Georgia Institute of Technology College of Sciences

Weighted Poincar\'e inequalities known for various laws such as the exponential or Cauchy ones are shown to follow from the "usual"  Poincar\'e inequality involving the non-local gradient.  A key ingredient in showing so is a covariance representation and Hardy's inequality.  

The framework under study is quite general and comprises infinitely divisible laws as well as some log-concave ones.  This same covariance representation is then used to obtain quantitative concentration inequalities of exponential type, recovering in particular the Gaussian results.  

Joint Work with Benjamin Arras.  

I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.

Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.