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Abstract

We study the isoperimetric problem in the plane with weighting or density e-1/r. The isoperimetric problem seeks to enclose prescribed weighted area with minimum weighted perimeter. For density e-1/r, isoperimetric curves are conjectured to pass through the origin. We provide numerical and theoretical evidence that such curves have an angle at the origin approaching 1 radian from above as area approaches zero and provide further estimates.

Author Bio

Paul Gallagher graduated in May 2014 from the University of Pennsylvania with a Bachelors and Masters in Mathematics. His mathematical interests lie in both differential geometry and analysis. Research for this paper was done at SMALL 2013 under the guidance of Frank Morgan. Outside of math, he is an avid singer and pianist. In Fall 2014 he enrolled at MIT in a Ph.D. program in pure mathematics.

David Hu graduated from Georgetown University in May 2014. He plans on continuing his studies at SUNY Stony Brook, where he will pursue a Ph.D. in Mathematics. His primary area of interest is differential geometry, especially its interface with analysis/differential equations. His work on this paper is the result of his participation in the 2013 Geometry Group of the SMALL REU at Williams College, advised by Professor Frank Morgan.

Zane Martin graduated from Williams College in 2013, where he majored with honors in mathematics. His undergraduate work focused on geometry, though he is also interested in topology and applied math. His work on this paper began at the 2012 SMALL REU at Williams College, and continued during the 2013 SMALL program. He currently works at a civil rights firm in New York City, and hopes to attend graduate school in mathematics this fall.

Maggie Miller is a senior math major at the University of Texas at Austin. She is currently applying to graduate programs in mathematics, and is primarily interested in knot theory and low-dimensional topology. Outside of class, she draws, paints, and crochets. Her work on this paper was advised by Dr. Frank Morgan during the 2013 SMALL REU at Williams College.

Byron Perpetua graduated in 2014 from Williams College, where he majored in mathematics and music, conducted research in geometry and commutative algebra, and directed the student orchestra. He is currently working as a data analyst at Achievement First, a charter school organization based in Brooklyn.

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