Geometry/Topology Seminar
3:30 - 4:30pm Tuesdays
425
Hill
Please contact Steve
Ferry, Konstantin
Mischaikow, or Xiaochun Rong
if you would like to speak or suggest a speaker.
Seminar Schedule --- Spring, 2012
(Please scroll down for abstracts.)
|
| |
|
|
| | |
|
| | |
|
| |
|
|
| | |
|
| | |
|
2:15-3:30 room 705 | | |
|
| |
|
|
| | |
|
| |
|
|
| | |
|
| | |
|
| | (joint with Herbert Edelsbrunner) |
|
| | |
|
| (U of Texas and IAS) | |
|
| |
Recovering continuous functions up to homology from dense samples |
|
| |
The asymptotic expansion of Bergman kernel for orbifolds |
|
| Dave Richeson | |
Schedule with abstracts
|
| |
| |
|
| | | We will discuss a theorem of Cohen-Jones-Segal that explains how to reconstruct the homotopy type of a manifold from a Morse function on that manifold. |
|
| | | The publication of the persistence
algorithm by Edelsbrunner, Letscher and Zomorodian (2000) was one of the
decisive breakthroughs in the field of applied topology. It instantly
made it possible to define and efficiently compute robust topological
invariants of a statistical data set. The first step is to represent the
data set by a nested family of simplicial complexes: the smallest being
just the point set itself, the largest being a contractible blob, with
the intermediate complexes carrying topological information at different
scales. The homology of this family of complexes is described by a
"barcode" or "persistence diagram", the length of each bar indicating
the range of of scales over which a particular feature is in evidence.
Two very subtle variations are the "extended persistence" of Cohen-Steiner, Edelsbrunner and Harer, and the "interval persistence" of Dey and Wenger. In different ways, these attempt to capture the changing topology of a family of spaces parametrised by the real line. For a long time, I found both of these constructions to be rather mysterious. Eventually, in collaboration with Carlsson and Morozov, we were able to understand their effectiveness in terms of a new theory called zigzag persistence. This is a generalisation of persistence which is powered by Gabriel's theory of "quiver" representations. I will describe all of this as I currently understand it. |
|
| |
| Conley Index theory provides information about the structure of invariant sets of a dynamical system. In particular, it is a shift equivalence class of an induced map on homology. We are interested in computing Conley Index, and hence induced map on homology. We show how under our usual assumptions the Vietoris-Begle theorem gives a straightforward method of accomplishing this by computing the homology of a relative graph complex furnished by a map on a relative pair. However, the computational space requirements for such an approach might not be acceptable. We present an algorithm (which actually furnishes a constructive proof of the Vietoris-Begle theorem in our setting) that manages to have lower space requirements by handling the graph complex in a fiber-wise manner. We also discuss the possibility of using Discrete Morse Theory techniques in order to speed up the necessary computation on the fibers. |
|
| | | The purpose of this talk is to produce obstructions in the Morel-Voevodsky motivic homotopy category for the existence of embeddings of algebraic varieties. |
|
| | | This will be a gentle introduction to simple homotopy theory. |
|
| | | Knot theory beyond the three-sphere has seen increased attention in recent years, and in this talk I'll focus on knot theory in Seifert fibered spaces. In particular, we'll consider Legendrian knots in Seifert fibered spaces equipped with a special contact form. This setting gives rise to both topological and contact geometric questions, and I'll describe some of the ingredients used to prove the Legendrian non-simplicity of an infinite family of knot types representing torsion homology classes. This is joint work with J. Sabloff. |
|
| | | |
|
| |
| The notion of a weak stability boundary has been successfully applied to design low energy trajectories from the Earth to the Moon. The structure of this boundary has been investigated in a number of studies, where partial results have been obtained. We propose a generalization of the weak stability boundary. In the context of the planar circular restricted three-body problem, under certain conditions on the mass ratio of the primaries and on the energy, we prove analytically that the weak stability boundary about the heavier primary coincides with a branch of the stable manifold of the Lyapunov orbit about one of the Lagrange points. Under more general conditions, we give a semi-numerical argument that the weak stability boundary about the lighter primary consists of points that lie on the stable manifolds of the Lyapunov orbits about two of the Lagrange points. |
|
| |
|
CTC Wall has written an 11 page introduction to simple-homotopy theory called "Formal Deformations" that contains lemmas that may be useful for computational algebraic topology. We will be going over this paper. |
|
| | |
Spring break |
|
| | |
No seminar |
|
| |
| Assume that f: X -> Y is a Lipschitz continuous function between compact Riemannian submanifolds of Euclidean space. Instead of precise knowledge X and Y, we are only given point sets P and Q sampled uniformly from X and Y respectively. Instead of precise knowledge of the images f(P), we can only assign to each p in P some point F(p) "near" f(p) in Y but we do not require that F(p) lie in the sample Q. We provide precise conditions under which knowledge of P, F(P) and Q suffices - with high confidence - to reconstruct the map f_*:H_*(X) -> H_*(Y) induced by f on the homology groups of X and Y. If time permits, we will also show that this reconstruction is robust to bounded sampling noise. |
|
| | | TBA |
|
| (U of Texas and IAS) | | TBA |
|
| Vidit Nanda | | Assume that f: X -> Y is a Lipschitz continuous function between compact Riemannian submanifolds of Euclidean space. Instead of precise knowledge X and Y, we are only given point sets P and Q sampled uniformly from X and Y respectively. Instead of precise knowledge of the images f(P), we can only assign to each p in P some point F(p) "near" f(p) in Y but we do not require that F(p) lie in the sample Q. We provide precise conditions under which knowledge of P, F(P) and Q suffices - with high confidence - to reconstruct the map f_*:H_*(X) -> H_*(Y) induced by f on the homology groups of X and Y. If time permits, we will also show that this reconstruction is robust to bounded sampling noise. |
|
| Xianzhe Dai | | The Bergman kernel in the context of several complex
|
|
| | | We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. The mapping of the index system mimics the expansion and contraction of hyperbolic maps on the tangent space and they may be used like Markov partitions to generate symbolic dynamics. However, because the elements of an index system may have nontrivial intersections, itineraries are not necessarily unique. Thus we discuss how to obtain entropy bounds from symbolic dynamics generated from partitions with overlapping elements. |