This is an introduction to graph theory, from a geometric viewpoint. A finite graph X is describe... more This is an introduction to graph theory, from a geometric viewpoint. A finite graph X is described by its adjacency matrix d ∈ M N (0, 1), which can be thought of as a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using d and linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups G(X) ⊂ G + (X), which must leave invariant the eigenspaces of d. Finally, we discuss similar questions for the quantum graphs, with these being again described by certain matrices d ∈ M N (C), but in a more twisted way.
This is an introduction to calculus, and its applications to basic questions from physics. We fir... more This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions f : R → R, with the notion of continuity, and the construction of the derivative f ′ (x) and of the integral b a f (x)dx. Then we investigate the case of the complex functions f : C → C, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, f : R N → R M or f : R N → C M or f : C N → C M , with general theory, integration results, maximization questions, and basic applications to physics.
A closed subgroup G ⊂ u U + N is called easy when its associated Tannakian category C kl = Hom(u ... more A closed subgroup G ⊂ u U + N is called easy when its associated Tannakian category C kl = Hom(u ⊗k , u ⊗l) appears from a category of partitions, C = span(D) with D = (D kl) ⊂ P , via the standard implementation of partitions as linear maps. The examples abound, and the main known subgroups G ⊂ 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 ⊂ U + N , as well as various generalizations of the formalism, known as super-easiness theories, and the unification problem for them.
The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We rev... more 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.
This is a joint introduction to classical and free probability, which are twin sisters. We first ... more 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.
This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can ... more 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 $X$.
This is an introduction to linear algebra and group theory. We first review the linear algebra ba... more 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.
We work out axioms for the duals G ⊂ U + N of the finite quantum permutation groups, F ⊂ S + N wi... more We work out axioms for the duals G ⊂ U + N of the finite quantum permutation groups, F ⊂ S + N with |F | < ∞, 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. Contents 4. Integration results, orbits and orbitals 12 5. Actions on finite graphs, in the dual setting 15 6. Infinite extensions and open problems 18 References 19 2010 Mathematics Subject Classification. 46L65.
A classical theorem of Frucht states that any finite group appears as the automorphism group of a... more 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.
The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ w... more The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ 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.
This is an introduction to classical and free probability, from a quantum algebra and random matr... more This is an introduction to classical and free probability, from a quantum algebra and random matrix perspective. We discuss the foundations, insisting on limiting theorems, and then we discuss a number of more specialized aspects. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to subfactors, from a quantum group perspective, focusing on the case of ... more This is an introduction to subfactors, from a quantum group perspective, focusing on the case of integer index. We discuss the foundational aspects of the theory, and then we discuss a number of more advanced topics. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to operator algebras, from a quantum algebra and quantum physics perspect... more This is an introduction to operator algebras, from a quantum algebra and quantum physics perspective. We discuss the foundational aspects of the theory, $C^*$-algebras and von Neumann algebras, and then more specialized topics. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to noncommutative geometry, from an operator algebra and quantum group vi... more This is an introduction to noncommutative geometry, from an operator algebra and quantum group viewpoint. We discuss the basics, axiomatization and classification, then we study our manifolds using algebraic and analytic methods. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to the Hadamard matrices, focusing on the complex case, and geometric and... more This is an introduction to the Hadamard matrices, focusing on the complex case, and geometric and analytic aspects. We discuss the Hadamard conjecture, and then the complex case, basic theory, and more specialized topics as well. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to quantum groups, focusing on the most basic examples, namely the closed... more This is an introduction to quantum groups, focusing on the most basic examples, namely the closed subgroups $G\subset U_N^+$. We discuss the foundational aspects, and then a number of more specialized topics, of algebraic and probabilistic nature. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using c... more 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.
An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More g... more An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.
This is an introduction to graph theory, from a geometric viewpoint. A finite graph X is describe... more This is an introduction to graph theory, from a geometric viewpoint. A finite graph X is described by its adjacency matrix d ∈ M N (0, 1), which can be thought of as a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using d and linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups G(X) ⊂ G + (X), which must leave invariant the eigenspaces of d. Finally, we discuss similar questions for the quantum graphs, with these being again described by certain matrices d ∈ M N (C), but in a more twisted way.
This is an introduction to calculus, and its applications to basic questions from physics. We fir... more This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions f : R → R, with the notion of continuity, and the construction of the derivative f ′ (x) and of the integral b a f (x)dx. Then we investigate the case of the complex functions f : C → C, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, f : R N → R M or f : R N → C M or f : C N → C M , with general theory, integration results, maximization questions, and basic applications to physics.
A closed subgroup G ⊂ u U + N is called easy when its associated Tannakian category C kl = Hom(u ... more A closed subgroup G ⊂ u U + N is called easy when its associated Tannakian category C kl = Hom(u ⊗k , u ⊗l) appears from a category of partitions, C = span(D) with D = (D kl) ⊂ P , via the standard implementation of partitions as linear maps. The examples abound, and the main known subgroups G ⊂ 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 ⊂ U + N , as well as various generalizations of the formalism, known as super-easiness theories, and the unification problem for them.
The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We rev... more 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.
This is a joint introduction to classical and free probability, which are twin sisters. We first ... more 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.
This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can ... more 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 $X$.
This is an introduction to linear algebra and group theory. We first review the linear algebra ba... more 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.
We work out axioms for the duals G ⊂ U + N of the finite quantum permutation groups, F ⊂ S + N wi... more We work out axioms for the duals G ⊂ U + N of the finite quantum permutation groups, F ⊂ S + N with |F | < ∞, 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. Contents 4. Integration results, orbits and orbitals 12 5. Actions on finite graphs, in the dual setting 15 6. Infinite extensions and open problems 18 References 19 2010 Mathematics Subject Classification. 46L65.
A classical theorem of Frucht states that any finite group appears as the automorphism group of a... more 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.
The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ w... more The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ 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.
This is an introduction to classical and free probability, from a quantum algebra and random matr... more This is an introduction to classical and free probability, from a quantum algebra and random matrix perspective. We discuss the foundations, insisting on limiting theorems, and then we discuss a number of more specialized aspects. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to subfactors, from a quantum group perspective, focusing on the case of ... more This is an introduction to subfactors, from a quantum group perspective, focusing on the case of integer index. We discuss the foundational aspects of the theory, and then we discuss a number of more advanced topics. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to operator algebras, from a quantum algebra and quantum physics perspect... more This is an introduction to operator algebras, from a quantum algebra and quantum physics perspective. We discuss the foundational aspects of the theory, $C^*$-algebras and von Neumann algebras, and then more specialized topics. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to noncommutative geometry, from an operator algebra and quantum group vi... more This is an introduction to noncommutative geometry, from an operator algebra and quantum group viewpoint. We discuss the basics, axiomatization and classification, then we study our manifolds using algebraic and analytic methods. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to the Hadamard matrices, focusing on the complex case, and geometric and... more This is an introduction to the Hadamard matrices, focusing on the complex case, and geometric and analytic aspects. We discuss the Hadamard conjecture, and then the complex case, basic theory, and more specialized topics as well. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to quantum groups, focusing on the most basic examples, namely the closed... more This is an introduction to quantum groups, focusing on the most basic examples, namely the closed subgroups $G\subset U_N^+$. We discuss the foundational aspects, and then a number of more specialized topics, of algebraic and probabilistic nature. These lecture notes consist of slides written in the Summer 2020. Presentations available at my Youtube channel.
This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using c... more 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.
An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More g... more An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.
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