Jordan Watts (University of Illinois Urbana--Champaign)
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We will begin by looking at what an orbifold is in the classical way. Then, fixing an orbifold, we forget all of the information contained in the orbifold structure except for the topology, stratification, and some integer labels on certain strata. From here, we reconstruct the entire orbifold structure.
This is important, because the "minimal data" contained in the topology, stratification, and labels can all be obtained from the ring of smooth functions on an orbifold. More categorically, there is an essentially injective functor from the "category" of orbifolds (which has different definitions depending on your perspective, not all equivalent) to the category of so-called differential spaces: sets equipped with sheaves of "smooth" functions with "smooth" maps between them. This a priori is unexpected, since in general the ring of smooth functions on a space built out of quotients, such as an orbifold, forgets the local quotient structure.
In the mid 1970s, Bernstein-Zelevinsky and Gelfand-Kazhdan proved that for any irreducible admissible representation \pi of GL(n,F), F a p-adic field, the dual of \pi is isomorphic to \pi twisted by the inverse-transpose map. The proof depends heavily on sheaf theory applied to distributions of totally disconnected locally compact groups. Since then, similar statements have been proved for other p-adic groups, all depending on the same algebro-geometric arguments. In 2006, Tupan gave an elementary proof of the statement for GL(n,F), depending only on linear algebra and topological properties. In joint work with Alan Roche (U. Oklahoma), we are able to extend Tupan's argument to the orthogonal and symplectic p-adic groups, and their groups of similitudes, using only linear algebra statements, p-adic topology, and some lie theory.
Dual representations of classical p-adic groups Sponsored by the Meyer Fund
Mar. 17, 2015 3pm (Math 220)
Functional Analysis
Palle Jorgensen (University of Iowa)
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The talk will have two sides, one tilted towards applications, and the other representation theory. The applications are from symbolic dynamics, fractals, and from computational harmonic analysis, e.g., discrete wavelet algorithms; both standard wavelets and wavelets on fractals. The representations appearing in the talk refer to classes of certain non-abelian algebras (one in particular, the Cuntz-algebra.) We will show that a number of questions which are natural from the point of representation theory (equivalence classes of irreducibles, restriction, induction, decompositions etc) are key to solving questions in the applied areas mentioned.
Atomic and non-atomic representations: algebras in analysis and in symbolic dynamics Sponsored by the Meyer Fund
