Can humans evolve (mature) fast enough to keep up with AI?

No.

There are multiple reasons for this.

Humans are limited to slow processing speed. At age 68 Bob Munden appeared in Stan Lee's Superhumans. In it, it was found out that his hand is withstanding 10 Gs of force when his weapon is drawn. In a demo, using a Colt .45 single-action revolver, he shot two balloons six feet apart in less than a tenth of second. The computer I am writing to you on has a processor speed measured in billions of instructions per second. By the time you can count to ten (by ones), my computer can to over one billion (by ones). This is a very simplistic comparison of thinking speeds.

The traditional problems of AI include reasoning, knowledge representation, planning, learning, natural language processing, perception and the ability to move and manipulate objects. General intelligence is among the field's long-term goals. Approaches include statistical methods, computational intelligence, and traditional symbolic AI. All this soon becomes beyond the scope of this post. So, I will refer you to several of today's demonstrations of AI.

In May 1997, (an updated version of) Deep Blue defeated Garry Kasparov 3½–2½ in a famous chess match. A documentary mainly about the confrontation was made in 2003, titled Game Over: Kasparov and the Machine.

AlphaGo is a computer program that plays the board game Go. It was developed by DeepMind Technologies which was later acquired by Alphabet Inc.'s Google. AlphaGo had three far more powerful successors, called AlphaGo Master, AlphaGo Zero and AlphaZero. Starting from a 'blank page', with only a short training period, AlphaGo Zero achieved a 100-0 victory against the champion-defeating AlphaGo, while its successor, the self-taught AlphaZero, is currently perceived as the world's top player in Go as well as possibly in chess.

Then there is Boston Dynamics:

https://www.youtube.com/watch?v=bmNaLtC6vkU
https://www.youtube.com/watch?v=hSjKoEva5bg
https://www.youtube.com/watch?v=_sBBaNYex3E


And this is just today. Imagine in 10 years from now. Or 100 years.

According to the most popular version of the singularity hypothesis, called intelligence explosion, an upgradable intelligent agent (such as a computer running software-based artificial general intelligence) will eventually enter a "runaway reaction" of self-improvement cycles, with each new and more intelligent generation appearing more and more rapidly, causing an "explosion" in intelligence and resulting in a powerful superintelligence that qualitatively far surpasses all human intelligence. As you referred to, Stephen Hawking (and Elon Musk has) expressed concern that full artificial intelligence could result in human extinction. The consequences of the singularity and its potential benefit or harm to the human race have been intensely debated.

There are many contemporary movies and television shows that are exploring these thoughts. To link this topic to us, there is currently a television series "Emergence" with an actress that very much pulls at my pedophilia strings. The actress is Alexa Swinton.

https://www.imdb.com/name/nm5216475/