| Location: Hill 425 |
| Time: Th 3:15-4:15 |
| Organizers: David Duncan and Ali Maalaoui |
| Email: david.duncan2[at]gmail.com |
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Abstract: In this talk I will start by sketching the construction of the equivariant homology and then we will compute it in some simple examples and then we will see how it is much richer then the singular homology and how in applications to some variational problems it gives us more information about the critical points.
Abstract: On a complex vector bundle, there is a correspondence between connections and holomorphic structures. There is a stratification on the space of holomorphic structures - which corresponds to the Morse stratification of the Yang Mills functional on the space of connections.
Abstract: N/A
Abstract: Gauss-Bonnet connects the curvature of a surface with any metric to toplological information, namely the Euler characteristic. It is a cute fact that this theorem can be derived from the Riemann-Roch, essentially by interpreting the various terms involved and some rudimentary complex analysis (the residue theorem). I will assume the Riemann-Roch as fact and then proceed to do some computations out of which will pop out the Gauss-Bonnet. If time permits, I will also hint at how one can apply Riemann-Roch to prove the holomorphic Hopf index theorem.
Abstract: In this talk I will introduce the space of legendrian curves and sketch the proof of some results of Smale about homotopy equivalence of the space of legendian curves to the full loop space. Also we will discuss some of the extensions of that result.
Abstract: For a given complex structure on a Kahler manifold, the set of Kahler forms compatible with that, form a convex cone in $(1,1)$ part of second cohomology which we call Kahler cone. One may ask whether the set of Kahler forms depends on the complex structure? As we deform the complex structure, the Kahler cone might change. In this talk, after a short introduction to the subject, we'll study the possible changes of Kahler cone on a Calabi-Yau Manifold.
Abstract: One strategy for understanding the differential topology of a manifold is by decomposing it into smaller pieces that are easier to understand. We will discuss various ways of doing this, along the way encountering Heegaard decompositions, Dehn twists, Morse functions, and Morse 2-functions. Time permitting we will discuss applications to TQFTs.
Abstract: A Pseudo-riemannian metric of signature (p,q) on a manifold is a smooth section of the bundle of symmetric bilinear forms such the quadratic form has signature (p,q). (For a Riemannian metric, it is positive definite). We give examples of pseudo-riemannian manifolds and look at their interesting properties, contrasting them with the Riemannian case.
Abstract: We'll introduce some basic language about symplectic manifolds, and then talk about moment maps and how they arise in classical mechanics. I will work through lots of examples to motivate the theory.
Abstract: The Ricci flow is a way of evolving the metric. It is known that a solution exists for a short time for any initial metric. But most initial metrics lead to singularities in finite time (which basically means the curvature blows up). Hamilton's theorem gives a criteria for extracting a convergent subsequence (in some sense) in terms of uniform lower bounds for the injectivity radius which is equivalent to volume not locally collapsing. Perelman's key analytic contribution was his no local collapsing theorem which cleared the major hindrance in carrying out Hamilton's program in solving Poincare conjecture.