Kempner Colloquium Abstracts

Fall 2007

For a field K, the Leavitt K-algebra L_K(1,n) was defined and investigated in the early 1960's, and has enjoyed an intense revival in interest over the past few years. When K = C (the field of complex numbers), the Leavitt K-algebra is intimately connected with the C^*-algebra O_n, the so-called Cuntz algebra.

Let R denote L_K(1,n). For any ring S, let M_t(S) denote the t × t matrix ring over S. We will first show (easily) that that if t ≡ 1 (mod (n-1)) then R ≅ M_t(R) as K-algebras. We will then show (less easily) that if t divides some power of n, then also R ≅ M_t(R). With these two observations as motivation, we will then show how to prove the following (recently completed joint work with P.N. Ánh and E. Pardo).

Theorem: R ≅ M_t(R) if and only if g.c.d.(t,n-1) = 1.

The proof of this result involves some straightforward (but apparently nonstandard) number theory, which we will describe in detail.

As a consequence of this theorem, we get information not only about Leavitt K-algebras, but we answer a longstanding question about isomorphisms between matrix rings over Cuntz C^*-algebras as well.

We discuss the large positive solutions of the nonlinear elliptic equation

-∇² u = r e^u in D,

u=0 on the boundary of D,

Note that this equation occurs in many places, including combustion theory and catalysis theory. In particular, we sketch how some results known for D a ball generalize to arbitrary domains. For example, we show that the problem has infinitely many bifurcation points.

The talk will start with examples of cycles in the homology of the space of long knots. The whole homology of this space is described by the Hochschild homology of the Poisson algebras operad. The proof and main techniques of the above result will be sketched. (joint work with Pascal Lambrechts and Ismar Volic)

In this talk, we study a complex manifold X with a strongly pseudoconvex boundary M. If u is a defining function for M, then -log u is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form

s = i ∂ ∂(-log u)

is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M.

We explain that the Poisson structure obtained by inverting s extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, we show that the Poisson structure near M is completely determined up to isomorphism by the contact structure on M.

In addition, using Poisson geometry we are able to prove that when -log u is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Englis for the Berezin-Toeplitz deformation quantization of X are smooth up to the boundary.

This is joint work with L. Boutet de Monvel, E. Leichtnam, and A. Weinstein.

Using an explicit cyclic cocycle on the Weyl algebra, we give an explicit construction of cyclic cocycles on formal deformations of proper etale groupoids associated to symplectic orbifolds. Associated to such cocycles is an abstract, higher index theorem. If time permits, we will discuss some possible applications of this index theorem.

As is well-known, every principal bundle is determined in a unique way by the gluing cocycle g_αβ. Hence, one should expect that all the invariants of this bundle can be expressed in terms of this cocycle. In my talk I will propose a simple algorithm that allows one to write down such formulas for arbitrary characteristic classes of the bundle. The algorithm is based on the notion of twisting cochain - an object well-known to topologists but rarely appearing in Geometry.

Starting from what Hopf algebras are, I will describe what Hopf symmetries are in terms of their actions and coactions on algebras. I will define what Hochschild cohomology is and describe its relation to differential forms. Then I will explain how one can incorporate the aforementioned Hopf-symmetry into the Hochschild (co)homology of an algebra which admits a Hopf action. I will also briefly mention how such symmetries affect the cyclic (co)homology of algebras.