Abstract 1.
Let f be a square-free polynomial. The root separation of f is the minimum of the pair-wise distance between the complex roots of f. A root separation bound is a lower bound on the root separation.
Finding a root separation bound is a fundamental problem, arising in numerous disciplines in  mathematics, science and engineering. Due to its importance, there has been extensive research on this problem, resulting in various celebrated bounds. 
However, the previous bounds are still very pessimistically small. Furthermore,  surprisingly, they are not compatible with the geometry roots:  for instance, when the roots are doubled, the bounds do not double. Worse yet, the bounds even become smaller.
 In this talk, we present another bound, which is "nicer" than the previous bounds in that(1) It is siginificanly bigger (hence better) than the previous bounds.(2) It is compatible with the geometry of the roots.

2.Connectivity in Semialgebraic Sets
If time allows, we will also describe a generalization to multivariate polynomials systems.This is a joint work with Aaron Herman and Elias Tsigaridas.
Abstract 2.
A semialgebraic set is a subset of real space defined by  polynomial equations and inequalities.  A semialgebraic set  is a union of finitely many maximally connected components.
In this talk, we consider the problem of deciding whether  two given points in a semialgebraic set are connected, that  is,  whether the two points lie in  a same connected  component. In particular, we consider the semialgebraic set defined by f not equal 0 where f is a given bivariate  polynomial.
The  motivation comes from the observation that many  important/non-trivial problems in science and engineering  can be often reduced to that of connectivity. Due to it  importance, there has been intense research effort on the  problem.
This is a joint work with James Rohal, Mohab Safey Eldin and Erich Schost. 
We will describe a method based on gradient fields and  provide a sketch of the proof of correctness based Morse  complex.  The method seems to be more efficient  than the previous methods in practice.We will also provide a bound on the length of the curves connecting the points.

3.Algorithm for Computing mu-bases of univariate polynomials
Abstract 3.
We present a new algorithm for computing a mu-basis of the syzygy module of n polynomials in one variable. The algorithm is conceptually  different from the previously-developed algorithms  by Cox, Sederberg, Chen, Zheng, and Wang for n=3, and by Song and Goldman for an arbitrary n. The algorithm involves computing a  ``partial'' reduced row-echelon  form of a certain matrix. The proof of the algorithm is  based on standard linear algebra and is completely self-contained. Comparison with Song-Goldman algorithm will be given.
This is a joint work with Irian Kogan, Zachary Hough.