But when might a coffee cup not be equivalent to a doughnut?
“Say we have a six-dimensional coffee cup, and we know we can stretch it into a doughnut, but if you’re only allowed to stretch at a constant rate, how long does that take? Manin said. “Can we get a computer to decide whether you can do it?”
In some cases it’s guaranteed that two topologically equivalent objects will easily transform into one another, he said, but in others it’s not possible for even computers to tell whether two are topologically the same. His work examines the transformation in terms of both the geometric complexity (the deformation) and the algorithmic complexity (the computer’s ability to discern equality or lack thereof).
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Every so often the work of an early career researcher becomes so important, it captures the attention of their professional peers. Recognizing these young scholars of outstanding promise, the Alfred P. Sloan Foundation awards Sloan Research Fellowships to support their future breakthroughs and careers as leaders in their fields.
This year, three of those 128 early-career innovators come from UC Santa Barbara. I congratulate Sung Soo Kim, Fedor Manin and Timothy Brandt on this exceptional honor, said Pierre Wiltzius, dean of mathematical, physical and life sciences. With these fellowships, they join a rarefied group of 126 extraordinarily talented young scholars whose research shows great promise. I look forward to learning about the valuable discoveries these awards will support.