The professor says "whether we like it or not" which could mean there will be many effects of globalization, some could be positive or negative and there would be reason for certain people to not like it. Also, "here to stay" as in coming into our lives to remain forever.
Doldrums are caused by solar radiation from the sun. Because of this the sunlight beams warms the air and causes it to rise straight up instead of side to side (horizontally).