[Newsletter.]
 

September 2003
Candidate: Zuoliang Hou Abstract: In this thesis, I studied the stability of local complex singularity exponents (lcse) for holomorphic functions whose zero sets have only isolated singularities. For a given holomorphic function f defined on a neighborhood of the origin in Cn, the lcse c0(f) is defined as the supremum of all positive real number [lambda] for which 1/|f|2[lambda] is integrable on some neighborhood of the origin. It has been conjectured that c0(f) should not decrease if f is deformed small enough. Using J. Mather and S.S.T. Yau's result on the classification of isolated hypersurface singularities, together with a well known result on the stability of c0(f) when f is deformed in a finite dimension base space, I proved that if the zero set of f has only isolated singularity at the origin, then c0(g) >= c0(f) for g close enough to f with respect to the C0 norm over a neighborhood of the origin, thus gave a partial solution to the conjecture. Using the stability results, I also computed the holomorphic invariant [alpha](M) for some special Fano manifold M.
Title: Local Complex Singularity Exponents for Isolated Singularities
Date: Monday, September 29, 2003
Time: 2:00 pm
Location: Room 3-343
Committee: Gang Tian, thesis advisor
Victor Guillemin
Jason Starr

December 2003
Candidate: Arvind Sankar Abstract: We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let \bar{A} be any matrix and let A be a slight random perturbation of \bar{A}. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed to perform Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).

       We then extend the result on the growth factor to the case of partial pivoting, and present the first analysis of partial pivoting that gives a sub-exponential bound on the growth factor. In particular, we show that if the random perturbation is Gaussian with variance [sigma]2, then the growth factor is bounded by (n/[sigma])O(\log n) with very high probability.

Title: Smoothed Analysis of Gaussian Elimination
Date: Tuesday, December 16, 2003
Time: 9:30 am
Location: Room 2-338
Committee: Daniel Spielman, thesis advisor
Alan Edelman
Shang-Hua Teng, Professor of Computer Science, Boston University
Michel Goemans
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