Just as a a soap bubble minimizes surface area, crystals minimize a more general energy depending on orientation with respect to the underlying crystal lattice, given by integrating some (continuous) norm on the unit normal. (One might drop the usual assumption that a norm is even.) The optimal shape is the unit ball of the dual norm, called the Wulff crystal (see [M2, Chapt. 16]). Continue reading ‘Clusters for General Norms’ »
Archive for the ‘Math’ Category.
Enjoying a conference on isoperimetric problems in Pisa. (Click on image to enlarge.)
I’m speaking at a CIME school at the truly Grand Hotel San Michele on the Italian coast, kindly organized by Alberto Farina and Enrico Valdinoci. The path from the hotel to the private beachclub is rather dramatic. It begins with a long descent down the front stairs and another long stairway under the highway.
A joint paper [C1] with my SMALL undergraduate research Geometry Group found least-perimeter pentagonal unit-area tiles, Cairo and Prismatic:
They proved that mixtures of unit-area convex pentagonal tiles can do no better, but found many examples of Cairo-Prismatic tilings that do equally well [C1, C2]. Since their work seven more have been discovered. The first four were found and rendered by one of the original coauthors, Maggie Miller. The first, “Double Pillbox,” adds p4 to p1, p2, p4g, and cmm as the fifth of the seventeen Wallpaper symmetry groups; will anyone find examples of the other twelve? We’d be happy to post any new examples here.
Double Pillbox River
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’ »
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?’ »