3:00 pm Wednesday, September 3, 2014
Group Actions & Dynamics : Recent progress on the Masser conjecture by Han Li (UT Austin) in RLM 9.166
Given two symmetric integral matrices (quadratic forms) A and B of degree n, we are interested in if they are equivalent or not, that is, whether there exists a unimodular integral matrix X such that A=X’BX where X’ is the transpose matrix of X. For definite forms a simple decision procedure can be constructed. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in 1972. In the late 1990s, D. W. Masser made the following conjecture: Let n be at least 3. Then, A and B are equivalent if and only if there exists a unimodular integral matrix X such that A=X’BX and ||X|| is bounded above by C(||A||+||B||)^k, where the constants C, k depend only on n (in contrast to the exponential bound which follows from Siegel’s work). In a recent joint work with Prof. G. A. Margulis, we have settled Masser's conjecture in its full generality. Our approach involves homogeneous dynamics, automorphic representations, reduction theory, etc. But in this talk, I would like to emphasize on the crucial role that the geometry of arithmetic groups has played in solving this problem. Submitted by
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