Student Research Colloquium 2009–10

Abstract: Finite-type knot invariants, originally studied by Vassiliev, are conjectured to approximate all knot invariants and hence to distinguish all knots. We will describe these invariants in elementary, purely combinatorial terms. We will then discuss the configuration space integrals of Bott and Taubes, which provide one way of constructing all finite-type invariants. These integrals have been used more generally to construct nontrivial real cohomology classes in spaces of knots. We will conclude by hinting at how to construct classes in cohomology with arbitrary coefficients using algebraic topology ideas from Math 215C.

Abstract: I will address (with lots of pictures) the very classical problem of studying complete properly embedded minimal surfaces in R³. I will first try and convince the audience that there are a large number of examples of such surfaces. Next, I will highlight what is known and what sorts of questions remain. Finally, I will try and draw a connection between such classification results and compactness theories for sequences of minimal surfaces.

Abstract: I will discuss qualitative properties of solutions of the (n+1)-dimensional wave equation and use the Fourier transform to construct explicitly the solution operator for the equation. I will then discuss the wave equation on curved spacetimes. I will indicate why solutions of these wave equations exhibit the same qualitative properties. Time permitting, I will discuss how the Fourier transform construction can be adapted to construct approximate solution operators for the wave equation on curved spacetimes.

Abstract: Here's the deal. Let M be a closed Riemann surface, (N, g) a Riemannian manifold, and let φ: M → N be a homeomorphism. There is a functional on the homotopy class of φ, the energy functional, that takes a map f and computes the integral of the square norm of its differential. Critical points of this functional are called harmonic maps.

Already you should be angry. The differential is a section of T^*M ⊗ TN, which has no norm unless one is chosen on M. Settle down. The energy integral is invariant under conformal change of the domain metric, so the functional is well-defined. This fact will be important later. Actually, it's important right now.

Theorems proven during the Reagan administration show that there is a harmonic diffeomorphism in the homotopy class of φ, which is unique if the genus of N is bigger than 1.

Under Obama, we have turned our attention to singular metrics. In this talk, I'll show you how to prove the old theorems above in the case of negatively curved targets, and then I'll discuss how to extend those techniques to prove the exact same stuff to spaces with conic singularities. If you're lucky, I'll say something about how one uses this to study the space of conformal structures on a surface of genus g, but I probably won't.

Abstract: Persistence homology is a technique that allows for the computation of homological invariants of a space when only given a large but finite set of points sampled from that space (i.e., point cloud data). I will give an introduction to persistence homology and briefly describe how it is computed.

Significant levels of noise in the data set can become prohibitive for persistence homology techniques and standard de-noising techniques are computationally inefficient on high-dimensional data sets. I will present a computationally efficient algorithm that allows for persistence homology computations on noisy high-dimensional point cloud data sets. I will show the results of the algorithm to synthetically generated noisy data sets and show the recovery of topological information impossible to obtain via established methods for topological data analysis. I will also apply this algorithm to natural image data in R⁸ and show the recovery of topological information previously available only with significant amounts of preprocessing. Time permitting, I will discuss future directions for improving this algorithm using the zig-zag persistence methods.

Abstract: I will describe briefly several motivational applications of moduli spaces of J-holomorphic curves assuming they are "nice" objects. Then I will explain when and why we can assume that.