Combinatorics and Arithmetic for Physics

19 nov. 2025, 09:00 → 21 nov. 2025, 17:40 Europe/Paris

Centre de conférences Marilyn et James Simons (IHES)

Combinatorics and Arithmetic for Physics
The meeting focuses on questions of discrete mathematics and number theory, emphasizing computability. Problems are drawn mainly from theoretical physics: renormalization, combinatorial physics, geometry, evolution equations (commutative and noncommutative), or related to its models, but not only. 
Computations, based on combinatorial structures (graphs, trees, words, automata, semirings, bases), or classical structures (operators, Hopf algebras, evolution equations, special functions, categories) are good candidates for computer-based implementation and experimentation.

Organized by: Gérard H. E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY and Karol A. PENSON.

Speakers:

• Cyril Banderier (LIPN)
• Marek Bożejko (Wroclaw University)
• Philippe Di Francesco (IPhT Saclay)
• Vladimir Dotsenko (Université de Strasbourg)
  
• Vladimir Fock (IRMA Strasbourg)
  
• Oleg Kaikov (LIST, Saclay)
  
• Arthemy Kiselev (University of Groningen)
  
• Maxim Kontsevich (IHES)
  
• Pierre-Vincent Koseleff (IMJ-PRG)
  
• Gleb Koshevoy (IITP, Moscow)
  
• Anastasia Matveeva (École polytechnique)
  
• Thomas Müller (LIPN, Paris)
• Hiroaki Nakamura (Osaka University)
  
• Toshiki Nakashima (Sophia University Tokyo)
  
• Lucas Pannier (Université de Versailles Saint-Quentin en Yvelines)
• Karol Penson (LPTMC, Sorbonne Université)
• Parham Radpay (Université Paris-Saclay)
  
• Kilian Raschel (Université d'Angers)
  
• Vladimir Roubtsov (Université d'Angers)
  
• Thomas Simon (Université de Lille)
  
• Reiko Toriumi (Okinawa Institute of Science and Technology)
  

   

Sponsors: 
IHES - Math-STIC - LIPN (UMR-7030) - LPTMC (Univ-Paris 6) -   INRIA - GDR EFI 

Scientific Committee:
Joseph Ben Geloun (LIPN-Paris XIII), Alin Bostan (INRIA), Marek Bożejko (Wroclaw University), Vincent Rivasseau (Orsay-CEA), Pierre Simonnet (Univ. Corse)

 

 

affiche_CAP_2025.pdf

mer. 19 novembre

09:00 → 10:00 Welcome Coffee

10:00 → 10:50 About Transfer Matrices

Orateur: Maxim Kontsevich (IHES)

Vidéo

10:50 → 11:40 On Universal Quadratic Inequalities for Minors of TNN Matrices

Based on the following arxiv reference
https://arxiv.org/pdf/2506.03754

Orateur: Gleb Koshevoy (IITP, Moscow)

Vidéo

11:40 → 12:00 Pause

12:00 → 12:50 An Effective Proof of the p-curvature Conjecture for First-order Differential Equations With Rational Coefficients

In 1974, Honda proved the $p$-curvature conjecture for order one differential equations with rational coefficients over a number field. He demonstrated that in this setting, the p-curvature conjecture was equivalent to a theorem due to Kronecker, providing a local-global criterion for the splitting of polynomials over the rational numbers. In 1985 the Chudnovskys published another proof of Honda’s theorem (and of Kronecker’s theorem) by means of Padé approximation and elementary number theory, thus paving the way to an effective version of these results. Here, by ”effective” we mean that we wish to obtain an explicit finite bound on the number of $p$-curvatures to be computed in order to decide the algebraicity of the solution of the differential equation. In this talk, I will explain how to obtain such a bound, and report on an implementation.
This is joint work with Florian Fürnsinn (University of Vienna).

Orateur: Lucas Pannier (Université de Versailles Saint-Quentin en Yvelines)

Vidéo

12:50 → 14:00 Lunch

14:00 → 14:50 On the Integrality of Some P-recursive Sequences

I revisit the integrality criterion for Motzkin-type sequences due to Klazar and Luca, and propose a unified approach for analysing global boundedness and algebraicity within a broader class of holonomic sequences. The central contribution is an algorithm that finds all algebraic solutions of certain second-order recurrence relations with linear polynomial coefficients. As algebraicity and global boundedness are shown to be equivalent in the special cases considered, the method detects all globally bounded solutions as well. This offers a systematic approach to deciding when a given P-recursive sequence is integral or almost integral - a question that arises naturally in combinatorics and differential algebra.
Based on joint work with Alin Bostan.

Orateur: Anastasia Matveeva (École polytechnique)

Vidéo

14:50 → 15:40 The Wronskians over Multidimension and Homotopy Lie Algebras

The Wronskian determinant of $N$ functions in one variable $x$ helps us verify that they are linearly independent on an interval $(a, b)$ in ${\mathbb R}$. Can we have an equally convenient procedure over multidimensional spaces ${\mathbb R}^d$ with Cartesian coordinates $x, y, z, . . .$ ? Yes we can; let us study the definition of Wronskians for functions in many variables, and let us explore which differential-algebraic identities these structures satisfy.
To see why the Wronskian determinants actually do satisfy a set of quadratic, Jacobi-type identities, we observe first that the Wronskian of size $2 \times 2$ results from commutation of vector fields on the real line ${\mathbb R}$. From differential geometry we know that vector fields are differential operators of strict order $p = 1$. By taking the alternated composition of $N = 2p$ differential operators of strict order $p > 0$ on the affine line, we obtain the operator of same order $p$ with the Wronskian determinant for coefficient. As we had the Jacobi identity for the Lie algebra of vector fields, so we establish the (table of) quadratic, Jacobi-type identities for higher-order Wronskians of $N > 1$ arguments. (In string theory, these identities govern homotopy deformations of Lie algebras, here of vector fields.) We prove that the new Wronskians over multidimensional base with $d$ coordinates $x, y, z,. . .$ do satisfy the (table of) identities for strongly homotopy Lie algebras.
The problem is to understand how fast the dimension grows under iterated $N$-ary brackets. We spot a countable chain of finite-dimensional homotopy Lie algebras that generalize the vector field realization of $sl(2)$ on ${\mathbb R}$; we explicitly calculate all the structure constants. Yet the four-dimensional analogue of $sl(2)$ over the plane ${\mathbb R}^2$ with Cartesian coordinates $(x, y)$, now with ternary bracket from the Wronskian, is so far the only known finite-dimensional homotopy Lie algebra of this type over base dimension $> 1$. The hunt is on; we conclude that Lie algebra $sl(2)$ is the prototype not only for semisimple complex Lie algebras encoded by root systems but also for countably many $N$-ary homotopy Lie algebras.

https://arxiv.org/abs/2511.03848

https://arxiv.org/abs/2510.02145

https://arxiv.org/abs/math/0410185

Orateur: Arthemy Kiselev (University of Groningen)

Vidéo

15:40 → 16:00 Pause

16:00 → 16:50 Persistence Probabilities for Auto-regressive Markov Chains

We investigate the first crossing time of zero of an auto-regressive Markov chain with atomless innovations, denoted by T. Under a log-concavity assumption on the innovation law, we show that the law of T is log-convex for positive drifts, which implies a Baxter-Spitzer factorization as in the case of random walks. We also show that the law of T is never log-convex for negative drifts. For positive drifts, we conjecture that the law of T is, in general, completely monotonic and that the discrete Baxter-Spitzer factorization is actually a continuous Wiener-Hopf factorization.

Orateur: Thomas Simon (Université de Lille)

Vidéo

16:50 → 17:40 Remarks on Certain Parametrized Algebraic Equations

We consider several instances of pairs of integer sequences $(c(p,n), a(p,n))$, for integer $n\ge0$, and rational parameters $p>0$, that are interrelated through:
$$\exp(\sum_{n=1}^{\infty}\,c(p,n).\frac{t^n}{n})=\sum_{n=0}^{\infty}\,a(p,n).t^n.$$ Exact statements (Kontsevich, Reutenauer...) indicate that the algebraicity of the ordinary generating function (gf) of $a(p,n)$'s implies the algebraicity of the gf of $c(p,n)$'s. We employ an experimental-type approach by guessing such algebraic gf's of $c(p,n)$'s that furnish algebraic gf's of $a(p,n)$'s. We use the $c(p,n)$'s in form containing various binomial coefficients parametrized by $p$, and this results in their gf's that are generalized hypergeometric functions. In certain situations we can recover the algebraicity of gf's of $a(p,n)$ even without the knowledge of an actual functional form of $a(p,n)$. In cases that the $a(p,n)$'s can be fully identified from given $c(p,n)$'s, we derive novel logarithmic identities between various generalized hypergeometric functions. If, in addition, we conceive $c(p,n)$ and $a(p,n)$ as moments, we solve their corresponding Hausdorff moment problems with the method of Meijer $G$-functions. The behaviour of solutions (weight functions) is studied both analytically and graphically.

*) In collaboration with: G. H. E. Duchamp, M. Kontsevich and G. Koshevoy.

Orateur: Karol Penson (LPTMC, Sorbonne Université)

Vidéo

jeu. 20 novembre

09:00 → 10:00 Welcome Coffee

10:00 → 10:50 On Recent Conjectures Concerning Free Jordan Algebras and Free Alternative Algebras

In 2019, Iryna Kashuba and Olivier Mathieu proposed a beautiful conjecture on Lie algebra homology which, if true, would supply a lot of new information on the structure of free Jordan algebras, one of most mysterious algebraic structures systematically appearing in different areas of mathematics and mathe matical physics. Their construction relies on a functorial version of the celebrated Tits-Kantor-Koecher construction. Motivated by their work, Shang proposed an analogue of this conjecture, which would in turn supply new information on the structure of free alternative algebras. In this talk, I shall explain why both of these conjectures are not true, discuss new computational data concerning free Jordan algebras, and, time permitting, outline some related open problems.
This is partially based on joint work with Irvin Hentzel.

Orateur: Vladimir Dotsenko (Université de Strasbourg)

Vidéo

10:50 → 11:40 Characterization of the Unit Object in Localized Quantum Unipotent Category

For the quiver Hecke algebra $R$, let $R$-gmod be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\mbox{-gmod}}$ be the localization of $R$-gmod, called ''localized quantum unipotent category``. It has been shown that the set of equivalence classes of simple objects up to grading shifts ${\rm Irr}(\widetilde{R\mbox{-gmod}})$ in $\widetilde{R\mbox{-gmod}}$ has a crystal structure, and ${\rm Irr}(\widetilde{R\mbox{-gmod}})$ is isomorphic to the so-called cellular crystal ${\mathbb B}_{\bf i}$ by M. Kashiwara and myself. This isomorphism induces a function $\varepsilon_i^*$ on ${\mathbb B}_{\bf i}$. We give an explicit form of $\varepsilon_i^*$, and using this, we give a characterization of the unit object of $\widetilde{R\mbox{-gmod}}$ for finite classical types, $A_n,\,B_n,\, C_n$ and $D_n$.
This is a joint work with Koh Matsuura.

Orateur: Toshiki Nakashima (Sophia University Tokyo)

Vidéo

11:40 → 12:00 Pause

12:00 → 12:50 Källén Function and Around

There are elementary functions that turn out to be ubiquitous. The Källén function is one of them. Of course, it is incomparably less well-known than the exponential function (although, in a certain sense, it is related to it).
Originally arising in "school-textbooks" mathematics, the Källén function was later "rediscovered" by physicists in the context of scattering amplitude calculations. It is directly connected with the famous combinatorial generating functions (for the Catalan numbers), appears in the study of solutions of classical ODEs (Bessel–type), and is related to the generalized hypergeometric functions of Appell and Kampé de Férie, to the Wigner-Bloch dilogarithm as well as to properties of discriminants and associativity of 2-valued multiplication laws.
I will try to tell some stories around of this remarkable function.

Orateur: Vladimir Roubtsov (Université d'Angers)

Vidéo

12:50 → 14:00 Lunch

14:00 → 14:50 Steinberg Symbol and Tau Function

Steinberg symbol is a multiplicative bicharacter of a ring satisfying additional (Steinberg) relation. Beilinson in 1982 suggested an explicit expression for a canonical symbol for the ring functions on a circle. This symbol can be interpreted as a multiplicative analogue of a residue and also as a cocycle defining the Heisenberg group. We will show how one can define the boson-fermion correspondence using the symbol and show that the finite-gap tau-function can be interpreted as an automorphic form.

Orateur: Vladimir Fock (IRMA Strasbourg)

Vidéo

14:50 → 15:40 Quantum Error Mitigation Driven by Classical Simulations and Evolution Equations

Analytical and classical numerical approaches can fail for significant regimes of certain physical systems, see, e.g., the sign problem in lattice Quantum Chromodynamics. Quantum computing presents a viable framework to perform calculations in such regimes. However, current quantum hardware is affected by noise, requiring quantum error mitigation (QEM). We present two QEM techniques: First, QEM driven by data obtained in classical simulations. This approach involves learning the properties of the quantum noise in a regime accessible by both noisy quantum and classical devices, and then using this for error mitigation in a regime accessible only by noisy quantum devices. Second, QEM driven by analytically computed evolution equations. This approach leverages the fact that the observables within the simulation of an evolved quantum system obey a system of coupled evolution equations. Using an appropriate subset of these equations allows to mitigate errors in the measurements obtained on noisy quantum hardware. We demonstrate the two QEM techniques on the example of the lattice Schwinger model with a topological θ term.
Based on joint work with Theo Saporiti, Vasily Sazonov, and Mohamed Tamaazousti: [Phys. Rev. A 111 (2025) 6, 062202], [arXiv:2507.06601 (2025)] and [Phys. Rev. A 112 (2025) 3, 032409], work in progress, respectively.

Orateur: Oleg Kaikov (Université Paris-Saclay, CEA-List)

Vidéo

15:40 → 16:00 Pause

16:00 → 16:50 Invitation to Random Tensor Models: from Random Geometry, Enumeration of Tensor Invariants, to Characteristic Polynomials

I will introduce random tensor models by first reviewing their motivation coming from random geometric approach to quantum gravity. Then, I will selectively present some of the interesting research results, by highlighting recent results on enumerations of graphs representing tensor invariants, and reporting our recent work on a new notion of characteristic polynomials for tensors via Grassmann integrals and distributions of roots of random tensors.
The latter two are based on arXiv:2404.16404[hep-th] and arXiv:2510.04068[math-ph]

Orateur: Reiko Toriumi (Okinawa Institute of Science and Technology)

Vidéo

16:50 → 17:40 Higher Dimensional Floorplans and Baxter $d$-permutations

A 2-dimensional mosaic floorplan is a partition of a rectangle by other rectangles with no empty rooms. These partitions (considered up to some deformations) are known to be in bijection with Baxter permutations. A $d$-permutation is a $(d-1)$-tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, J. Integer Sequences 25 (2022), Baxter $d$-permutations generalising the usual Baxter permutations were introduced.
In this talk, I will introduce the $d$-floorplans which generalise the mosaic floorplans to arbitrary dimensions. Then, I will present the construction of their generating tree. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Finally, I will present a bijection between the $2^{d-1}$-floorplans and $d$-permutations characterised by forbidden vincular patterns. Surprisingly, this set of $d$-permutations is strictly contained within the set of Baxter $d$-permutations.

This is a joint work with Nicolas Bonichon and Adrian Tanasa (Université de bordeaux), this talk is based on arXiv:2504.01116.

Orateur: Thomas Müller (LIPN)

Vidéo

ven. 21 novembre

09:00 → 10:00 Welcome Coffee

10:00 → 10:50 Phase Transitions and Mittag-Leffler Functions for Critical Schemes Under the Gibbs Model

Composition schemes are ubiquitous in combinatorics, number theory, statistical mechanics and probability theory.
We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of $q$-enumeration (models where objects with a parameter of value $k$ have a Gibbs measure/Boltzmann weight $q^k$).
For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value $q=q_c$, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime ($q>q_c$), and it is a Boltzmann distribution in the subcritical regime ($0 We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons.
(Based on the article https://arxiv.org/abs/2311.17226 by Cyril Banderier, Markus Kuba, Stephan Wagner, Michael Wallner).

Orateur: Cyril Banderier (LIPN)

Vidéo

10:50 → 11:40 Łukasiewicz Logic and Tsallis Entropy Connected with Free Projections in the Free and Conditionally Free Probability

In my talk, we consider the following topics:

1. Free and C-free probability and completely positive maps.

2. Free independent projections as a model of Jozef Łukasiewicz $n$-valued logic, $n > 2$, and also a model of continuous logic of Łukasiewicz–Tarski.

3. Main Theorem: If $q$ is a real number and $x, y$ are from interval $(0, 1)$, then the Tsallis entropy is defined as
   $$ T_q (x, y) = (x_1-q + y_1-q-1)^{\frac{1}{1-q}} $$ Then we have: If $P$ and $Q$ are free independent in some probability space $(A, tr)$ with trace $tr$ state on $A$, and $tr(P) = x$, $tr(yQ) = y$, then $tr(P^Q) = T_0(x, y)$. If $P$ and $Q$ are Boolean independent, then $tr(P Q) = T_2(x, y)$ and relations with Dagum distributions, which are called log-logistic distributions in many statistics models. If $P$ and $Q$ are classically independent then $tr(P^Q) = T_1(x, y) = lim_{q\to 1}T_S (x, y)$, as t tends to 1. Here the projection $P^Q$ is the smallest projection on the closed linear span of $Im(P)$ and $Im(Q)$. The generalizations of cases of Tsallis entropy $T_q$, for $q$ in $(0, 1)$, we will use conditionally free independent projections.

4. Remarks on the free product of quantum channels.

References:

1. M. Bożejko, Positive definite functions on the free group and the noncommutative Riesz product, Boll. Un. Mat. Ital. (6) 5-A (1986), 13–21.

2. M. Bożejko, Remarks on free projections, Heidelberg Seminar 1999.

3. W. Mlotkowski, Operator-valued version of conditionally free product, Studia Math. 153:13–30, (2002).

4. M. Bożejko, Projections in free and Boolean probability with applications to J. Lukasiewicz logic, Conference on Quantum Statistics and Related Topics, Lodz, 10 pp., 2018.

5. M. Bożejko, Conditionally free probability, in Signal Proceeding and Hypercomplex Analysis, 139–147, 2019.

Orateur: Marek Bożejko (Wroclaw University)

Vidéo

11:40 → 12:00 Pause

12:00 → 12:50 Combinatorics and Arithmetic of Lissajous 3-braids (Remote)

We introduce a classification of braids arising from 3-body motions along Lissajous curves on the plane and discuss illustrations through the lenses of associated continued fractions, cutting sequences of checkered Farey tesselation and/or symbolic frieze patterns.
This is a joint work with Eiko Kin and Hiroyuki Ogawa.

Orateur: Hiroaki Nakamura (Osaka University)

Vidéo

12:50 → 14:00 Lunch

14:00 → 14:50 Persistence Probabilities for Random Walks and Related Processes

In this talk, we will discuss the persistence probabilities of various random processes, including random walks, autoregressive sequences, and moving average sequences. By studying their generating functions, we will establish connections between these persistence probabilities and several special functions, such as deformed exponential functions and generating functions of certain combinatorial polynomials.

https://arxiv.org/abs/2507.04427
https://arxiv.org/abs/2112.03016

Orateur: Kilian Raschel (Université d'Angers)

Vidéo

14:50 → 15:40 Minimal Polynomial Parameterization of Rational Knots

A polynomial parameterization of a knot $K$ in $S^3$ is a polynomial map $\gamma : \mathbf{R} \to \mathbf{R}^3$ whose closure of the image in $S^3$ is isotopic to $K$. Every knot admits a polynomial parametrisation, and we are interested in determining the lexicographic degree of a knot $K$, i.e. the minimal degree for the lexicographic order of a polynomial parametrisation of $K$.
We give the lexicographic degree of all two-bridge knots with 12 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.
Joint work with E. Brugallé an D. Pecker

Orateur: Pierre-Vincent Koseleff (IMJ-PRG)

Vidéo

15:40 → 16:00 Pause

16:00 → 16:50 Renormalization Group Methods for Quantum Spin Glasses

In this talk we study the dynamics of a quantum p-spin glass. For this purpose a nonperturbative renormalization group approach based on Wetterich-Morris formalism is adopted with two approximation schemes to address the model's non-locality, arising from the replica trick. The model is investigated in the symmetric and symmetry-broken phases. Finally, the results are briefly compared with previous results on 2+p spin glasses based on a less conventional coarse-graining scheme.

Orateur: Parham Radpay (Université Paris-Saclay)

Vidéo

16:50 → 17:40 Macdonald Functions and Quantum Dilogarithm

The theory of q-Whittaker functions for classical types is known to have a (quantum) cluster algebra realization. In this framework, a natural connection with the quantum dilogarithm is known. We show how this extends to the more general case of Macdonald theory in type A. We propose new Givental-like and Mellin-Barnes-like expressions for the Mac-donald functions, and explore their properties. These involve heavy use of quantum dilogarithms.
(ongoing collaboration with M. Bershtein, J.-E. Bourgine, R. Kedem, V. Pasquier and J. Shiraishi).

Orateur: Philippe Di Francesco (IPhT Saclay)

Vidéo