Georgia Institute of Technology College of Sciences

 Let (X,Y) be a random couple with unknown distribution P, X being an observation and Y - a binary label to be predicted. In practice, distribution P remains unknown but the learning algorithm has access to the training data - the sample from P. It often happens that the cost of obtaining the training data is associated with labeling the observations while the pool of observations itself is almost unlimited. This suggests to measure the performance of a learning algorithm in terms of its label complexity, the number of labels required to obtain a classifier with the desired accuracy. Active Learning theory explores the possible advantages of this modified framework.We will present a new active learning algorithm based on nonparametric estimators of the regression function and explain main improvements over the previous work.Our investigation provides upper and lower bounds for the performance of proposed method over a broad class of underlying distributions. 

An elliptic curve is the set of solutions to a cubic equation in two variables and it has a natural group structure---you can add two points to get another. I'll explain why this is so, give some examples of the different types of groups that can arise (depending on the ground field), and mention some recent results on curves with many points. The are some nice thesis problems in this area which will be discussed in a follow-up talk later this semester in the algebra seminar.

Various exact results in two-dimensional percolation are presented. A method for finding exact thresholds for a wide variety of systems, which greatly expands previously known exactly solvable systems to such new lattices as "martini" and generalized "bowtie" lattices, is given. The size distribution is written in a Zipf's-law form in terms of the enclosed- area distribution, and the coefficient can be written in terms of the the number of hulls crossing a cylinder. Additional properties of hull walks (equivalent to some kinds of trajectories) are given. Finally, some ratios of correlation functions are shown to be universal, with a functional form that can be found exactly from conformal field theory.

In this talk, we consider the n-dimensional bipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. In 1-D case, when the difference between the initial electron mass and the initial hole mass is non-zero (switch-on case), the stability of nonlinear diffusion wave has been open for a long time. In order to overcome this difficulty, we ingeniously construct some new correction functions to delete the gaps between the original solutions and the diffusion waves in L^2-space, so that we can deal with the one dimensional case for general perturbations, and prove the L^\infty-stability of diffusion waves in 1-D case. The optimal convergence rates are also obtained. Furthermore, based on the results of one-dimension, we establish some crucial energy estimates and apply a new but key inequality to prove the stability of planar diffusion waves in n-D case, which is the first result for the multi-dimensional bipolar hydrodynamic model of semiconductors, as we know. This is a joint work with Feimin Huang and Yong Wang.