short, information-focused introduction to quantum computation. The models of quantum computation will be then introduced from the point of view of monoids. Several possible definitions of computation are possible in each case, two of which are: 1. Non-deterministic computation, which is equivalent to an ability to efficiently calculate certain traces. 2. Standard computation, which corresponds to the ability to sample from probability distributions associated with the representation of the monoid. The known results about the power of these models will be overviewed. Contact person: Doug Moore |
| December 6, 2001 |
Abstract: The recent sequencing of the human and other genomes opens many possibilities for understanding the evolutionary relatioships among organisms, the functioning of living cells, and the mechanisms of disease. The speaker will describe problems of mathematical modeling and computation that arise within these investigations, including genome annotation, the construction of evolutionary trees and the use of clustering and classification techniques to analyze gene expression. Contact person: Doug Moore |