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Events happening on March 15, 2016

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Mar. 15, 2016 1pm (MATH 220)	Dmitriy Zhuk (Moscow State Unuversity) X The standard way to parameterize interesting subclasses of the constraint satisfaction problem (CSP) is via finite relational structures. The main problem is to classify those subclasses that are tractable (solvable in polynomial time) and those that are NP-complete. It was conjectured that if a core of a relational structure has a weak near-unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. This conjecture was proved by Andrei Bulatov for 3-element domain, by Petar Markovic for 4-element domain. We present the proof of this conjecture for 5-element domain. Precisely, we prove the remaining part of this conjecture showing that a CSP instance can be solved in polynomial time if the relational structure has a weak near-unanimity polymorphism. The proof of CSP Dichotomy Conjecture for 5-element domain Sponsored by the Meyer Fund
Mar. 15, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X Abstract: As mentioned in the first lecture, the quantum Heisenberg manifold can be realized as a crossed product by a Hilbert $C$ -bimodule which is the special case of Cuntz-Pimsner algebras. In this lecture (and the next lecture), we introduce Cuntz-Pimsnef algebras and crossed products by Hilbert $C$ -modules, and discuss how we realize the quantum Heisenberg manifold as those $C$ algebras using the classical Picard groups. Cuntz-Pimsner algebras and crossed products by $C$ -bimodules. Sponsored by the Meyer Fund
Mar. 15, 2016 2pm (MATH 220)	Dmitriy Zhuk (Moscow State University) X The main problem in clone theory is to get a description of all clones. In 1941 Emil Post obtained an amazing description of all clones on 2 elements, but in 1959 it was proved that there exists a continuum of clones on k elements if k > 2. The first part of my talk is devoted to the efforts to describe uncountable lattices of clones, in particular, I show how uncountable lattice of all clones of self-dual operations on 3 elements can be described. However, obtaining such description for other maximal clones seems to be too cumbersome and complicated. This motivated me to focus on the following questions: how to check properties of a clone and how to find finite sublattices. Despite the fact that clone theory has been studied for many years, we still don't know how to check simple properties of clones. For example we don't know how to check whether a clone defined by a relation is finitely generated, or how to decide how many clones contain a given operation. In the second part of the talk I try to collect such open problems and present some partial results. Open problems in clone theory Sponsored by the Meyer Fund
Mar. 15, 2016 3pm (Math 350)	Christopher Seaton (Rhodes College) X Let $G$ be a compact Lie group and $V$ a unitary representation of $G$ . Treating $V$ as a real symplectic manifold and choosing the homogeneous quadratic moment map, we can form the symplectic quotient, the quotient $M0 = Z/G$ where $Z$ is the zero fiber of the moment map. The space $M0$ has a variety of structures, including that of a symplectic stratified space and a semialgabraic set, and its algebra of smooth functions has a graded subalgebra $R[M0]$ of regular functions defined as a quotient of the real polynomial invariants of the $G$ -action on $V$ . In this talk, we will discuss the use of the polynomial invariants of the $G$ -representation to construct symplectomorphisms and distinguish between symplectic quotients. In particular, we will study the problem of which symplectic quotients are symplectomorphic or diffeomorphic to symplectic orbifolds. We demonstrate that in some cases, $R[M0]$ has many of the nice properties of a ring of orbifold invariants and describe explicit computations of invariants of $R[M0]$ . Along the way, we will see some recent results in computational invariant theory that are relevant in other contexts. (Joint work with Joshua Cape, Carla Farsi, Hans-Christian Herbig, Daniel Herden, Ethan Lawler, and Gerald Schwarz) Constructing symplectomorphisms between singular linear symplectic quotients Sponsored by the Meyer Fund
Mar. 15, 2016 4pm (Math 220)	Shawn Burkett (CU Boulder) X Supercharacter theory was originally developed to describe the representation theory of algebra groups, a family of groups in which unimodular groups belong. Essentially, supercharacter theories are certain types of central subalgebras which approximate the usual character theory of a finite group. In this talk, we will introduce supercharacter theories from multiple persepectives and discuss a few ways to use the structure of a given finite group to construct nontrivial supercharacter theories. Time permitting, I will discuss some recent work of myself and Jonathan Lamar (CU Boulder) on induced actions on the set of supercharacter theories of a finite group. Random talk on a supercharacter theory

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