Abstract: Can you find a right triangle all of whose side lengths are whole numbers? Reaching back in your memory, you might recall from an algebra or geometry class that the side lengths have to satisfy the equation a^2 + b^2 = c^2, and that whole number solutions to these equations are called Pythagorean triples. So the answer is yes! And, in fact, there are infinitely many such triangles. However, there are many closely related equations that seem to have no (positive) whole number solutions. For example, there are no nonzero whole number solutions to the equation a^n + b^n = c^n whenever n is at least 3, and we don't know of any whole number solutions to the so-called rational box problem: Does there exist a rectangular box where the distance between any two corners is a whole number? In this talk, I will explain our current guess as to why there is such a difference in the whole number solutions between the pythagorean triple equation and these other examples, describe some of the state-of-the-art results in this direction, and highlight some of the new frontiers in this research.
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Abstract: Just as a current impacts the effort a swimmer must make, so too does the research atmosphere in a community or conference affect research output. In this talk, I will discuss various examples of this, both long-standing programs of others, and many examples that I have experienced or witnessed. In particular, I will discuss different branches of my research program and how their development was impacted by the atmosphere in conferences, seminars, and research communities, and also what I have learned from times when my actions have created counter currents for others. This talk will include some descriptions of harassment.