i have the new Mac Mini with the M1 chip and want to connect it to a 2013 imac and have it used as a display. Will this work? Apple store says it will , but still uncertain.

Not Target Display Mode, but you can screen share the M1 mini onto the 2013 iMac and control it via that iMac keyboard and mouse. Ideally, if it is to be done at all, should be over fast 1000BaseT (or faster). I have a 32-inch display on my M1 mini so have no need to display it onto my Late-2013 27-inch iMac, but screen sharing is possible, as I just tested it.

On the M1 mini, you visit System Preferences > Sharing, and select Screen Sharing. If your firewall is on, it may be automatically set to block all incoming connections, and in System Preferences > Security & Privacy, you will want to change the Firewall options to:

and on the 2013 iMac, you want to drag and drop the /Systems/Library/CoreServices/Applications/Screen Sharing.app to your Dock. When you launch it, you will want to enter the hostname of the M1 mini with the .local suffix. Once the M1 mini screen is up on the 27-inch iMac, you can drag/drop files between them.

Apple has the Share the screen of another Mac for reference.

No because the 2020 M1 Mac Mini can not use the 2013 iMac as a display.

please carefully read > https://support.apple.com/en-us/HT204592

Believe what others here are saying. I own a late 2013 27" iMac and there is zero possibility of getting it to work in Target Display Mode.

Depending on what you are doing and are using an ethernet connection, Screen Sharing may be acceptable. Things that would be unacceptable in terms of performance: gaming, any 3D work doing real time manipulation, video editing, photo editing due to the fact that you cannot calibrate the shared screen to the Mini. WiFi performance will be too laggy unless you have a strong signal and ludicrous speed WiFi 6 router.

Even with all that, you still need a display, mouse and keyboard directly attached to the Mini to initially set it up

Bottomline, do yourself a favor and get an actual display and either repurpose the 2013 iMac or donate it to some charity that helps people without computers.