Hello there!
I am Daniel Hartman, a graduate student at the University of Georgia under the direction of David Gay. I like to think about problems in low dimensional topology, primarily around dimension four, and always smooth (if I can help it).
Here is a fun problem. Can you turn the picture on the left into the picture on the right without cutting the circle or puncturing the surface?
A question I'm asked a lot is "how do you see four dimensions". One way to do this is to think ok 4 as 3+1. The 3 is the familiar three dimensions that we are all used to visualizing in. The one is a movie parameter. To the right is a video of just this. One thinks of the whole four dimensional space as a movie in 3 dimensions.
(The other way just image n-dimensions, then set n equal to four)