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Events for November 16, 2017 from General and Seminar calendars

Algebra and Combinatorics Seminar

Time: 2:45PM - 3:45PM

Location: BLOC 628

Speaker: X. Shawn Cui, Stanford University

Title: On Two Invariants of Three Manifolds from Hopf Algebras

Abstract: We prove a conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury (TVBW) invariant and the Witten-Reshetikhin-Turaev (WRT) invariant, respectively. By a classical result relating TVBW and WRT, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radford invariant for the Drinfeld double of the Hopf algebra. However, whether the relation holds for non-semisimple Hopf algebras has remained open, partly because the introduction of framings in this case makes the Kuperberg invariant significantly more complicated to handle. We give an affirmative answer to this question. An important ingredient in the proof involves using a special Heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the Heegaard diagram.

Colloquium - Fritz Gesztesy

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Fritz Gesztesy, Baylor University, Mathematics Department

Title: Eigenvalue counting for Krein-von Neumann extensions of elliptic operators

Abstract: We start by providing a historical introduction into the subject of Weyl-asymptotics for Laplacians on bounded domains in n-dimensional Euclidean space, and a brief introduction into the basic principles of self-adjoint extensions.

Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets Omega in R^n. (No assumptions on the boundary of Omega are made; the coefficients are supposed to satisfy certain regularity conditions.)

Our technique relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n. We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension.

This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.