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Events happening on March 9, 2023

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Thu, Mar. 9 2:30pm (MATH 3…	Functional Analysis Levi Lorenzo (CU Boulder) X K-theory, and its dual K-homology, are the generalized homology and cohomology theories on the category of C-algebras that undergird noncommutative algebraic topology. Further, the two functors admit a bilinear pairing to the integers, the Index Pairing. For a smooth manifold, M, the K-homology of C_0(M) has the additional property of being uniformly p-summable on the subalgebra C_0^\infty (M), for each p larger than the dimension of M. This is desirable because finitely summable (p < \infty) K-homology cycles allow one to compute the Index Pairing in a more straightforward formula, via cyclic cohomology. Given a C-algebra, we can ask whether it has a dense subalgebra on which all of its K-homology classes have p-summable representatives for some p < infinity. In general the answer is no, but there are few known counterexamples and not so many positive results either. In this talk, we will define K-homology, discuss what it means for K-homology to be finitely summable, argue why we should care, and discuss what we know about the answer to this question and directions of where we might find further answers. Finitely Summable K-homology
Thu, Mar. 9 3:35pm (Online)	Probability Ewain Gwynne (University of Chicago) View Online:https://cuboulder.zoom.us/j/96074949744 X A permuton is a probability measure on $0,1]2$ whose two coordinate marginals are Lebesgue measure. Permutons describe the large-scale geometry of random permutations. I will discuss a geometric construction of a certain class of random permutons using a pair of random space-filling curves called Schramm-Loewner evolution (SLE) and a random measure arising from Liouville quantum gravity (LQG). This class includes the limits of various types of random pattern-avoiding permutations as well as the conjectural limit of meandric permutations (permutations arising from a simple loop which crosses a line a specified number of times). I will then discuss some results about random permutations which can be proven using SLE and LQG, concerning, e.g., the length of the longest increasing subsequence and the fractal dimension of the support. I will not assume any prior knowledge about permutons, SLE, or LQG. Based on joint work with Jacopo Borga and Xin Sun. Permutons, meanders, and SLE-decorated Liouville quantum gravity
Thu, Mar. 9 4pm (MATH 220)	Init Conditions Henry Maher Universal Properties