OK, so I stacked bolts of fabric on the mat overnight. Some minor warping remains.
I shot off an email to Quilter's Rule (the maker of the mats) and they got right back to me that day! So I will follow their kindly advice and get back to you all on how it works:
"Good morning, Janet,
I am sorry that your mat is not lying flat after unrolling it from the box. It should. We store our mats flat until the day of shipping, but storage reasons prevent some of our wholesale and distributor customers from storing the mats flat, so they roll and box them upon receipt and store the mats rolled. You may have purchased a mat that has been rolled for quite some time. Sometimes, there is a random mat or two, that just doesn’t quite unfurl properly, though, once it is unboxed.
One best remedy to flatten a mat is with a hot water bath. Run the hottest water you can in a tub and then place the entire mat in the tub – or as much as you can get into the tub. Let it soak for a good 30 minutes or more. Add hot water, as needed. The goal is to loosen as many fibers of the mat as possible. Once the mat has soaked, remove the mat, dry it off, and place it upside down with heavy objects on top of it. Leave it like this for 2-3 days. Sometimes furling the mat in the opposite direction right after the hot water bath (before lying the mat flat) helps, too.
Some people have used carpet tape under a slightly furled mat to keep it adhered flatly to the table-top.
Do not use an iron on the mat.
I hope this information helps!
If you have any additional questions or concerns, please call or email. Quilter’s Rule is open until 11:30am (Central Standard Time) on Fridays. Our office hours on Mondays through Thursdays are from 7:30am-4:30pm. Our phone number is 1-800-343-8671. "