Department of

Mathematics


Seminar Calendar
for events the day of Monday, March 30, 2015.

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Monday, March 30, 2015

12:00 pm in 341 AH,Monday, March 30, 2015

Cylindrical Contact Homology: A Retrospective

Jo Nelson (IAS / Columbia University)

Abstract: Cylindrical contact homology is arguably one of the more notorious Floer theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations have tarnished its claim to being a well-defined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of intersection theory, non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to $SH^+$ under suitable assumptions, though does not require a filling of the contact manifold. By making use of family Floer theory we obtain a $S^1$-equivariant theory defined over $\mathbb{Z}$-coefficients, which when tensored with $\mathbb{Q}$ yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.

3:00 pm in 243 Altgeld Hall,Monday, March 30, 2015

Irregular singularities of linear ODE and the HOMFLY homology of their links

Vivek Shende (Berkeley)

Abstract: The asymptotic behavior of solutions to a linear meromorphic ODE can be encoded by legendrian links.  The Riemann-Hilbert correspondence in this context can be interpreted to show that the "wild" part of the wild character varieties are the open Bott-Samelson varieties corresponding to these links.  In particular, for wild character varieties on the projective line with a single irregular singularity, the cohomology of the corresponding moduli spaces encodes a certain part of the HOMFLY homology of the link.  This is related to the conjectures of Oblomkov, Rasmussen, and myself regarding Hilbert schemes and HOMFLY homology by a wild version of the P=W conjecture of de Cataldo, Hausel, and Migliorini, which asserts a comparison between the weight filtration on the cohomology of character varieties, and a perverse Leray filtration on the cohomology of the Hitchin system. 

5:00 pm in 241 Altgeld Hall ,Monday, March 30, 2015

Approximation properties for quantum groups, monoidal equivalence, and Drinfeld doubles. Part 1

Michael Brannan (UIUC Math)

Abstract: In this series of lectures I plan to go over De Commer, Freslon and Yamashita's proof of the Haagerup property and the completely bounded approximation property for the operator algebras associated to the universal compact quantum groups. Using the notion of monoidal equivalence for compact quantum groups, these problems can be studied in terms of Woronowicz' deformed SU(2) quantum group. We will then make some connections to the Drinfeld double construction and (time permitting) discuss some recent connections to Kazhdan's property (T) for locally compact quantum groups due to Y. Arano.