Jo Nelson - Introduction to Lie Groups
Following John M. Lee's Introduction to Smooth Manifolds.
Ed Hanson - Dynkin Diagrams
Root systems and their Dynkin diagrams, plus bonus topics like their connection to singularity theory.
Links
- Schedule
- Current Abstracts
- Abstracts, Fall 2007
- Abstracts, Spring 2007
Christine Lien - Lie Algebra Cohomology
Following Bott's Homogeneous Vector Bundles.
Nick Addington - A Topologist goes to Jordan's Class
Jordan showed the other day that an algebraic morphism of elliptic curves (an "isogeny") is automatically a group homomorphism and a covering space, and that the Galois group, the kernel, and the group of deck transformations are all the same. Today we will start with isogenies of Lie groups. Then we will change gears and discuss the analogy between field extensions and covering spaces, and try to say something precise about it in the world of complex manifolds.
Noah Kieserman - Vector Fields on the Circle
Jo said we could study manifolds with action by a smooth group, by linearizing the group and its action. This is what Lie algebras do, and they give us lots of information about our favorite matrix groups. On the other hand, a very useful method of generating Lie algebras is the Lie algebra of derivations. Derivations of what? Of your favorite ring, including the ring of functions on a smooth manifold. In other words, vector fields, and what better place to start than the circle?
Stefan Müller - The Flux Homomorphism
In this talk, we look at examples of infinite-dimensional Lie groups, namely, subgroups of the group of diffeomorphisms of a smooth manifold. While the group of isometries of a Riemannian metric is always finite-dimensional, the groups of symplectic and Hamiltonian diffeomorphisms are infinite-dimensional (for reasons explained in the talk). Following Banyaga and Thurston, we study the 'difference' between the latter groups, using the so called flux homomorphism. After giving two different definitions of this homomorphism (one more geometric, the other more algebraic), we illustrate the flux by looking at my all-time favorite manifolds, the even-dimensional tori (which are themselves finite-dimensional Lie groups).
Garrett Alston - Hamiltonian Group Actions
I will talk about Lie groups that act on symplectic manifolds via Hamiltonian diffeomorphisms. This will lead us to the moment map and a construction known as symplectic reduction. Applications of this construction include toric varieties and Hamiltonian dynamics. Time permitting, I will give a dynamics application, mention action-angle coordinates and tell a funny story. The material will be elementary (i.e. requires no fancy theory).
Hanjun Kim - Symplectic Toric Manifolds
As a continuation Garrett's talk last week, we will continue to think about Hamiltonian actions on symplectic manifolds. But we will restrict ourselves on the case that the Lie group is a torus. This special case has a lot of interesting things. Especially, in most cases, they are algebraic varieties. Plus, by its convexity theorem of moment polytope, one can translate lots of geometric information into combinatorial one, which is good for computational purpose. If time permits, we will also study the algebraic side of this stuff.
Patrick Curran - Introduction to Homogeneous Spaces
No abstract.
Tarje Bargeer - Cultivating Operads in String Topology
I will attempt at providing an idea of how operads get into the picture in the discipline of string topology. This involves an explanation of what an operad is, and what it is good for. I shall furthermore outline an explicit map, that gives us a homological equivalence (originally due to A. Voronov) between the cacti operad and the framed little disc operad.
Michael Childers - Stable Homotopy Theory
I will introduce stable homotopy theory of spaces. I will also show that πk(SO(n)) stabilizes for large n using the fiber bundle SO(n) → SO(n+1) → Sn. In particular, π1(SO(n)) = Z/2Z if n ≥ 3. Finally, I will discuss how Gr(n,∞) is a BO(n).