amazing ideas

Talk is cheap, idea is valuable. There are so many talks at Berkeley and Simons in particular and most of them are forgotten within a few weeks. The best talks convey one fresh and engaging idea. It is such ideas that sustain new research. They are rare and we can spent days and weeks foraging at conferences before encountering a new one. And when we do, they should be treasured.

Here are some recent ideas:

1. Brain computer interface. By recording neuronal activity of several dozens of neurons in our brain as we perform a specific task (think picking up a cup of water and bringing it to our lips), we can train a mapping from physical activity to neural signals. We can invert this mapping, so that given a set of brain signals, we can infer what activity it correspond to. This is useful for people with physical disability for example. If they can think, “pick up the cup”, and we can decipher it, then we can pick up the cup for them. This works to some extent, but not very well. The reason is somatosensory perception, which, as far as I can tell, is the idea the we are aware of our body. Touch is an example of somatosensation, and apparently normal individuals who have temporarily lost touch have a hard time picking up small objects even if their motor system is not impaired. The idea is that brain computer interface without feedback to the brain that somehow mimic somatosensation would be inaccurate. Ok so let’s build better neural UI by stimulating some neurons to imitate such feedback.

2. EPAS1 gene and high altitude adaptation. One reason that Tibetans are well adapted to high altitude is because of several mutations in the EPAS1 gene. It’s clearly under strong positive selection. Is it selection on a de novo mutation or a standing variation? Actually turned out to be an introgression from neanderthals at EPAS1 that was present in the initial population and then spread due to selection. A clean (and very rare) example of beneficial gene flow from neanderthals to a human population. Very cool!

3. Godel’s incompleteness theorem, as told by the incomparable Christos. My favorite Simons lunch! This statement is unprovable. This is the type of self-referential statement that drives logicians to the wall. If true then the logical system is incomplete. If false, then we can “prove” that this statement is unprovable and the framework is inconsistent. Hmmm. So to show that a mathematical system, say integer arithmetic, is incomplete then we want to encode into mathematical language this self-referential statement. To do this rigorously, the key idea is arithmetization. We can assign each symbol to a hexadecimal number. So that each logical statement (e.g. for all x there exits y such that x = 2y or x=2y+1) now corresponds to a number. Moreover each proof, which is just a string of logical statements, is also just a number! Then we can encode a self-referential statement as a number.

4. Grobner basis. Given a multivariate polynomial f and a set of polynomials g1, g2, …, is f in the ideal generated by {g’s}? This is equivalent to asking can we write f as a linear combination of the g’s where the coefficients are other polynomials. The straightforward thing to do is to take f, divide by g1 as much as possible, and the divide the remainder by g2 and so on. If in the end, the remainder is 0 then f is in the ideal. The problem is that the outcome depends on the order of the g’s, so we might need to try all the orderings before conclusively concluding where f is in the ideal or not! Let I = <g’s> be the ideal. A Grobner basis of I is a set of polynomials h in I such the <leadterm(h)> = leadterm(I). Moreover, <h> = I. These h’s form a nice basis of I meaning that the order of h does not matter in the division. So we can test for ideal membership easily. In the worst case, it’s doubly exponential to find a Grobner basis (not unique), but works in practice.




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