Kempner Colloquium Abstracts

Spring 2009

The JLO version of the Connes-Chern character is a non-commutative analogue of the classical Chern character. On the one hand it provides a natural transformation from K-theory to cyclic (co)homology. On the other hand it comes naturally with a time parameter which generalizes the heat trace of an elliptic operator. Therefore there is a strong connection to local index theory and short and large time limits of the JLO Connes-Chern character provide interesting information.

In a current ongoing project with Henri Moscovici and Markus Pflaum we are investigating these issues in a relative context. More precisely we study the Connes-Chern character for manifolds with boundary using the so called b-calculus of Melrose. Our main results identify the short and large time limits of the Connes-Chern character in this context.

In this talk, we show that the representation variety of the fundamental group of a 2n-punctured S² with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link), and the symplectic Floer homology of links (knots). I will explain the basic construction of these invariants and relation to topological quantum field theory.

Almost every problem or conjecture in pure mathematics has a potentially applied version which can be decided by a finite if often formidably huge computation. Applicable problems at any historical period (e.g., in the year 2009) often require a smaller computation to achieve a realistic fully useful technological check. In other cases we may use a version of the Monte Carlo method invented by Ulam to establish the probability of truth or falsehood of the conjecture and let the technology community decide if the probabilty is sufficient for certain or even all current applications. The relevant parameters are the size of the largest integer used in technological application and the density of the grid of rational points that affect the technology outcome at a given historical time. Let us call the size of the integer above h and the density of the grid 1/h. The second parameter is more difficult to describe and should measure maximum available computing power on a global scale. Let us assume that we may describe it by another huge integer k. As technology progresses both h and k will clearly grow. The resulting methodology could be called evolving technological pure and applied mathematics or either etpam(h,k) or etpam(date) for short. our talk will include examples from number theory: for example, Fermat's last theorem prior to Wiles' proof, the Riemann Hypothesis with its status today, Diophantine equations and pseudo primes. From our specialty, algebraic geometry, we shall include problems including the resolution of singularities in characteristic p and problems about Zariski surfaces. We shall also include applications to partial differential equations especially as related to the work of Alfred Rosenblatt and his school in Peru.

The topological entropy of a dynamical system was introduced in the sixties as a quantity that is invariant under continuous changes of coordinates. The topological entropy of a map measures in some sense the complexity of the corresponding dynamical system, by counting how many "very different" orbits of arbitrarily large length we can find. The question is whether entropy can be computed (in theory, and if possible in practice) up to an error which can be made arbitrarily small. For piecewise monotone interval mappings, probably the simplest interesting dynamical systems, there is an effective computation that depends only on being able to order finitely many forward images of the critical points.

In this talk, I will define the topological entropy and the kneading data of a map, then illustrate their behavior on a subspace of quartic polynomials that are compositions of two logistic maps. In the parameter space P^Q of such maps, the entropy level-sets turn out to be connected. This result is made possible by the correspondence between P^Q and a simpler model subfamily of maps P^T , and is based on the fact that the kneading-data of a map determines its entropy.

I will also be discussing a working algorithm which gives a good entropy estimate in P^Q. A complex search for kneading-data in a model space of sawtooth maps turns out to be computationally fast and reliable, delivering good entropy estimates.

In the past few years, automated reasoning tools have proven to be very effective in resolving open problems in the theory of quasigroups and loops. In this talk I will survey some of these results. No background in quasigroup or loop theory is needed; what is necessary will be developed in the course of the talk.

In this talk, we give an introduction and brief history to the study of interated function systems (IFSs) and their associated equilibrium measures. In particular, we will concentrate on Bernoulli IFSs on the real line. These consist of two affine functions with the same scaling factor λ ∈ (0,1).

One may ask whether the measure μ_λ is spectral, i.e. whether the Hilbert space L²(μ_λ) has an orthonormal basis of exponential functions. We will give a description of what is and is not known about this question.

We examine the nonlinear differential equations satisfied by certain classical modular forms on the upper half plane. The standard example is the triple of Eisenstein series E₂, E₄, E₆, associated to the full modular group PSL(2,Z). They satisfy a nonlinear first-order differential system, and by elimination each satisfies a nonlinear third-order equation. It was recently realized that Ramanujan's alternative elliptic integrals can be viewed as classical modular forms associated to genus-zero congruence subgroups Γ₀(N), N=2,3,4. To each subgroup, a triple of modular forms is naturally associated, leading to similar nonlinear third-order differential equations ("generalized Chazy equations"). The derivation of these equations yields, as an interesting byproduct, many identities relating weighted divisor functions.

We describe a general framework in which we use KMS states for circle actions on a C^*-algebra to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense.