It is the purnima (full moon) day today, a day when, typically, all the sannyasis and brahmacharis living in the Math (monastery) shave their heads completely bald (except for a small tuft of hair, called a sikha, at the top of the back of the head).
After one of the devotees shaves my head, I shave off my beard too. The last time I had a shave was in Moscow, two weeks ago, the day before I came to India. (If you are clean-shaven and neatly dressed, you seldom have problems with Customs and Immigration.) My sikha is now long enough to tie a knot in it, to keep it tidy.
Why do we shave our heads?
One of the reasons is that a shaved head symbolizes renunciation of the material way of life and dedication to matters of the spirit. Of course, a shaved head does not a renunciant make! Shaving one's head and dressing in the saffron-colored cloth of a renunciant are external: the outer trappings do not make one renounced or give one devotion. That comes from within.
Still, the ancient Vedic tradition of shaving one's head and wearing simple robes does help the younger celibate students (brahmacharis) and older renounced monks (sannyasis) to cultivate humility and freedom from vanity.
And the sikha (amongst other things) distinguishes us as personal theists, as opposed to the Buddhist voidist and Mayavadi impersonalist monks.