Ivan Contreras | Amherst College (original) (raw)

Papers by Ivan Contreras

arXiv: Mathematical Physics, 2017

In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacian... more In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacians. Our motivation comes from the discrete formulation of quantum mechanics and thermodynamics in the case of finite graphs, which suggest a natural extension to simplicial and CW-complexes. With this motivation, we also define generalizations of the odd Laplacian which is specific to hypergraphs arising from CW-complexes. We then provide a combinatorial interpretation for the powers of these Laplacians.

arXiv: Combinatorics, 2020

We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the disc... more We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the discrete de Rham cohomology of a finite graph Gamma\GammaGamma. This action produces a splitting of Aut$(\Gamma)$ that depends on the cycles of Gamma\GammaGamma. We also prove some graph-theoretical analogues of standard results in differential geometry, in particular, a graph version of Stokes' Theorem and the Mayer-Vietoris sequence in cohomology.

manuscripta mathematica, 2021

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a ... more We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible Kähler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a Kähler structure to generalise some classical result on co-adjoint orbits.

Linear and Multilinear Algebra

Discrete versions of the Laplace and Dirac operators haven been studied in the context of combina... more Discrete versions of the Laplace and Dirac operators haven been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graphtheoretic versions of the Schrödinger and Dirac equation. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras.

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms ... more We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a Lie algebroid we discuss the existence of an abelian integration of its abelianization. W... more Given a Lie algebroid we discuss the existence of an abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in CFMb. We also show that the abelianization can be obtained by a path-space construction, similar to the Weinstein groupoid of CF1, but where the underlying homotopies are now supported in surfaces with arbitrary genus. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an... more A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for L to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace C of a weak symplectic space V which imply that the induced canonical relation L_C from V to C/C^ω is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.

We functorially characterize groupoids as special dagger Frobenius algebras in the category of se... more We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets and relations, and locally cancellative regular semigroupoids. Finally, we study a universal passage from the former setting to the latter.

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mech... more In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to ... more We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra in Rel of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.

Pacific Journal of Mathematics

We introduce the notions of relational groupoids and relational convolution algebras. We provide ... more We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group G and a given normal subgroup H. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.

Journal of Mathematical Physics

The main idea of this note is to describe the integration procedure for poly-Poisson structures, ... more The main idea of this note is to describe the integration procedure for poly-Poisson structures, that is, to find a poly-symplectic groupoid integrating a poly-Poisson structure, in terms of topological field theories, namely via the path-space construction. This will be given in terms of the poly-Poisson sigma model (P P SM) and we prove that every poly-Poisson structure has a natural integration via relational poly-symplectic groupoids, extending the results in [8] and [26]. We provide familiar examples (trivial, linear, constant and symplectic) within this formulation and we give some applications of this construction regarding the classification of poly-symplectic integrations, as well as Morita equivalence of poly-Poisson manifolds. Contents 1. Introduction 1 2. Poly-Poisson manifolds 3 2.1. Examples 4 2.2. Some special submanifolds 5 2.3. Reduction by symmetries 7 3. Poly-Poisson structures and their integration 9 4. The sigma model 11 4.1. The PSM 12 4.2. The PPSM 13 4.3. Examples of PPSM integration 16 5. Relational groupoids 17 5.1. Examples 18 5.2. Lagrangian submanifolds of Poly-symplectic structures 18 6. Further Applications 19 6.1. Other integrations 19 6.2. Integration of Lie algebroids via poly-Poisson integration 20 6.3. Weinstein map and Morita equivalence 20 References 21

Entropy

We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemi... more We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemically-restricted toy theories formalized by Spekkens (Spekkens, 2016). In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.

This note introduces the construction of relational symplectic groupoids as a way to integrate ev... more This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed.

Abstract. We functorially characterize groupoids as special dagger Frobenius algebras in the cate... more Abstract. We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets and relations, and locally cancellative regular semigroupoids. Finally, we study a universal passage from the former setting to the latter. 1.

Annales Henri Lebesgue

A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an... more A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for L to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace C of a weak symplectic space V which imply that the induced canonical relation L C from V to C/C ω is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations. Résumé.-Un sous-espace Lagrangien L d'un espace vectoriel symplectique faible est appelé Lagrangien scindé s'il a un complément isotrope (donc Lagrangien). Lorsque la structure symplectique est forte, il suffit que L ait un complément fermé, qui peut ensuite être déplacé pour devenir isotrope. Le but de cette note est de développer la théorie des compositions et des réductions des relations canoniques scindées pour les espaces vectoriels symplectiques. Nous donnons des conditions sur un sous-espace coisotrope C d'un espace symplectique faible V qui impliquent que la relation canonique induite L C de V à C/C ω est scindée, et en déduisons des conditions suffisantes pour que les relations canoniques scindées soient composables. Nous prouvons que les relations canoniques résultant du modèle sigma de Poisson dans l'approche lagrangienne de la théorie des champs sont scindées, donnant une description des groupïdes symplectiques intégrant les variétés de Poisson en termes de relations canoniques scindées.

arXiv: Mathematical Physics, 2016

A large part of operational quantum mechanics can be reproduced from a classical statistical theo... more A large part of operational quantum mechanics can be reproduced from a classical statistical theory with a restriction which implies a limit on the amount of knowledge that an agent can have about an individual system [6, 17]. These epistemic restrictions have recently been restated via the symplectic structure of the underlying classical theory [18]. Starting with this symplectic framework, we obtain C*-algebraic formulation for the epistemically restricted theories. In the case of continuous variables, the groupoid quantization recipe of E. Hawkins provides us a twisted group C*-algebra which is the usual Moyal quantization of a Poisson vector space [11].

Reviews in Mathematical Physics

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target s... more In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

Pacific Journal of Mathematics

The objective of this note is to provide an interpretation of the discrete version of Morse inequ... more The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics [8], adapted to finite graphs, as a particular instance of Morse-Witten theory for cell complexes [4]. We describe the general framework of graph quantum mechanics and we produce discrete versions of the Hodge theorems and energy cutoffs within this formulation.

Linear and Multilinear Algebra

We study two different types of gluing for graphs: interface (obtained by choosing a common subgr... more We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing even and odd Laplacians of graphs obtained by gluing, as well as their spectra. We subsequently discuss applications to quantum mechanics and bounds for the Fiedler value of the gluing of graphs.

arXiv: Mathematical Physics, 2017

In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacian... more In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacians. Our motivation comes from the discrete formulation of quantum mechanics and thermodynamics in the case of finite graphs, which suggest a natural extension to simplicial and CW-complexes. With this motivation, we also define generalizations of the odd Laplacian which is specific to hypergraphs arising from CW-complexes. We then provide a combinatorial interpretation for the powers of these Laplacians.

arXiv: Combinatorics, 2020

We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the disc... more We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the discrete de Rham cohomology of a finite graph Gamma\GammaGamma. This action produces a splitting of Aut$(\Gamma)$ that depends on the cycles of Gamma\GammaGamma. We also prove some graph-theoretical analogues of standard results in differential geometry, in particular, a graph version of Stokes' Theorem and the Mayer-Vietoris sequence in cohomology.

manuscripta mathematica, 2021

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a ... more We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible Kähler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a Kähler structure to generalise some classical result on co-adjoint orbits.

Linear and Multilinear Algebra

Discrete versions of the Laplace and Dirac operators haven been studied in the context of combina... more Discrete versions of the Laplace and Dirac operators haven been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graphtheoretic versions of the Schrödinger and Dirac equation. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras.

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms ... more We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a Lie algebroid we discuss the existence of an abelian integration of its abelianization. W... more Given a Lie algebroid we discuss the existence of an abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in CFMb. We also show that the abelianization can be obtained by a path-space construction, similar to the Weinstein groupoid of CF1, but where the underlying homotopies are now supported in surfaces with arbitrary genus. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an... more A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for L to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace C of a weak symplectic space V which imply that the induced canonical relation L_C from V to C/C^ω is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.

We functorially characterize groupoids as special dagger Frobenius algebras in the category of se... more We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets and relations, and locally cancellative regular semigroupoids. Finally, we study a universal passage from the former setting to the latter.

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mech... more In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to ... more We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra in Rel of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.

Pacific Journal of Mathematics

We introduce the notions of relational groupoids and relational convolution algebras. We provide ... more We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group G and a given normal subgroup H. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.

Journal of Mathematical Physics

The main idea of this note is to describe the integration procedure for poly-Poisson structures, ... more The main idea of this note is to describe the integration procedure for poly-Poisson structures, that is, to find a poly-symplectic groupoid integrating a poly-Poisson structure, in terms of topological field theories, namely via the path-space construction. This will be given in terms of the poly-Poisson sigma model (P P SM) and we prove that every poly-Poisson structure has a natural integration via relational poly-symplectic groupoids, extending the results in [8] and [26]. We provide familiar examples (trivial, linear, constant and symplectic) within this formulation and we give some applications of this construction regarding the classification of poly-symplectic integrations, as well as Morita equivalence of poly-Poisson manifolds. Contents 1. Introduction 1 2. Poly-Poisson manifolds 3 2.1. Examples 4 2.2. Some special submanifolds 5 2.3. Reduction by symmetries 7 3. Poly-Poisson structures and their integration 9 4. The sigma model 11 4.1. The PSM 12 4.2. The PPSM 13 4.3. Examples of PPSM integration 16 5. Relational groupoids 17 5.1. Examples 18 5.2. Lagrangian submanifolds of Poly-symplectic structures 18 6. Further Applications 19 6.1. Other integrations 19 6.2. Integration of Lie algebroids via poly-Poisson integration 20 6.3. Weinstein map and Morita equivalence 20 References 21

Entropy

We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemi... more We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemically-restricted toy theories formalized by Spekkens (Spekkens, 2016). In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.

This note introduces the construction of relational symplectic groupoids as a way to integrate ev... more This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed.

Abstract. We functorially characterize groupoids as special dagger Frobenius algebras in the cate... more Abstract. We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets and relations, and locally cancellative regular semigroupoids. Finally, we study a universal passage from the former setting to the latter. 1.

Annales Henri Lebesgue

A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an... more A Lagrangian subspace L of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for L to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace C of a weak symplectic space V which imply that the induced canonical relation L C from V to C/C ω is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations. Résumé.-Un sous-espace Lagrangien L d'un espace vectoriel symplectique faible est appelé Lagrangien scindé s'il a un complément isotrope (donc Lagrangien). Lorsque la structure symplectique est forte, il suffit que L ait un complément fermé, qui peut ensuite être déplacé pour devenir isotrope. Le but de cette note est de développer la théorie des compositions et des réductions des relations canoniques scindées pour les espaces vectoriels symplectiques. Nous donnons des conditions sur un sous-espace coisotrope C d'un espace symplectique faible V qui impliquent que la relation canonique induite L C de V à C/C ω est scindée, et en déduisons des conditions suffisantes pour que les relations canoniques scindées soient composables. Nous prouvons que les relations canoniques résultant du modèle sigma de Poisson dans l'approche lagrangienne de la théorie des champs sont scindées, donnant une description des groupïdes symplectiques intégrant les variétés de Poisson en termes de relations canoniques scindées.

arXiv: Mathematical Physics, 2016

A large part of operational quantum mechanics can be reproduced from a classical statistical theo... more A large part of operational quantum mechanics can be reproduced from a classical statistical theory with a restriction which implies a limit on the amount of knowledge that an agent can have about an individual system [6, 17]. These epistemic restrictions have recently been restated via the symplectic structure of the underlying classical theory [18]. Starting with this symplectic framework, we obtain C*-algebraic formulation for the epistemically restricted theories. In the case of continuous variables, the groupoid quantization recipe of E. Hawkins provides us a twisted group C*-algebra which is the usual Moyal quantization of a Poisson vector space [11].

Reviews in Mathematical Physics

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target s... more In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

Pacific Journal of Mathematics

The objective of this note is to provide an interpretation of the discrete version of Morse inequ... more The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics [8], adapted to finite graphs, as a particular instance of Morse-Witten theory for cell complexes [4]. We describe the general framework of graph quantum mechanics and we produce discrete versions of the Hodge theorems and energy cutoffs within this formulation.

Linear and Multilinear Algebra

We study two different types of gluing for graphs: interface (obtained by choosing a common subgr... more We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing even and odd Laplacians of graphs obtained by gluing, as well as their spectra. We subsequently discuss applications to quantum mechanics and bounds for the Fiedler value of the gluing of graphs.