I just had a great time at the amazing conference for 800 top Indian high school science students. Here I provide some contacts and information for my young new Indian friends and all; some pictures, including some wonderful pentagonal tilings I found on the path to the Guest House, the ever busy and calm organizer Kaushal Verma, and my student host Devang Rammohan, who met me at the airport at 5 am and took me back at midnight. I’m grateful to all the organizers and participants for their role in this inspiring vision of the future of science in India and beyond. Continue reading ‘Indian Science Camp’ »
Emanuel Milman [M2, 2013] provides very general sharp lower bounds on perimeter to enclose prescribed volume, including the following special case of convex bodies in Rn. Here we give an alternative proof for that special case as suggested by Milman [M2, §7.2]. For sharp upper bounds see my post [Mo1] on the Convex Body Isoperimetric Conjecture. Continue reading ‘Sharp Isoperimetric Bounds for Convex Bodies’ »
This beautiful Sunday in Williamstown began with a bike ride to the celebration of Alpaca Day at Sweet Brook Farm. Then Fall Festival at Hopkins Forest, including homemade cider and apple butter, music, and many fair representatives of the Williams Math/Stats Department (see photos below). Finally a delightful harpsichord concert by Professor Emeritus Victor Hill at the Clark. Continue reading ‘Beautiful Sunday in Williamstown’ »
In Barcelona, Robert McCann talked about his work with Jonathan Korman and Christian Seis on optimal transportation with a constraint h(x,y) on the flow from x to y. A constant constraint h means that an x must be spread out over at least fraction 1/h of the target; there is not the capacity to send it all to the most desirable spot. Here we present a simplified extension of some of their results. Continue reading ‘Optimal Transportation with Constraint’ »
Sixty mathematicians and students gathered in Barcelona at the Center for Research in Mathematics CRM for a Conference on Qualitative and Geometric Aspects of Elliptic PDEs, proficiently organized by Xavier Cabré, Daniele Castorina, Manel Sanchón, and Enrico Valdinoci.
In my talk I mentioned a new isoperimetric theorem by Xavier Cabré and his students Xavier Ros-Otón and Joaquim Serra, which they describe in this video. Continue reading ‘Geometry and PDEs in Barcelona’ »
The famous Wallet Paradox invites two similar individuals to lay their wallets on the table, the one with the lesser amount of money to win both. Paradoxically, each might reason: “I have the advantage, because if I lose, I lose just what I have, but if I win, I win more than I have.” A follow-up analysis assumes that each has the same expected amount of money and asks for the best probability distribution or “best strategy” with that given mean. The following note is based on a senior colloquium talk. Continue reading ‘Pradham ’13 on Wallet Paradox’ »
Given a smooth Riemannian manifold, the isoperimetric profile I(V) gives the infimum perimeter of smooth regions of volume V.
Proposition 1. In a compact smooth Riemannian manifold of dimension at least two, the isoperimetric profile is continuous. Continue reading ‘Isoperimetric Profile Continuous?’ »
My mom and I just had a great time on the first ACBL Regional at Sea, an Alaskan cruise with the inimical Billy Miller. My mom won enough gold points to become a Life Master. In that fateful session Thursday afternoon, July 18, 2013, as West she played one hand at 3N which should only make 2N and she somehow made 5N:
Jian Ge’s recent ArXiv post on “Comparison theorems for manifolds with mean convex boundary,” Theorem 0.1, has a generalization to manifolds with density, here within a factor of 2 of sharp for constant density: Continue reading ‘Distance to Boundary of Manifold with Density’ »