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Events happening on October 10, 2017

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Oct. 10, 2017 10am (Math 350)	Functional Analysis Arlan Ramsay (CU Boulder) Decompositions of Polish Spaces and Measures on Them
Oct. 10, 2017 11am (MATH 350)	Number Theory Robert Benedetto (Amherst College) X Let be a number field, let be a rational function of degree , and let . The roots of are the -th preimages of under , and they have the natural structure of a -ary rooted tree . There is a natural Galois action on the tree, inducing a representation of the absolute Galois group of in the automorphism group of . In many cases, it is expected that the image of this arboreal Galois representation has finite index in the automorphism group, but in some cases, such as when f is postcritically finite (PCF), the image is known to have infinite index. In this talk, we will present the first complete calculation of the arboreal Galois group of a PCF map, namely , that is not induced by an endomorphism of an algebraic group. The arboreal Galois group of a PCF cubic polynomial
Oct. 10, 2017 12:10pm (MATH …	Kempner Robin Deeley (CU Boulder) X The Atiyah-Singer index theorem represents a landmark in 20th century mathematics. Since its proof in the 1960s, index theory has developed in many directions and found many applications. Understanding the index theorem is still nontrivial. However, there are a number of useful frameworks which make the Atiyah-Singer index theorem somewhat more intuitive. I will discuss one such framework, K-homology. K-homology is the homology theory dual to the cohomology theory K-theory. However, its usefulness in index theory comes from its various realizations (e.g., Baum-Douglas cycles, Fredholm cycles, extensions, etc), explicit isomorphisms between such realizations, and explicit pairings between such cycles and K-theory cocycles. Our main goal is to view the Atiyah-Singer index theorem as a corollary of the isomorphism between geometric and analytic K-homology. The Atiyah-Singer index theorem via K-homology: An introduction
Oct. 10, 2017 1pm (MATH 220)	Grad Algebra/Logic Peter Mayr (CU Boulder) X Deciding whether the vertices of a given graph can be colored by 3 colors is well-known to be NP-complete. But how hard is it to distinguish between whether a graph can be colored by 3 colors or it cannot even be colored by 30 colors? This is an example of a Promise Constraint Satisfaction Problem (PCSP). There are only partial results on the complexity of PCSPs in general. We will present an algebraic approach for studying PCSP that uses a Galois connection between pairs of relational structures and their polymorphisms that is due to Brakensiek-Guruswami (2017) and Pippenger (2002). Promises and constraints 1
Oct. 10, 2017 2pm (MATH 350)	Lie Theory Mandi Schaeffer-Fry (Metro State) X Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the behavior of the ordinary irreducible characters of G under a specific Galois automorphism. Navarro-Tiep-Vallejo have conjectured a similar statement regarding groups whose Sylow 2-normalizers contain a single irreducible 2-Brauer character. Thanks to reduction theorems proved by myself and Navarro-Vallejo, respectively, a large part of the proofs of these conjectures is to understand the action of this Galois automorphism on characters of groups of Lie type. I will discuss the recent proof of these conjectures, including a description of the action of Galois automorphisms on characters of groups of Lie type. (Some portions of these results are joint work with J. Taylor.) On Galois Automorphisms Acting on Characters and Navarro's Sylow 2-Normalizer Conjectures
Oct. 10, 2017 2pm (MATH 220)	Peter Mayr (CU Boulder) Promises and constraints 2
Oct. 10, 2017 3pm (MATH 350)	Topology Leo Herr Topological Andre-Quillen homology

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