The picture, as usual, tells a 1000 words.
You have a Dynanic PET board with 32k of 4116 DRAM. Of which the upper 16k are in sockets. And your 2114 video RAMs not in sockets.
The glaringly obvious thing is that the CPU is in the wrong socket (assuming the IC without a label on it is a 6502 CPU at all), it should be in socket UC4 and the Rockwell 6522 should be in socket UC5. That should be remedied immediately.
A mistake like that suggests somebody has been pulling out IC's and just putting them back into sockets without thinking. Your ROM's could be mixed up too, so check carefully. It pays to check that you have the correct ROMs in the correct places by checking on Zimmers.
If still not working after that:
Check the simple things first, oddly the jumpers on the two jumper blocks are incorrect for a 32k PET. On the 6 way block jumper D should be closed and on the 10 way block jumper k should be closed. You can solder those cut jumpers back together, the rest look correct.
There are a few sources of unreliability on these boards which may or may not be affecting your board.
One is the white single wipe IC sockets. The are usually ok though if the IC's are re-seated so do that initially.
Most PETs repaired on the forum have had at least one faulty 2114 video RAM IC.
Often the ROM's are defective too. You can check these against the Zimmer's files. You need to be able to read the ROMs, a GQ-4x programmer can do it and can create new ROM's using the 2532JL and the 2716 UV Eproms.
The blue Tant capacitors are notorious for shorting out. And occasionally the VIA & PIA IC's are defective.
The best thing initially, is to run the board with the upper 16 k DRAM removed from the sockets, this removes some variables. Check the power supply voltages on the pcb from the 4 voltage regulators. Check that the 555 timer is producing a 1 second reset pulse at turn on and that there is activity on the CPU's sync pin 7.
After the basic checks are done, most people approach the repair by removing the VIA and PIA IC's to rule out more initial variables and running a NOP generator and PETTESTER system. Other fault finding methods can work depending on how familiar you are with digital logic and how handy you are with a scope.