The use of THE

They are both correct.

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The beginning is referring to a point in time. Let's say that the "age of computer" started in 1980.

You could write, "Since 1980, computer technological advances have increased 10 fold."

"The beginning of the age of computers" *merely* marks a point in time.

>

This is slightly different. It is referring when Dr. Joe started his research and continuing to today.

You could replace the word "beginning" with "starting".

"Since starting his research, Dr. Joe has collected...rocks"

To summarize:

"The beginning" references a specific point in time.

"Since beginning" something implies that the "doer" has begun something and continues to do it.

I hope that helps.