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# Upcoming Seminars

A curious property of randomized log-space search algorithms is that their outputs are often longer than their workspace. One consequence is that there is no clear way to reproduce the same output when running the algorithm twice on the same input. It is not feasible to store the random bits (or the output) of the previous run in log-space, and using new random bits in another execution can result in a different output. This leads to the question: how can we reproduce the results of a randomized log space computation of a search problem?

We will give a precise definition of this notion of "reproducibility". Then we will show that every problem in search-RL has a randomized log-space algorithm where the output can be reproduced. Reproducibility can be thought of as an extension of pseudo-determinism. Indeed, for some problems in search-RL we show pseudo-deterministic algorithms whose running time significantly improve on known deterministic algorithms.

Joint work with Yang Liu.

An extremal metric, as defined by Calabi, is a canonical Kahler metric: it minimizes the curvature within a given Kahler class. According to the Yau-Tian-Donaldson conjecture, polarized Kahler manifolds admitting an extremal metric should correspond to stable manifolds in a Geometric Invariant Theory sense.

In this talk, we will explain that a projective extremal Kahler manifold is asymptotically relatively Chow stable. This fact was conjectured by Apostolov and Huang, and its proof relies on quantization techniques. We will explain various implications, such that unicity or splitting results for extremal metrics.

Joint work with Yuji Sano ( Fukuoka University).

Multi-finger caging offers a robust approach to object grasping. To securely grasp an object, the fingers are first placed in caging regions surrounding a desired immobilizing grasp. This prevents the object from escaping the hand, and allows for great position uncertainty of the fingers relative to the object. The hand is then closed until the desired immobilizing grasp is reached.

While efficient computation of two-finger caging grasps for polygonal objects is well developed, the computation of three-finger caging grasps has remained a challenging open problem. We will discuss the caging of polygonal objects using three-finger hands that maintain similar triangle finger formations during the grasping process. While the configuration space of such hands is four dimensional, their contact space which represents all two and three finger contacts along the grasped object's boundary forms a two-dimensional stratified manifold.

We will present a caging graph that can be constructed in the hand's relatively simple contact space. Starting from a desired immobilizing grasp of the object by a specific triangular finger formation, the caging graph is searched for the largest formation scale value that ensures a three-finger cage about the object. This value determines the caging regions, and if the formation scale is kept below this value, any finger placement within the caging regions will guarantee a robust object grasping.

Understanding the ways in which the Brain gives rise to the Mind (memory, behavior, cognition, intelligence, language) is the most challengingproblem facing science today. As the answer seems likely to be at least partly computational, computer scientists should work on this problem --- except there is no obvious place to start. I shall recount recent work (with W. Maass and S. Vempala) on a model for the formation and association of memories in humans, and reflect on why it may be a clue about language.