Spend any amount of time around a topologist and you’re likely to hear the words “nice topological space”. It’s a certainly a handy phrase to exclude counterexamples that might break your proof, but what properties make a space “nice”? How “nice” does a space have to be to be a “nice space”?
We will begin with some motivating examples of spaces that are definitely not nice, and then consider different ways of ruling out these pathologies. Eventually, we will introduce CW complexes and talk about their role in topology, especially homotopy theory. (The basics of homotopy will be defined.)
If time permits, we’ll talk about what makes a "nice" category of spaces, but no categorical background knowledge is needed.
How (Not) to Be Nice: Some Pathological Spaces and an Introduction to CW-Complexes
Feb. 28, 2018 5pm (Math 350)
MathClub
Josh Grochow (CU Boulder: Department of Computer Science)
X
Although computational complexity was motivated by the rise of digital computers, it turns out to have deep relationships with both pure mathematics and the natural sciences. The computational complexity properties of a mathematical object - how computationally difficult it is to compute various functions on the object, or to answer various questions about it - can be as fundamental as its other properties, such as the measure of a set, the homology of a topological space, or the dimension of a ring or group. Computational complexity has also revealed unexpected interactions between seemingly disparate mathematical areas: for example, there's a precise sense in which questions of propositional logic, knot theory, systems of integer (Diophantine) equations, and graph coloring are all isomorphic*. Computational complexity can also shed light on some of the central questions in mathematics, for example, about classification of various objects such as groups, topological spaces, or metrics on a given manifold. This talk will start from the beginning (what is computational complexity?) and take us on a brief tour of some of these beautiful and surprising connections.
* - And if you've seen NP-completeness before, it's not *just* that these problems are all NP-complete! The really is a much stronger notion of *isomorphism* between them.