Leonard Huang (CU Boulder) Fields of Generalized Fixed-Point Algebras for Proper Groupoid Actions on $C}^{*$-Algebras
Tue, Oct. 17 1pm (MATH 220)
Grad Algebra
Peter Mayr (CU Boulder) Promises and constraints 3
Tue, Oct. 17 2pm (MATH 350)
Lie Theory
Henry Kvinge (CSU) The Kirillov-Reshetikhin crystal $B}^{1,1$ and cyclotomic quiver Hecke algebras
Tue, Oct. 17 3pm (MATH 350)
Topology
Robin Deeley (CU Boulder) Geometric K-homology with Z/k-coefficients
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Khovanov-Lauda-Rouqiuer (KLR) algebras (also known as quiver Hecke algebras) were invented to categorify the negative half of quantum Kac-Moody algebras. It was later shown by Lauda-Vazirani that the simple modules of the cyclotomic KLR algebra, $R}^{\Lambda$, carry the structure of the highest weight crystal $B(\Lambda )$. It follows from this that combinatorial properties of $B(\Lambda )$ should be the shadow of module-theoretic properties of simple $R}^{\Lambda$-modules.
In classical affine type, highest weight crystals (which are infinite) have the remarkable property that they can be constructed from the tensor product of the much more tractable perfect crystals (which are finite). In this talk I will describe the algebraic analogue of this phenomenon in terms of simple $R}^{\Lambda$-modules in the case where the perfect crystal is the Kirillov-Reshetikhin crystal $B}^{1,1$ and $\Lambda$ is the fundamental weight $\Lambda}_{i$.
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In the 1980s, Baum and Douglas defined a geometric model for K-homology using spin^c manifolds. Through the isomorphism from geometric K-homology to analytic K-homology one can obtain the Atiyah-Singer index theorem. In the first half of the talk I will introduce both realizations of K-homology and this isomorphism.
In the second half, an extension of this theory will be discussed. The starting point is Sullivan's notion of a Z/k manifold. Freed and Melrose proved an index theorem for operators acting on such spaces. We will consider a geometric model for K-homology with Z/k-coefficients where we replace the spin^c manifold theory used in the Baum-Douglas model with the corresponding theory for Z/k manifolds. The Freed-Melrose index theorem then plays the role that the Atiyah-Singer index theorem plays in the Baum-Douglas model.