How can I lower my gaze in certain situations, such as interacting with a female university professor, without causing embarrassment or appearing inappropriate?

The Prophet, peace and blessings be upon him, informed us that Islam would return to being strange, just as it began, but he gave glad tidings to the strangers and urged adherence to the religion in times of estrangement. Although lowering the gaze in times of intermingling is difficult, it is made easy for those for whom Allah makes it easy. A Muslim should be mindful of his Lord and strive for His pleasure. For whoever pleases Allah by displeasing people, Allah will suffice him against the people; and whoever displeases Allah by pleasing people, Allah will leave him to them. The basic rule regarding looking at non-mahram women is prohibition and the obligation to lower the gaze, except in cases of necessity or legitimate need, or the first, accidental glance.