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Fall 2010 Tutorial
Ricci flow on surfacesDescription: The Ricci flow is a useful and powerful tool in studying geometric and topological problems. Most notably, it was the key to the recent proof of the Poincaré conjecture, which had been one of the most notorious unsolved problems in mathematics. While the Poincaré conjecture concerned manifolds of dimension 3, this tutorial will examine the Ricci flow in dimension 2. Here it offers an accessible and interesting blend of differential equations (related to the heat equation) and the geometry of surfaces (their Gaussian curvature). A goal of the tutorial will be to show how Ricci flow can be used to prove a famous result about the geometry of surfaces, namely the uniformization theorem, which says that every surface admits a conformal metric of constant Gaussian curvature.
Prerequisites: Some basic knowledge of differentiable geometry for surfaces (e.g. Math 136) will be required; some familiarity with PDE would be helpful but not required.
Contact: Yi Li (firstname.lastname@example.org)
Spring 2011 Tutorial
Operators on Hilbert spaces : analysis and applicationDescription: The origins of operator theory can be found in the works of Fredholm; he was interested in integral equations coming from mathematical physics. Soon Hilbert realized that you can consider these equations as kernels of certain integral "operators" and made a detailed study of them. It was then recognized that these operators are best viewed as linear transformations on a Hilbert space. To put it simply, these operators are infinite analogues of the matrices that we see in linear algebra. Today the theory of operators has remarkable and amazing connections with every field of mathematics - from partial differential equations and analysis to knot theory, and from topology and geometry to representation theory, number theory and physics. Not only it is very useful but also very exciting and full of surprises. This tutorial will serve as an introduction to this circle of ideas. We will start by understanding some important classes of operators, the Fredholm operators. We will see that the "index" of these operators is a topological quantity : it is invariant under small purturbations. We will then prove the Toeplitz Index Theorem relating the winding number (another topological quantity) of non-vanishing continous functions to the index of Toeplitz operators. Our goal will be to finish with the proof and some applications of the classical and beautiful theorem of Gelfand and Naimark that says that a commutative C*-algebra can be identified with the algebra of functions on a compact Hausdorff space. This is the starting point of Connes's non- commutative geometry program that adopts the idea that general C*- algebras should be thought of as algebras of functions on "non- commutative spaces".
Prerequisites: Analysis at the level of Math 114. You will need to be familiar with Hilbert spaces (inner products, bounded operators and their adjoints, closed subspaces and projections etc.)
Contact: Stergios Antonakoudis (email@example.com)