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1. Noncrossing arithmetics

Author

Ebrahimi-Fard, Kurusch, Foissy, Loïc, Kock, Joachim, and Patras, Frédéric

Subjects
Mathematics - Combinatorics, 05A18, 46L54, 11A05, 06A11, 18N50
Abstract

Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in free probability, it also possesses an elegant, purely combinatorial core that is of independent interest. The present article aims at offering a simple account of various aspects of higher-order Kreweras complementation on the basis of elementary arithmetic, (co)algebraic, categorical and simplicial properties of noncrossing partitions. The main idea is to see noncrossing partitions as providing an interesting noncommutative analogue of the interplay between the divisibility poset and the multiplicative monoid of positive integers. Just as the divisibility poset can be regarded as the decalage of the multiplicative monoid, we exhibit the lattice of noncrossing partitions as the decalage of a partial monoid structure on noncrossing partitions encoding higher-order Kreweras complements. While our results may be considered known, some of the viewpoints can be regarded as novel, providing an efficient approach both conceptually and computationally., Comment: For the CATMI proceedings volume. 19pp

Published
2024

2. Commutative B\_infty -algebras are shuffle algebras

Author

Foissy, Loïc and Patras, Frédéric

Subjects
Mathematics - Combinatorics
Abstract

We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.

Published
2024

3. Posets, their Incidence Algebras and Relative Operads, and the Cohomology Comparison Theorem

Author

Jacky, Vane, Mbatchou, Batkam, Patras, Frédéric, and Tcheka, Calvin

Subjects
Mathematics - Algebraic Topology
Abstract

Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem.The theorem classically asserts that there is a cochain equivalence between the usual singular cochain complex of a simplicial complex and the relative Hochschild complex of its incidence algebra, and a quasi-isomorphism with the standard Hochschild complex. Here, we will be mostly interested in its application to arbitrary posets (or, equivalently, finite topologies) and their incidence algebras. We construct various structures, classical and new, on the above two complexes: cosimplicial, differential graded algebra, operadic and brace algebra structures and show that the comparison theorem preserves all of them. These results provide non standard insights on links between the theory of posets, incidence algebras, endomorphism operads and finite and combinatorial topology. By non standard, we refer here to the use of relative versions of Hochschild complexes and operads., Comment: arXiv admin note: text overlap with arXiv:math/0112177

Published
2023

4. Effective Action in Free Probability

Author

Ebrahimi-Fard, Kurusch and Patras, Frederic

Subjects
Mathematical Physics, Mathematics - Combinatorics, Mathematics - Probability, Mathematics - Rings and Algebras
Abstract

Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective action, which is closely related to the large deviation rate function in classical probability and one-particle irreducible correlation functions in quantum field theories. The central aim is to understand how it can be defined and studied in free probability. In this respect, we introduce a suitable diagrammatic formalism.

Published
2023

5. Algebraic structures underlying quantum independences : Theory and Applications

Author

Chetrite, Raphael and Patras, Frederic

Subjects
Mathematical Physics, Quantum Physics
Abstract

The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture introducing jointly to several fields of applications, many of which are probably not all familiar (at leat at the same time and in the form we use to present them) to the readers. Lastly, we take the opportunity to present various results obtained recently that use group and bialgebra techniques to handle notions such as cumulants or Wick polynomials in the various noncommutative probability theories., Comment: 46 pages. Krzysztof Gawedzki's special issue

Published
2022

6. Algebraic groups in non-commutative probability theory revisited

Author

Chevyrev, Ilya, Ebrahimi-Fard, Kurusch, and Patras, Frédéric

Subjects
Mathematics - Rings and Algebras, Mathematics - Operator Algebras, Mathematics - Probability, 16T05, 46L54 (Primary) 16T10, 16T30, 17A30, 46L53 (Secondary)
Abstract

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Sch\"urmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behaviour of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Sch\"urmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of noncrossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Sch\"urmann's aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature., Comment: 27 pages. Minor revisions. To appear in Infinite Dimensional Analysis, Quantum Probability and Related Topics

Published
2022
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7. Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

Author

Ebrahimi-Fard, Kurusch, Patras, Frédéric, Tapia, Nikolas, and Zambotti, Lorenzo

Subjects
Mathematics - Probability, Mathematics - Operator Algebras, 16T05, 16T10, 16T30, 17A30, 46L53, 46L54
Abstract

We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.

Published
2022
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8. A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Author

Celestino, Adrian and Patras, Frédéric

Subjects
Mathematics - Combinatorics, Mathematics - Probability, Mathematics - Rings and Algebras
Abstract

Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-Lie algebra and in a pre-Lie algebra of words originating in free probability. The latter example provides combinatorial formulas relating the different brands of cumulants in non-commutative probability., Comment: 30 pages

Published
2022

9. The Homes of Thought

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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10. Axioms and Intuitions

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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11. To the Meeting of Reality

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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12. The Origins of Modern Mathematics

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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13. Structures and Categories

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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14. From Plato to Husserl

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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15. The Structuralist Current

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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16. Style in Mathematics

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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17. Introduction

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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18. Conclusion

Author

Patras, Frédéric, Bueno, Otávio, Editor-in-Chief, Brogaard, Berit, Editorial Board Member, French, Steven, Editorial Board Member, Dutilh Novaes, Catarina, Editorial Board Member, Rowbottom, Darrell P., Editorial Board Member, Ruttkamp, Emma, Editorial Board Member, Miller, Kristie, Editorial Board Member, and Patras, Frédéric

Published
2023
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19. Russell and Carnap or Bourbaki? Two Ways Towards Structures

Author

Cantù, Paola, Patras, Frédéric, Heinrich-Ramharter, Esther, Series Editor, Kusch, Martin, Series Editor, Schiemer, Georg, Series Editor, Stadler, Friedrich, Series Editor, Carrier, Martin, Advisory Editor, Cartwright, Nancy, Advisory Editor, Creath, Richard, Advisory Editor, Ferrari, Massimo, Advisory Editor, Friedman, Michael, Advisory Editor, Galavotti, Maria Carla, Advisory Editor, Galison, Peter, Advisory Editor, Hacohen, Malachi, Advisory Editor, Hegselmann, Rainer, Advisory Editor, Heidelberger, Michael, Advisory Editor, Howard, Don, Advisory Editor, Hoyningen-Huene, Paul, Advisory Editor, Jabloner, Clemens, Advisory Editor, Kox, Anne J., Advisory Editor, Lennox, James G., Advisory Editor, Mormann, Thomas, Advisory Editor, Morscher, Edgar, Advisory Editor, Mulligan, Kevin, Advisory Editor, Nemeth, Elisabeth, Advisory Editor, Nida-Rümelin, Julian, Advisory Editor, Niiniluoto, Ilkka, Advisory Editor, Pfersmann, Otto, Advisory Editor, Rédei, Miklós, Advisory Editor, Richardson, Alan, Advisory Editor, Schurz, Gerhard, Advisory Editor, Sluga, Hans, Advisory Editor, Sober, Elliott, Advisory Editor, Soulez, Antonia, Advisory Editor, Spohn, Wolfgang, Advisory Editor, Stöltzner, Michael, Advisory Editor, Uebel, Thomas E., Advisory Editor, Wagner, Pierre, Advisory Editor, Waters, C. Kenneth, Advisory Editor, Wolters, Gereon, Advisory Editor, Zeilinger, Anton, Advisory Editor, Essler, Wilhelm K., Honorary Editor, Holton, Gerald, Honorary Editor, Janik, Allan S., Honorary Editor, Kamlah, Andreas, Honorary Editor, Köhler, Eckehart, Honorary Editor, Manninen, Juha, Honorary Editor, Oeser, Erhard, Honorary Editor, Schuster, Peter, Honorary Editor, Šebestík, Jan, Honorary Editor, Sigmund, Karl, Honorary Editor, Thiel, Christian, Honorary Editor, Weingartner, Paul, Honorary Editor, Woleński, Jan, Honorary Editor, Stoppelkamp, Bastian, Honorary Editor, and Cantù, Paola, editor

Published
2023
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20. 'Poincaré: The Philosopher' by Léon Brunschvicg: A Perspective

Author

Patras, Frédéric, Heinrich-Ramharter, Esther, Series Editor, Kusch, Martin, Series Editor, Schiemer, Georg, Series Editor, Stadler, Friedrich, Series Editor, Carrier, Martin, Advisory Editor, Cartwright, Nancy, Advisory Editor, Creath, Richard, Advisory Editor, Ferrari, Massimo, Advisory Editor, Friedman, Michael, Advisory Editor, Galavotti, Maria Carla, Advisory Editor, Galison, Peter, Advisory Editor, Hacohen, Malachi, Advisory Editor, Hegselmann, Rainer, Advisory Editor, Heidelberger, Michael, Advisory Editor, Howard, Don, Advisory Editor, Hoyningen-Huene, Paul, Advisory Editor, Jabloner, Clemens, Advisory Editor, Kox, Anne J., Advisory Editor, Lennox, James G., Advisory Editor, Mormann, Thomas, Advisory Editor, Morscher, Edgar, Advisory Editor, Mulligan, Kevin, Advisory Editor, Nemeth, Elisabeth, Advisory Editor, Nida-Rümelin, Julian, Advisory Editor, Niiniluoto, Ilkka, Advisory Editor, Pfersmann, Otto, Advisory Editor, Rédei, Miklós, Advisory Editor, Richardson, Alan, Advisory Editor, Schurz, Gerhard, Advisory Editor, Sluga, Hans, Advisory Editor, Sober, Elliott, Advisory Editor, Soulez, Antonia, Advisory Editor, Spohn, Wolfgang, Advisory Editor, Stöltzner, Michael, Advisory Editor, Uebel, Thomas E., Advisory Editor, Wagner, Pierre, Advisory Editor, Waters, C. Kenneth, Advisory Editor, Wolters, Gereon, Advisory Editor, Zeilinger, Anton, Advisory Editor, Essler, Wilhelm K., Honorary Editor, Holton, Gerald, Honorary Editor, Janik, Allan S., Honorary Editor, Kamlah, Andreas, Honorary Editor, Köhler, Eckehart, Honorary Editor, Manninen, Juha, Honorary Editor, Oeser, Erhard, Honorary Editor, Schuster, Peter, Honorary Editor, Šebestík, Jan, Honorary Editor, Sigmund, Karl, Honorary Editor, Thiel, Christian, Honorary Editor, Weingartner, Paul, Honorary Editor, Woleński, Jan, Honorary Editor, Stoppelkamp, Bastian, Honorary Editor, and Cantù, Paola, editor

Published
2023
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22. Quasi-shuffle algebras in non-commutative stochastic calculus

Author

Ebrahimi-Fard, Kurusch and Patras, Frédéric

Subjects
Mathematics - Probability
Abstract

This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of It{\^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{\^o} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{\^o} stochastic exponential of matrix-valued semimartingales.

Published
2020

23. From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras

Author

Ebrahimi-Fard, Kurusch and Patras, Frederic

Subjects
Mathematics - Rings and Algebras, Mathematical Physics
Abstract

Gian-Carlo Rota mentioned in one of his last articles the problem of developing a theory around the notion of integration algebras, which should be dual to the one of differential algebras. This idea has been developed historically along three lines: using properties of iterated integrals and generalisations thereof, as in the theory of rough paths; using the algebraic structures underlying chronological calculus such as shuffle and pre-Lie products, as they appear in theoretical physics and control theory; and, more generally, using a particular operator identity which came to be known as Rota-Baxter relation. The recent developments along each of these lines of research and their various application domains are not always known to other communities in mathematics and related fields. The general aim of this survey is therefore to present a modern and unified perspective on these approaches, featuring among others non-commutative Rota-Baxter algebras and related Hopf and Lie algebraic structures such as descent algebras, algebras of simplices as well as shuffle, pre- and post-Lie algebras and their enveloping algebras. Accordingly, two viewpoints are proposed. A geometric one, that can be derived using combinatorial properties of Euclidean simplices and the duality between integrands and integration domains in iterated integrals. An algebraic one, provided by the axiomatisation of integral calculus in terms of Rota-Baxter algebra.

Published
2019
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27. Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

Author

Ebrahimi-Fard, Kurusch, Foissy, Loïc, Kock, Joachim, and Patras, Frédéric

Subjects
Mathematics - Combinatorics, Mathematics - Probability, Mathematics - Rings and Algebras, 16T05, 16T10, 16T30, 18M60, 46L53, 46L54
Abstract

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of M\"obius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions., Comment: 41 pages. v4: Fixed a small mistake on p.23; it does not affect the rest of the paper. Final version, to appear in Adv. Math

Published
2019
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30. Combinatorial Hopf Algebras, Twisted Structures, and Species

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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31. Algebraic Topology

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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32. Group Theory

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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33. Renormalization

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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34. Graded Hopf Algebras and the Descent Gebra

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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35. Pre-lie Algebras

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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36. Structure Theorems

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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37. Hopf Algebras and Groups

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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38. Coalgebras, Duality

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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39. Introduction

Author

Cartier, Pierre, Patras, Frédéric, Broué, Michel, Series Editor, Fialowski, Alice, Series Editor, Friedlander, Eric, Series Editor, Gordon, Iain, Series Editor, Greenlees, John, Series Editor, Hiß, Gerhard, Series Editor, Moerdijk, Ieke, Series Editor, Schweigert, Christoph, Series Editor, Teicher, Mina, Series Editor, Cartier, Pierre, and Patras, Frédéric

Published
2021
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40. Cumulant-Cumulant Relations in Free Probability Theory from Magnus' Expansion

Author

Celestino, Adrian, Ebrahimi-Fard, Kurusch, Patras, Frédéric, and Perales, Daniel

Subjects
Combinatorial probabilities -- Analysis, Geometric probabilities -- Analysis, Algebra -- Analysis, Probabilities -- Analysis, Mathematics
Abstract

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker-Campbell-Hausdorff formula due to A. Murua., Author(s): Adrian Celestino [sup.1], Kurusch Ebrahimi-Fard [sup.1], Frédéric Patras [sup.2], Daniel Perales [sup.3] Author Affiliations: (1) grid.5947.f, 0000 0001 1516 2393, Department of Mathematical Sciences, Norwegian University of Science and [...]

Published
2022
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41. A group-theoretical approach to conditionally free cumulants

Author

Ebrahimi-Fard, Kurusch and Patras, Frederic

Subjects
Mathematics - Probability
Abstract

In this work we extend the recently introduced group-theoretical approach to moment-cumulant relations in non-commutative probability theory to the notion of conditionally free cumulants. This approach is based on a particular combinatorial Hopf algebra which may be characterised as a non-cocommutative generalisation of the classical unshuffle Hopf algebra. Central to our work is the resulting non-commutative shuffle algebra structure on the graded dual. It implies an extension of the classical relation between the group of Hopf algebra characters and its Lie algebra of infinitesimal characters and, among others, the appearance of new forms of adjoint actions of the group on its Lie algebra which happens to play a key role in the new algebraic understanding of conditionally free cumulants., Comment: 24 pages

Published
2018
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42. Surjections and double posets

Author

Patras, Frédéric and Foissy, Loïc

Subjects
Mathematics - Combinatorics, Mathematics - Rings and Algebras
Abstract

The theory and structure of the Hopf algebra of surjections (known as WQSym, the Hopf algebra of word quasi-symmetric functions) parallels largely the one of bijections (known as FQSym or MR, the Hopf algebra of word quasi-symmetric functions or Malvenuto-Reutenauer Hopf algebra). The study of surjections from a picture and double poset theoretic point of view, which is the subject of the present article, seems instead new. The article is organized as follows. We introduce first a family of double posets, weak planar posets, that generalize the planar posets and are in bijection with surjections or, equivalently, packed words. The following sections investigate their Hopf algebraic properties, which are inherited from the Hopf algebra structure of double posets and their relations with WQSym

Published
2018

43. Hopf-algebraic deformations of products and Wick polynomials

Author

Ebrahimi-Fard, Kurusch, Patras, Frédéric, Tapia, Nikolas, and Zambotti, Lorenzo

Subjects
Mathematics - Probability, 16T05, 16T15, 60C05 (Primary), 16T30, 60H30 (Secondary)
Abstract

We present an approach to classical definitions and results on cumulant--moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalise the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures., Comment: Revised and improved Section 9. 29 pages

Published
2017
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44. Phenomenology of Numbers

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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45. Cantor and Set Theory

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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46. Generalized Numbers II

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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47. Numbers and Magnitudes

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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48. Mathematics and Reality

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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49. The Third Man Argument

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
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50. The One and the Multiple

Author

Patras, Frédéric, Morel, Jean-Michel, Editor-in-Chief, Teissier, Bernard, Editor-in-Chief, Baur, Karin, Series Editor, Brion, Michel, Series Editor, De Lellis, Camillo, Series Editor, Figalli, Alessio, Series Editor, Huber, Annette, Series Editor, Khoshnevisan, Davar, Series Editor, Kontoyiannis, Ioannis, Series Editor, Kunoth, Angela, Series Editor, Mézard, Ariane, Series Editor, Podolskij, Mark, Series Editor, Serfaty, Sylvia, Series Editor, Vezzosi, Gabriele, Series Editor, Wienhard, Anna, Series Editor, and Patras, Frédéric

Published
2020
Full Text
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