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作者：Colding, Tobias Holck
作者单位：Massachusetts Institute of Technology (MIT)
摘要：We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated monotonicity formula for the Ricci flow. We will...
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作者：Namazi, Hossein; Souto, Juan
作者单位：University of Texas System; University of Texas Austin; University of British Columbia
摘要：We give a complete proof of the Bers-Sullivan-Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence...
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作者：Viklund, Fredrik Johansson; Lawler, Gregory F.
作者单位：Columbia University; University of Chicago; University of Chicago
摘要：The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm-Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.
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作者：Barbasch, Dan; Ciubotaru, Dan; Trapa, Peter E.
作者单位：Cornell University; Utah System of Higher Education; University of Utah
摘要：We define an analogue of the Casimir element for a graded affine Hecke algebra , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H (D) (X) of an -module X, and show that H (D) (X) carries a representation of a canonical double cover of the Weyl group . Our main result shows that the -structure on the Dirac cohomology of an irreducible -module X determines the central character of X in a precise way. This can be interpreted as p-a...
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作者：Klainerman, Sergiu; Rodnianski, Igor
作者单位：Princeton University
摘要：In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with...
