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Events happening on January 30, 2018

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Jan. 30, 2018 11am (MATH 350)	Number Theory David McKinnon (University of Waterloo, Canada) X Of all the algebraic curves around, the rational ones have the most rational points. (And yes, the terminology is mostly accidental!) So maybe if an algebraic variety of higher dimension has lots of rational points, then it ought to have lots of rational curves too? The answer is "probably yes", but is complicated enough that I can spin it out to a whole seminar's worth of talking. Vojta's Conjectures, the Batyrev-Manin Conjectures, and Diophantine approximation are but three of the threads that I will weave into the narrative. Rational curves and rational points
Jan. 30, 2018 12:10pm (MATH …	Kempner Jesse Wolfson (UC Irvine) X Algebraic topology was invented by Poincare in 1895 to study the behavior of algebraic functions. In his seminal ICM address 5 years later, Hilbert posed a fundamental challenge to the field: find a topological obstruction to reducing the solution of the general degree 7 polynomial to an expression in functions of two or fewer variables. In this talk, I'll review some of the beautiful history of algebraic topology and algebraic functions, discuss Hilbert's problem(s), and outline ongoing work in applying the topology of braids and algebraic functions to this problem. This is joint work with Benson Farb. Algebraic Topology and Hilbert's 13th Problem(s)
Jan. 30, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) Supernilpotent algebras 2
Jan. 30, 2018 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (CU Boulder) X By using the action of a Galois group on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, that each Galois character is a positive linear combination of the Galois irreducible characters. For some family of groups, we show that whenever the complex characters of the groups produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups also produce a PSH. We will classify the Galois characters and Galois classes of the general linear groups over finite fields. In the end, we will precisely indicate an isomorphism between the PSH of Galois characters and a certain tensor product of Hopf algebra of symmetric functions. Galois Character Theory
Jan. 30, 2018 2pm (MATH 220)	William DeMeo (CU Boulder) A Tutorial Introduction to the Lean Prover: Part 2
Jan. 30, 2018 3pm (MATH 350)	Topology Jesse Wolfson (University of California Irvine) X Abstract: Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood. Title: Coincidences of homological densities, predicted by arithmetic

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