פעילויות השבוע

The theory of non-commutative (nc) rational functions which are regular at 0 is well known and studied, in terms of their minimal realizations: any such function admits a unique minimal realization centred at 0 and the domain of the function coincides with the invertibility set of the (resolvent of the) realization. In addition, a nc power series around the origin will be the power series expansion of a nc rational function if and only if a given Hankel matrix built from the coefficients of the given power series has a finite rank (Fliess-Kronecker).

In this talk, we present generalizations of these ideas to the case where the centre is non-scalar. In particular, we prove the existence and uniqueness of a minimal Fornasini-Marchesini realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity, and show that using this realization, one can evaluate the function on all of its domain (of matrices of all sizes).

Unlike the case of a scalar centre, the coefficients of the realization can not be chosen arbitrarily. We present necessary and sufficient conditions (called the linearized lost-abbey conditions) on the coefficients of a minimal realization centred at a matrix point, such that there exists a nc rational function which admits the realization.

This is a joint work with Victor Vinnikov.

The commutative DG rings in the title are more commonly known as ”nonpositive strongly commutative unital differential graded cochain K-algebras“, where K is a commutative base ring. In the literature the standard assumption is that K is a field of characteristic zero - but one of our themes in this talk is that this assumption is superfluous (K = Z works just as well).

There are two kinds of derived categories ralated to commutative DG rings. First, given a DG ring A, we can consider D(A), the derived category of DG A-modules, which is a K-linear triangulated category. This story is well understood by now, and I will only mention it briefly.

In this talk we shall consider another kind of derived category. Let DGRng denote the category whose objects are the commutative DG rings (the base K is implicit), and whose morphisms are the DG ring homomorphisms. The derived category of commutative DG rings is the category D(DGRng) gotten by inverting all the quasi-isomorphisms in DGRng. (In homotopy theory the convention is to call it the ”homotopy category“, but this is an unfortunate historical accident.)

I will define semi-free DG rings, and prove their existence and lifting properties. Then I will introduce the quasi-homotopy relation on DGRng, giving rise to the quotient category K(DGRng), the ”genuine“ homotopy category. One of the main results is that the canonical functor from K(DGRng) to D(DGRng) is a faithful right Ore localization.

I will conclude with a theorem on the existence of the left derived tensor product inside D(DGRng), and with the pseudofunctor from D(DGRng) to the TrCat, sending a DG ring A to the triangulated category D(A).

Next semester I will talk about the geometrization of these ideas: ”The Derived Category of Sheaves of Commutative DG Rings“.

The main result in this talk is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset S of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $\mu_1$ has the property that the random walk with initial distribution $\mu_1$ evolved by the action of S equidistributes very fast. The second measure $\mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.

The hydrogen atom system is a fundamental example of a quantum mechanical system. Symmetry plays the main role in our current understanding of the system. In this talk I will describe a new type of algebraic symmetry for the system. I will show that the collection of all regular solutions of the Schrödinger equation is an algebraic family of representations of different algebras. Such a family is known as an algebraic family of Harish-Chandra modules. The algebraic family has a canonical filtration from which the physically relevant solutions and the spectrum of the Schrödinger operator can be recovered.

If time permits I will relate the spectral theory of the Schrödinger operator to the algebraic family. No prior knowledge about quantum mechanics or representation theory will be assumed.