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Events happening on April 25, 2017

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Apr. 25, 2017 11am (MATH 350)	Number Theory Hershey Kisilevsky (Concordia University) X The (open) Inverse Problem of Galois Theory asks if an arbitrary finite group G can be realized as the Galois group of a finite extension K of the rational field Q. Among the solutions K/Q with Galois group G, the minimally ramified realizations are those with the least number of primes of Q ramified in K. We study minimally ramified extensions of Q for various groups G, and consider the possible decomposition configurations of the primes of Q ramified in such extensions. In the process we are led to consider some novel combinatorial problems. Decomposition of Ramified primes in Minimally Ramified extensions
Apr. 25, 2017 12:10pm (MATH …	Kempner Boris Hanin (MIT) X Consider a polynomial p_N(z) in one complex variable. The Gauss-Lucas Theorem says that the critical points of p_N lie inside the convex hull of its zeros. But how are critical points actually distributed inside the convex hull if p_N is chosen at random? The purpose of this talk is to explain that in fact each critical point of p_N typically comes paired with a single zero. The distance between a critical point and its paired zero is on the order of N^{-1}, which is much smaller than the typical N^{-1/2} spacing between order of N independently selected points on the sphere. In the first part of my talk, I will give a heuristic interpretation for this pairing by relating zeros and critical points to electrostatics on the Riemann sphere. In the second part, I explain what rigorous theorems are now available and state a few open problems. Pairing between zeros and critical points of random polynomials
Apr. 25, 2017 1pm (MATH 220)	Trevor Jack (CU Boulder) X An element x in a semigroup S is regular if there exists some y in S such that xyx = x. We show that given a semigroup S of transformations by its generators, checking whether an element in S is regular is PSPACE-complete. Checking regularity in semigroups
Apr. 25, 2017 2pm (MATH 350)	Lie Theory Marty Walter (CU) X We redevelop our previous results in a way which is in some ways more complicated, and in other ways, simpler. The goal was/is to solve a problem, which was/is successful. Abelian Groups Revisited
Apr. 25, 2017 2pm (Math 220)	Steve Weinell (CU) Jonsson lattices
Apr. 25, 2017 3pm (MATH 350)	Topology Jonathan Wise (CU Boulder) Infinity Categories
Apr. 25, 2017 3pm (Math 220)	Functional Analysis Nathan Davidoff (CU Boulder) X Hawkins et al. showed how to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set. We show how the Bunce-Deddens algebra arises, up to strong Morita equivalence, from a $ℤ$ -action on a certain rooted tree. Does the construction of spectral triples on graph algebras and crossed products then yield new spectral triples on the Bunce-Deddens algebra? Good question. Spectral Triples and the Bunce-Deddens Algebra

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