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Events happening on November 17, 2015

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Nov. 17, 2015 10am (Math350)	Jonathan Lamar (Colorado) X Combinatorial Hopf algebras (CHAs) are part of a nascent field, and as such have no accepted definition; rather several attempts to classify a growing set of algebras that we all agree to be examples of CHAs.� In this talk, I will "define" a CHA and provide a few examples, including my favorite object in all of mathematics.� If I manage to get through that in less than 50 minutes, then I will talk about recent work (Pang et al, 2014ff) using bases of CHAs as the state spaces of random walks with the increment distribution induced by the Hopf-square. In particular, this approach admits analysis of the Gilbert--Shannon--Reeds model of riffle shuffling by Hopf-algebraic techniques. Combinatorial Hopf Algebras (and Markov chains)
Nov. 17, 2015 11am (MATH 350)	Number Theory Hao Chen (University of Washington) X We start by defining fully homomorphic encryption and the Ring Learning-With-Errors (RLWE) problem. Then we introduce an attack to RLWE based on the chi-square statistical test, and give examples of Galois RLWE instances vulnerable to our attack. We prove a search-to-decision reduction for Galois fields with unramified modulus. Also, we analyze the vulnerability of cyclotomic fields to our attack, and show that they are safe in general, except when the modulus is totally ramified. We will use algebraic number theory and Fourier analysis on finite fields. In particular, no cryptographic background is necessary for the talk. Attacks on Search-RLWE
Nov. 17, 2015 1pm (MATH 220)	Grad Algebra/Logic Jakub Bulin (CU Boulder) The Bounded Width Theorem, Part 2
Nov. 17, 2015 2pm (MATH 220)	Keith Kearnes (CU) Tame Congruence Theory
Nov. 17, 2015 3pm (Math 350)	Jordan Watts (CU Boulder) X This week, we look at a different problem on orbit spaces of proper Lie group actions, as well as symplectic quotients. We first recall two facts: 1) If a proper action is free, the de Rham complex on the orbit space matches the de Rham complex of basic forms upstairs on the original manifold. 2) It is a result of Koszul and Palais that even in the non-free case, the singular cohomology of the orbit space matches the cohomology of the basic forms upstairs. Question: is there a de Rham complex of differential forms on the orbit space in the non-free case that yields a cohomology matching the singular cohomology? (The orbit space is not a manifold, typically.) The answer is yes, but we will not use the language of Sikorski differential spaces to answer this question (for which the answer seems to be false). Instead, we will introduce a "dual" smooth structure called diffeology, give an idea of the proof, and then consider a similar question for the symplectic quotient. The de Rham Complex on Orbit Spaces and Symplectic Quotients

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