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1-2-3 Seminar: What is a moduli space?

Arkamouli Debnath, University of Washington
Friday, February 2, 2024 - 2:30pm to 3:30pm
PDL C-401

One usually encounters moduli spaces in Algebraic or Complex Geometry, but the concept itself is categorical and can hence be used in almost every branch of mathematics. We will first introduce the idea of a "family" which vaguely speaking is a map from X ---> Y such that each fibre has some special property P. It turns out that we would like to have "families" of various objects of interest, because instead of studying each of them individually, studying them collectively is more systematic and gives stronger understanding. A moduli space, will then be defined as a very special "family" such that each point on it parameterizes a certain object of interest. In fact one is usually introduced to the projective space as something that parameterizes lines in the plane passing through the origin.

No background knowledge in any of the above will be required to understand examples 1 and 2 of my talk. Example 3, time permitting, will be related to complex geometry, with very little background assumed.

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