Activities This Week

It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space$\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space.

In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative functions with the target set $\mathcal L ( \mathcal Y)$ of $K$ replaced by $\mathcal L (\mathcal A, \mathcal L (\mathcal Y))$ where $\mathcal A$ is a $C^*$-algebra. In my talk, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.

See attached file

Zilber showed that a compact complex manifold $M$, equipped with the structure generated by the collection of all complex analytic subsets of each $M^n$, is well-behaved from a logical perspective, forming a Zariski geometry in the sense of Hrushovski and Zilber. This has led to fruitful model-theoretic developments, including a classification of definable groups, the isolation of the canonical base property, and a theory of generic automorphisms.

Motivated by this example, we will examine the possibility of emulating this theory in the setting of an almost complex manifold, a real manifold equipped with a smoothly-varying complex vector space structure on each tangent space. In particular, we will define the notion of a pseudoanalytic subset of an almost complex manifold. We develop some rudimentary almost complex analytic geometry, including an identity principle for almost complex maps, and an analysis of the singular part of a pseudoanalytic subset under some algebraic conditions. The lack of a true algebraic theory means that geometric methods, including pseudoholomorphic curves and almost complex connections, have to pick up the slack. These results hint at routes towards an almost complex analogue of Zilber’s theorem.

Joint work with Pierre Simon. I will present a criterion that ensures that Aut(M) has a 2-generated dense subgroup when M is a countable structure (which holds in many examples), and discuss related subjects.

A finite set has an interesting numerical invariant - its cardinality. There are two natural generalizations of “cardinality” to an (homotopy) invariant for (suitably finite) spaces. One, is the classical Euler characteristic. The other is the Baez-Dolan “homotopy cardianlity”. These two invariants, both natural from a certain perspective, seem to be very different from each other yet mysteriously connected. The question of the precise relation between them was popularized by John Baez as one of the “mysteries of counting”. Inspired by this, we show that (p-locally) there is a unique common generalization of these two invariants satisfying some desirable properties. The construction of this invariant relies on a certain l-adic continuity property of the sequence of Morava-Euler characteristics of a given space, which seems to be an interesting “trans-chromatic” phenomenon by itself.