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{\Large Department of Mathematics, BGU}

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{\Huge Colloquium}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, May 26, 2015}
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\textbf{At} \emph{14:30 -- 15:30}
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\textbf{In} \emph{Math -101}

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{\large\scshape Emanuel Milman 
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  (Technion)
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will talk about
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{\Large\bfseries Curvature-Dimension Condition for Non-Conventional Dimensions\par}
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\textsc{Abstract:}
Given an n-dimensional Riemannian manifold endowed with a probability density, we are interested in studying its isoperimetric, spectral and concentration properties. To this end, the Curvature-Dimension condition CD(K,N), introduced by Bakry and Emery in the 80's, is a very useful tool. Roughly put, the parameter K serves as a lower bound on the weighted manifold's ``generalized Ricci curvature'', whereas N serves as an upper bound on its ``generalized dimension''. Traditionally, the range of admissible values for the generalized dimension N has been confined to {[}n,infty{]}. In this talk, we present some recent developments in extending this range to N \textless{} 1, allowing in particular negative (!) generalized dimensions.
We will mostly be concerned with obtaining sharp isoperimetric inequalities under the Curvature-Dimension condition, identifying new one-dimensional model-spaces for the isoperimetric problem. Of particular interest is when curvature is strictly positive, yielding a new single model space (besides the previously known N-sphere and Gaussian measure): the sphere of (possibly negative) dimension N\textless{}1, which enjoys a spectral-gap and improved exponential concentration.

Time permitting, we will also discuss the case when curvature is only assumed non-negative. When N is negative, we confirm that such spaces always satisfy an N-dimensional Cheeger isoperimetric inequality and N-degree polynomial concentration, and establish that these properties are in fact equivalent. In particular, this renders equivalent various weak Sobolev and Nash inequalities for different exponents on such spaces.








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