Frank Morgan Visit
Webster Atwell Professor of Mathematics, Williams College
Wednesday, January 25, 2012 7pm CUE 202 Reception to follow in Hacker Lounge on the second floor of Neill Hall (Mathematics Building).
All students (undergraduate and graduate), faculty and interested members of the community are invited to attend the talk.
Title: Densities: from Geometry to the Poincaré Conjecture
Abstract: Many insights in geometry, including Perelman's 2003 proof of the Poincaré Conjecture, come from putting a positive weighting or "density" on volume and perimeter. The talk will include some open questions and progress by undergraduates. No prerequisites, students welcome.
About the speaker: Frank Morgan is Webster Atwell Professor of Mathematics at Williams College and is a champion of mathematics teaching. He works in minimal surfaces, studies the behavior and structure of minimizers in various dimensions and settings, and has published over 150 papers. His proof with colleagues and students of the Double Bubble Connecture is featured at the NSF Discoveries site. He has six books: Geometric Measure Theory: a Beginner’s Guide (4th ed. 2009), Calculus Lite (2001), Riemannian Geometry: a Beginner’s Guide (1998), The Math Chat Book (2000) based on his live, call-in Math Chat TV show and Math Chat column, Real Analysis (2005), and Real Analysis and Applications(2005).
Further Bio is hosted on his website, http://math.williams.edu/morgan/bio
READINGS FOR THOSE SO MOTIVATED
Good place to start for understanding the difficulties of defining the notion of surface. What is a surface? - Frank Morgan
Chapter 18 of Geometirc Measure Theory: A Beginner's Guide by Frank Morgan - Manifolds with Densities and Perelman's Proof of the Poincaré Conjecutre. For a copy of the text, see Kevin Vixie
THE POINCARÉ CONJECTURE
The Poincaré Conjecture was considered one of the most difficult open problems in mathematics at the turn of the millennium. It is the only solved Millennium Prize Problem of the Clay Mathematics Institute. Link: http://www.claymath.org/millennium/
Asked in 1904, the Poincaré Conjecture states: Every simply connected, closed n-manifold (n>1) is homeomorphic to the n-sphere. Poincaré proved the n=2 case and proposed the question for n=3. This case was in fact the most difficult case. n>4 was solved in 1961, and n=4 in 1982. The extreme difficulty of n=3 made the Poincaré Conjecture the fifth Millennium Prize problem, whose solution carried a $1,000,000 bounty. In much amazement, Grigori Perelman submitted his solution in 2003, which was confirmed in 2006. Perelman was awarded the Fields Medal and Millennium prize for his work, however, he declined both. Links on bottom for the complete story of Perelman.
The Poincaré Conjecture is quite simple to state and explain. An n-manifold is a space which locally looks like n-dimensional Euclidean space. Intuitive notions of surfaces of R3 are 2-manifolds, like a Pringles chip (hyperbolic paraboloid) or the surface of a doughnut (torus). Simply connected means that any loop on the manifold may be shrunk into a single point. The hyperbolic paraboloid is simply connected, but the torus isn't; take a loop around the "doughnut hole".
The conjecture may be rephrased as: If every loop in an n-manifold can be shrunk into a single point, then the manifold can be deformed (stretched and contorted) into a 3-sphere.
Two important concepts were used in Perelman's proof, ricci flow and manifolds with densities. The latter is relevant to Dr. Morgan's colloquium.