Student Research Colloquium 2009–10

Abstract: In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |ζ(1/2 + it)| << exp(C log t/log log t). In this talk, I will show how we can find this constant C, and get the best constant for existing methods of bounding ζ(1/2 + it).

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.

Abstract: Given an irreducible component X of the variety of square-zero upper-triangular matrices, a combinatorial formula developed by Rothbach gives a stratification of X into orbits of the Borel group. Specializing to the complex numbers and imposing a rank condition motivated by the Halperin-Carlsson conjecture on the free ranks of products of spheres, we consider a coarser stratification into orbits of the parabolic group, which facilitates a homotopy-theoretic description of X as the homotopy colimit of simpler spaces more amenable to cohomology calculations.

Abstract: We will discuss the geometric motivation behind some of the algebraic operations in string topology, including the Chas-Sullivan loop product, and indicate briefly how to make those ideas rigorous using homotopy theory. We will then introduce Hochschild homology and cohomology as a certain type of homology theory for algebras and their bimodules, and we will see how they relate to loop spaces, string topology operations, and a version of Poincaré duality with fancy coefficients.

Abstract: One of the great ideas in symplectic geometry was to study solutions to an ODE (orbits of a Hamiltonian system) by considering solutions to a PDE (pseudoholomorphic curves). I will provide a brief overview of some of the problems of interest, and give some examples of how these can be approached by pseudoholomorphic curve methods.

Abstract: A prehomogeneous vector space is defined to be a vector space with an algebraic group action that is "nearly transitive", i.e., has a single Zariski open orbit. Work of Gauss, Delone-Faddeev, Bhargava, and others established that the Z-orbits of such vector spaces often parameterize interesting arithmetic objects.

In 1972, Shintani proved that one can naturally associate a zeta function to the space of binary cubic forms. We will discuss Shintani's zeta function from an analytic point of view (where are the zeroes?), and we will discuss what kind of consequences we might hope to derive from such investigations.

Abstract: I'll start by introducing the notion of a knot invariant and giving some example of non-homological invariants. I'll then introduce the idea of a homology theory invariant for knots, with a brief description of some of the important knot homologies currently in circulation. Time permitting, I'll discuss Heegaard Floer knot homology in a bit more detail.