Yes, you can map the “Assigned to” field with the ticket creator. This is done with a SharePoint workflow that is run when the ticket is generated.
Use SharePoint Designer to create the workflow.
Open SharePoint Designer and from the Menu pane on the left, select File>Open Site
Give the SharePoint site URL in the dialog and log in.
Now you can see all the lists and pages on the selected site. Click on the “List Workflow” button under the SITE tab in the ribbon and select the list you are using with E-mail Converter from the dropdown.
Add a name for the workflow and click “OK”. The functions are same for SharePoint 2010 and SharePoint 2013, so you can use either. NOTE that steps 8 and 9 are only for the SharePoint 2013 workflow.
Go to Workflow settings and check the box “Start the workflow automatically when an item is created” under Start Options. Click “Save”.
Click on the Edit Workflow button under the WORKFLOW SETTINGS tab. Click on Stage 1 and then click the Action button in the ribbon and select the action “Set Field in Current Item” under List Actions. Now “field”and “value” are inserted in Stage 1.
Go to Stage 1 and select “Assigned to” from the “field” dropdown and “User who created current item” from the “Value” dropdown. Click “ADD” and then click OK.
For a SharePoint 2013 workflow, click on the Action button and select “Go to a stage”. It will be added to the Transition to Stage step.
For a SharePoint 2013 workflow, select “End of workflow” from the “Go to a stage” dropdown
Now all e-mails that are converted into tickets in the selected list will have the name of the ticket creator in the “Assigned to” field.
Click on the Check for Errors button under the WORKFLOW tab in the ribbon. If the workflow is fine, save it and publish it by clicking the Publish button.
Refer to the SharePoint Workflow manual for more workflows.