*Title*: **When are two things the same?**

*Abstract*: Under Leibniz's characterization of equality: x and y are equal if they satisfy all the same properties,
so that P(x) is true if and only if P(y) is true, for every statement P. Upon further reflection, it may seem strange that a large part of mathematics involves
proving that two things are the same. After all, we define the objects, how could we not know they are the same? Do we care if Juror #8 is Davis?
Clearly, it is a revelation to the Joker that the Batman is Bruce Wayne. Is the angle-side-angle condition for congruence just as revealing?
We will discuss sameness in various contexts with an eye towards probability theory and the isomorphism problem in ergodic theory.
I promise that this will be a fun talk and the everyone will learn something cool.

Smith Colloquium Fall 2018