The Cloud Gate Dance Theatre is extremely influential in its homeland of Taiwan — so influential that there’s a nationally recognized Cloud Gate Day, and the street where the company’s office is located is called Cloud Gate Lane. And when a fire destroyed their rehearsal studio and archives in 2008, $12 million was donated to help rebuild it. The organization was founded in the early 1970s by choreographer Lin Hwai, and Cloud Gate 2, a subsidiary of the company, tours throughout the world. Tonight they’ll be at Vanderbilt, as the first of 2012’s Great Performances. The last dance performance the series hosted, by the Gallim Dance Company back in October, was so powerful that many in the crowd were wiping away tears once the stage lights went up. It’s a tough act to follow, but judging by the quality of other recent Great Performances, they’re up to the task — November’s one-act by New Zealand’s India Ink Theatre Company was so inspiring I swore to my date that, had I seen it at age 14, I might have taken up a career in playwriting. Expect a mix of traditional Taiwanese aesthetics, martial arts and modern dance.