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Events happening on April 24, 2018

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Apr. 24, 2018 11am (MATH 350)	Number Theory Comp exam scheduled No seminar today
Apr. 24, 2018 12:10pm (MATH …	Kempner Robert S. Maier (University of Arizona) X The most familiar special functions are hypergeometric functions, which appear in many areas of mathematics. They include the Legendre functions, which arise in harmonic and numerical analysis, and elliptic integrals. All these functions have one or more free parameters, and the parameter values for which they can be expressed in closed form (e.g., because they become algebraic, with finite monodromy) have in many cases been characterized: for the original Gauss function, by Schwarz and Klein; for generalizations, by Beukers and Heckman in the 1980s. But there is a shortage of algorithms for constructing explicit closed-form expressions, and even of transformation formulas that relate such functions to ones with different parameter values. In this talk, I sketch recent results: (i) algebraic-geometric uniformizations of certain generalized hypergeometric functions, (ii) transformation formulas for elliptic integrals, extending ones found by Ramanujan, and (iii) Lie-theoretic recurrences satisfied by the octahedral Legendre functions. Explicit formulas for hypergeometric and Legendre functions
Apr. 24, 2018 1pm (MATH 220)	Trevor Jack (CU Boulder) X We introduce a method of investigating semigroup properties using graph theory. Given a transformation semigroup generated by transformations $a1,...,ak$ on n letters, we produce a graph with $n^{d}$ vertices and directed edges corresponding to the generators acting on the vertices component-wise. The number of components, d, depends upon which semigroup property we are investigating. We will sketch several results obtained thus far, including: (a) whether a semigroup is nilpotent, (b) whether a semigroup is a band, and (c) whether a semigroup satisfies a given equation $φ(x1,...,xm) = ψ(x1,...,xm)$ . Investigating semigroup properties using graphs
Apr. 24, 2018 2pm (MATH 350)	Lie Theory Henry Kvinge (CSU) X For a given tower of algebras the associated Heisenberg category captures the combinatorial structure of induction/restriction functors between these algebras via a graphical calculus of string diagrams. It can also be shown that the center of this category (the closed diagrams) controls the centers of all algebras in the tower simultaneously. In this talk we will discuss examples of this phenomenon, focusing on the case where the tower of algebras is either group algebras of symmetric groups or the Sergeev superalgebras. We will show how known (inhomogeneous) bases of the symmetric functions appear naturally from the diagrammatics in each case, and explore properties of these symmetric functions diagrammatically. Finally, we will discuss a surprising connection to Markov processes on branching graphs for these towers of algebras. Symmetric functions, towers of algebras, and centers of Heisenberg categories
Apr. 24, 2018 3pm (MATH 350)	Topology Paul Lessard (CU Boulder) X Week 1 of an Algebraic Topology Course from Another World In this talk we will describe the homotopy category of the category of spaces as the localization of \Cat at the smallest class of morphisms which is weakly saturated and can be said to know that: - cones are trivial - Quillen's theorem A holds. From this description of the category of spaces we will unify all the ways in which categories of presheaves, e.g.: simplicial sets, cubical sets, globular sets, etc. model spaces by way of the universal property of the Yoneda embedding. Quillen's Theorem A and Grothendieck's Asphericite

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