Special CCMS Colloquium

Cube Complexes, 3-manifolds, and the Virtual Haken Theorem

04/29/2014 - 4:15pm
Timothy Hsu (San Jose State University)

The Virtual Haken Conjecture was, until recently, probably the biggest open problem in $ 3 $--manifolds
($ 3 $-dimensional geometry).  Then in March 2012, Ian Agol proved the conjecture by completing a key part of Dani Wise's program of studying nonpositively curved cube complexes.  So how did questions in $ 3 $-dimensional geometry end up being resolved using spaces made from (very high--dimensional) cubes?  We'll give an overview explaining the connection and describe the speaker's joint work with Wise that is part of the emerging and rapidly growing subject of cube complexes.

Background: This talk is meant to be accessible to students who have had one semester of abstract algebra.  No background in topology is required, and we will give at least cartoon definitions of the relevant technical terms (e.g., $ 3 $-manifold).

Shanahan 3460 (the SkyCube), Harvey Mudd College
Misc. Information: 

Refreshments at 3:45 p.m. 3rd Floor North Patio of Shanahan & wine and cheese after the talk on the 3rd Floor North Patio of Shanahan

Banach function algebras and approximate identities

03/11/2014 - 4:15pm
03/11/2014 - 5:15pm
H. G. Dales (Lancaster University)

Let $ K $ be a locally compact space. Then $ C_o(K) $ is the Banach algebra of all continuous,
complex-valued functions on $ K $ that vanish at infinity, taken with the pointwise algebraic
operations and uniform norm $ |\cdot|_K $. The maximal modular ideals of $ C_o(K) $ have the
form $ M_x =\{ f\in C(K) \ \colon \ f(x) = 0\} $, where $ x\in K $. We see that each of these maximal
ideals has a contractive approximate identity: this is a net $ (f_\alpha) $ in
$ M_x $ with $ |f_\alpha|_K\leqslant 1 $ and $ \lim_\alpha ff_\alpha=f $ for each $ f\in M_x $.

Now suppose that $ A $ is a Banach function algebra on $ K $ such that each maximal modular
ideal has such a contractive approximate identity. Must we have $ A=C_o(K) $? Or can you
think of such an algebra $ A $ with $ A \not= C_o(K) $? How close does the group algebra of
a locally compact group come to having the above properties?

This is joint work with Ali  Ülger of Istanbul

Seeley G. Mudd Science Library 126, Pomona College
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