Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

Abstract: We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts.

Localization in Disordered, Nonlinear Dynamical Systems

Abstract: We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrodinger equations, or randomly coupled rotors.

Planar realizations of nonlinear Davenport-Schinzel sequences by segments

Abstract: LetG={l1,...,ln} be a collection ofn segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions ofx, letYG be their lower envelope (i.e., pointwise minimum).YG is a piecewise linear function, whose graph consists of subsegments of the segmentsli. Hart and Sharir [7] have shown thatYG consists of at mostO(n?(n)) segments (where?(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a setG ofn segments for whichYG consists ofΩ(n?(n)) subsegments, proving that the Hart-Sharir bound is tight in the worst case. Another interpretation of our result is in terms of Davenport-Schinzel sequences: the sequenceEG of indices of segments inG in the order in which they appear alongYG is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements ofEG are equal andEG contains no subsequence of the forma ...b ...a ...b ...a. Hart and Sharir have shown that the maximal length of such a sequence composed ofn symbols is ?(n?(n)). Our result shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope ofn straight segments, thus settling one of the main open problems in this area.