The window for filing for the next fiscal year opens on April 1, 2013 with start date of employment of October 1, 2013.
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You will be subject to the CAP ... thus, you can not begin work prior to October 1, 2013.
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I agree with my colleagues that you will likely be subject to the H-1B cap. I think that the only possible exception to this would be if you have been counted against the cap within the past six years. In other words, before you started working for the University you worked in H-1B status for a cap-subject employer. If this is the case, then you may not be subject to the cap.