Found 13 relevant results in 3.30s where lecturer="Tom Ilmanen"
Calculus II
Analysis II
Calculus in several variables; differential equations
Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
Geometric heat equations: the Mean Curvature Flow and the Ricci flow.These equations make geometric objects evolve in time -- submanifolds and Riemannian manifolds respectively. At first, the object smooths out. Later, singularities and even topological changes may occur. With luck, as t -> infty, the object settles down to a static or self-replicating configuration with an optimal geometry.
Geometric measure theory studies detailed properties of irregular sets and functions in R^n. Some central notions are: Hausdorff measure, rectifiable and unrectifiable sets, covering theorems, varifolds and currents, first variation.Applications include minimal surfaces with singularities, and singularities of nonlinear PDE. The class will be strongly oriented towards solving exercises.
Linear Algebra I
Lineare Algebra I
Introduction to the theory of vector spaces for students of mathematics or physics: Basics, vector spaces, linear transformations, solutions of systems of equations, matrices, determinants, endomorphisms, eigenvalues, eigenvectors.
Linear Algebra II
Lineare Algebra II
Introduction to the theory of vector spaces for mathematicians and physicists including solutions of linear equations, linear transformations, determinants, eigenvalues and eigenvectors, bilinear forms, canonical forms for matrices, and selected applications, part II.
Linear Algebra II
Lineare Algebra II
Introduction to the theory of vector spaces for mathematicians andphysicists including solutions of linear equations, linear transformations,determinants, eigenvalues and eigenvectors, bilinear forms, canonical forms for matrices, and selected applications, part II.
The mean curvature flow is a parabolic evolution equation for submanifolds M_t of R^n. Called the "heat equation for submanifolds", the equation reduces the area of the submanifold, and makes it evolve toward a minimal surface. The equation arises in physical problems with surface tension, and has many applications in differential geometry.
Crucial in the understanding of the behavior of the Ricci flow is the notion of a "Ricci soliton" -- a metric that continually reproduces itself under the Ricci flow (possibly changing scale or sliding around on the manifold). The work of Perelman has revealed a relationship between Ricci solitons and "manifolds with density". We will study classic and recent papers on these topics.
Topics in Geometric Analysis
Topics in geometric analysis
Seminar in geometric heat equations including the linear heat equation,mean curvature flow and Ricci flow, particularly examples, the evolutionof geometric quantities, blowup criteria, the Li-Yau and related Harnackinequalities, and the Perelman functionals.