Related papers: Free function theory through matrix invariants
This is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. Functional analysis takes us beyond finite matrices, allowing us to work with infinite sets of…
In this paper, we study free pluriharmonic functions on noncommutative balls, and their boundary behavior. The main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which…
We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral…
We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
Every rotationless outer automorphism of a finite rank free group is represented by a particularly useful relative train track map called a CT. The main result of this paper is that the constructions of CTs can be made algorithmic. A key…
This is a continuation of our previous work in bi-free harmonic analysis for commuting left and right variables. Here we analyze the bi-free partial S-transform and use the results to study limit theorems and infinite divisibility relative…
The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only…
We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free…
This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R<x> in free (freely…
Identifying the trade-offs between model-based and model-free methods is a central question in reinforcement learning. Value-based methods offer substantial computational advantages and are sometimes just as statistically efficient as…
Aperture synthesis observations with full polarisation have long been used to study the magnetic fields of synchrotron emitting sources. Recently proposed closure invariants give us a powerful method for extracting information from measured…
Using evaluation at appropriately chosen points, we propose a Gr\"obner basis free approach for calculating the secondary invariants of a finite permutation group. This approach allows for exploiting the symmetries to confine the…
Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete…
We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. We also define similar variants of this map, that regards alternative models for the modified Macdonald…
We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are…
In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like…
We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…
This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and…
A theory with a mass scale and yet Weyl invariant is presented. The theory is not invariant under all diffeomorphisms but only under transverse ones. This is the reason why Weyl invariance does not imply scale invariance in a free falling…