Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, January 18, 2005.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 18, 2005

11:00 am in Altgeld Hall 241,Tuesday, January 18, 2005

#### Elliptic Cohomology: An Overview

###### Jacob Lurie (Harvard)

1:00 pm in 241 Altgeld Hall,Tuesday, January 18, 2005

#### Cosh-like arithmetic functions having an average value

###### Harold Diamond (UIUC)

2:00 pm in 243 Altgeld Hall,Tuesday, January 18, 2005

#### No meeting this week.

4:00 pm in 245 Altgeld Hall,Tuesday, January 18, 2005

#### How efficiently do 3-manifolds bound 4-manifolds?

###### Dylan Thurston (Harvard University)

Abstract: It known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the 4-manifold. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2), for reasonable definitions of complexity''. (For instance, one notion of complexity is the number of tetrahedra in a triangulation of M.) It is an open question whether this quadratic bound can be replaced by a linear bound. The proof uses shadow surfaces, a representation of 3- and 4-manifolds that generalizes many other representations of these manifolds. (Joint work with Francesco Costantino.)