arXiv subjects

Teo Banica

Publications and source records attributed to Teo Banica.

At least 19 recordsLinked to original sources

arXiv2026

Normal random variables

This is an advanced introduction to the various types of normal random variables, with most of the needed preliminaries included. We first discuss the probability basics, standard central limits, and the theory of the usual, real normal variables, with examples, illustrations and numerous formulae. Then we go on a similar discussion regarding the complex normal variables, and the Rayleigh variables too. We then move to arbitrary dimensions, with a discussion regarding the Gaussian vectors, and related probability laws, featuring some functional analysis, and geometry and physics too. Finally, we provide an introduction to the quantum versions of the normal variables, and notably to those coming from free probability and random matrices.

math.PR↗

arXiv2026

Continuous random variables

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.

math.PR↗

arXiv2025

Invitation to finite groups

This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic, dihedral and symmetric groups, and the structure result for finite abelian groups. Then we study the complex reflection groups, with general theory and examples, classification results, and with a look into braid groups too. We then go into the study of representation theory, and of more advanced aspects, such as Tannakian duality, Brauer theorems and Clebsch-Gordan rules. Finally, we discuss, using representation theory methods, a number of advanced analytic aspects, for the most in relation with questions coming from probability.

math.RT↗

arXiv2025

Basic quantum algebra

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields $F$. These algebras $A$ satisfy a commutativity type condition, and the general idea is that of lifting this condition, and calling quantum spaces the underlying space-like objects $X$. One problem comes from the fact that different fields $F$ lead, via different algebras $A$, to different classes of quantum spaces $X$. Our aim here is to identify and put at the center of the presentation those quantum spaces $X$ which do not depend on the choice of $F$.

math.QA↗

arXiv2025

Advanced linear algebra

This is an introduction to advanced linear algebra, with emphasis on geometric aspects, and with some applications included too. We first review basic linear algebra, notably with the spectral theorem in its general form, and with the theory of the resultant and discriminant. Then we discuss the Jordan form and its basic applications to physics, and other advanced decomposition results for the matrices. We then go into positivity topics, involving matrices and bilinear forms, and with a look into curved space-time, and discrete Laplacians. Finally, we discuss the various groups of matrices, with a look at reflection groups, Lie groups, spin matrices and random matrices.

math-ph↗

arXiv2025

Measure and integration

This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability, that we develop in detail, as a preliminary to general measure theory. Then we discuss measure theory, integration and function spaces, all developed in a standard way, and with emphasis on the explicit computation of various integrals. Finally, we come back to probability, discrete and continuous, with a more advanced discussion, of quantum flavor.

math-ph↗

arXiv2024

Graphs and their symmetries

This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using $d$, and various linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups $G(X)\subset G^+(X)$, which must leave invariant the eigenspaces of $d$, with the quantum symmetry group $G^+(X)$ being in general bigger than the classical symmetry group $G(X)$.

math.QA↗

arXiv2024

Calculus and applications

This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and of the integral $\int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:\mathbb C\to\mathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:\mathbb R^N\to\mathbb R^M$ or $f:\mathbb R^N\to\mathbb C^M$ or $f:\mathbb C^N\to\mathbb C^M$, with general theory, integration results, maximization questions, and basic applications to physics.

math.HO↗

arXiv2023

Easy quantum groups

A closed subgroup $G\subset_uU_N^+$ is called easy when its associated Tannakian category $C_{kl}=Hom(u^{\otimes k},u^{\otimes l})$ appears from a category of partitions, $C=span(D)$ with $D=(D_{kl})\subset P$, via the standard implementation of partitions as linear maps. The examples abound, and the main known subgroups $G\subset U_N^+$ are either easy, or not far from being easy. We discuss here the basic theory, examples and known classification results for the easy quantum groups $G\subset U_N^+$, as well as various generalizations of the formalism, known as super-easiness theories, and the unification problem for them.

math.QA↗

arXiv2022

Introduction to quantum groups

This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group $U_N$ has a free analogue $U_N^+$, whose standard coordinates $u_{ij}\in C(U_N^+)$ are allowed to be free, and the closed subgroups $G\subset U_N^+$ can be thought of as being the compact quantum Lie groups. There are many interesting examples of such quantum groups, for the most designed in order to help with questions in quantum mechanics and statistical mechanics, and some general theory available as well, including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten integration. We discuss here the basic aspects of all this.

math.OA↗

arXiv2022

Methods of free probability

This is a joint introduction to classical and free probability, which are twin sisters. We first review the foundations of classical probability, notably with the main limiting theorems (CLT, CCLT, PLT, CPLT), and with a look into examples coming from Lie groups and random matrices. Then we present the foundations and main results of free probability, notably with free limiting theorems, and with a look into examples coming from quantum groups and random matrices. We discuss then a number of more advanced aspects, in relation with free geometry and with subfactor theory.

math.PR↗

arXiv2022

Principles of operator algebras

This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mainly interested in the von Neumann algebras, which are stable under taking adjoints, $T\to T^*$, and are weakly closed. When the algebra has a trace $tr:A\to\mathbb C$, we can think of it as being of the form $A=L^\infty(X)$, with $X$ being a quantum measured space. Of particular interest is the free case, where the center of the algebra reduces to the scalars, $Z(A)=\mathbb C$. Following von Neumann, Connes, Jones, Voiculescu and others, we discuss the basic properties of such algebras $A$, and how to do algebra, geometry, analysis and probability on the underlying quantum spaces $X$.

math.OA↗

arXiv2022

Linear algebra and group theory

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices $G\subset U_N$, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where $G\subset U_N$ is compact, we explain how the Weingarten integration formula works, and we present some basic $N\to\infty$ applications.

math.CO↗

arXiv2021

Quantum partial automorphisms of finite graphs

The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $σ:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$, which contains the automorphism group $G(X)$. We discuss here the quantum analogue of this construction, with a definition and basic theory for the quantum semigroup of quantum partial automorphisms $\widetilde{G}^+(X)$, which contains both $G(X)$, and the quantum automorphism group $G^+(X)$. We comment as well on the case $N=\infty$, which is of particular interest, due to the fact that $\widetilde{G}^+(X)$ is well-defined, while its subgroup $G^+(X)$, not necessarily, at least with the currently known methods.

math.OA↗

arXiv2021

The Frucht property in the quantum group setting

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

math.OA↗

arXiv2021

Duals of finite quantum permutation groups

We work out axioms for the duals $G\subset U_N^+$ of the finite quantum permutation groups, $F\subset S_N^+$ with $|F|<\infty$, and we discuss how the basic theory of such quantum permutation groups partly simplifies in the dual setting. We discuss as well some potential extensions to the infinite case, in connection with the well-known question of axiomatizing the discrete quantum group actions on the infinite graphs.

math.QA↗

arXiv2020

Affine noncommutative geometry

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$ do have free analogues $S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$. There are many examples of real algebraic submanifolds $X\subset S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $\int:C(X)\to\mathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.

math.QA↗

arXiv2020

Quantum permutation groups

The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We review the known facts regarding $S_N^+$, and notably its easiness property, Weingarten calculus, and the isomorphism $S_4^+=SO_3^{-1}$ and its consequences. We discuss then the structure of the closed subgroups $G\subset S_N^+$, and notably of the quantum symmetry groups of finite graphs $G^+(X)\subset S_N^+$, with particular attention to the quantum reflection groups $H_N^{s+}$. We also discuss, more generally, the quantum symmetry groups $S_Z^+$ of the finite quantum spaces $Z$, and their closed subgroups $G\subset S_Z^+$, with particular attention to the quantum graph case, and to quantum reflection groups.

math.QA↗