# Mathematics Colloquia and Seminars

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### Speakers: Brian Munson, Stanford and Tim Cochran, Rice University

**Geometry/Topology**

Speaker: | Bay Area Topology Seminar, Fall meeting at Stanford |

Location: | 383 Stanford Mat |

Start time: | Tue, Nov 9 2004, 2:30PM |

Time: 2:30 pm Both talks will be in 383 N (3rd floor math dept). Speaker: Brian Munson, Stanford Title: Recent progress in the study of smooth embeddings Abstract: The problem of finding an embedding in the regular homotopy class of an immersion of a smooth manifold M in a smooth manifold N is a problem first studied by Hassler Whitney in the 40s and later by Andre Haefliger in the 60s. Whitney showed that when the dimension of M is less than half the dimension of N, there is no obstruction. In the case when twice the dimension of M is equal to the dimension of N, he gave a complete obstruction to finding such an embedding, which is measured completely by the double point set of M. Haefliger greatly generalized Whitney's methods, and proved that if the dimension of M is less than about 2/3 the dimension of M, then the double point set describes the complete obstruction to finding an embedding in the regular homotopy class of such an immersion. When N is Euclidean n-space, my contribution to this subject was to describe a complete obstruction when the dimension of M is about 3/4 of n. I will indicate how the Calculus of Functors unifies the above work and gives a window into how these problems relate to aspects of homotopy theory such as the Freudenthal suspension theorem and Whitehead products. I will also discuss a current project, joint with Greg Arone, which seeks to use some of Arone's computations in Michael Weiss' Orthogonal Calculus to gain further geometric insight into this problem. Time: 4:10 pm Speaker: Tim Cochran, Rice University Title: Homology and Derived Series of groups Abstract: I will discuss joint work with Shelly Harvey. In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. His work has had important applications in topology via questions related to homology cobordism of manifolds. We establish a similar relationship between the low-dimensional homology of a group and its derived series. I will discuss the theorem and sketch some applications. We also define a "solvable completion" of a group that is analogous to the Malcev completion, with the role of the lower central series replaced by the derived series. We prove that the solvable completion is invariant under rational homology equivalence.

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