Combinatorics and Arithmetic for Physics

26 nov. 2026, 09:00 → 27 nov. 2026, 17:40 Europe/Paris

Centre de conférences Marilyn et James Simons (IHES)

Combinatorics and Arithmetic for Physics
The meeting’s focus is about questions of discrete mathematics and number theory with an emphasis on computability. Problems are drawn mainly from theoretical physics : renormalisation, combinatorial physics, geometry, evolution equations (commutative and noncommutative) or related to their models. 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:

• Ariana Chin (UCLA)
• Lucia Di Vizio (Université de Versailles-St Quentin)
• Florian Fürnsinn (Universität Wien)
• Stéphane Gaubert (INRIA)
  
• Gleb Koshevoy (IITP, Moscow)
  
• Toshiki Nakashima (Sophia University, Tokyo)
  
• Robert Penner (University of California & IHES)
  
• Darya Polyakova (Hamburg University)
  
• Volodya Roubtsov (Université d'Angers)
  
• Theo Saporiti (Université Paris-Saclay)
• Travis Scrimshaw (Hokkaido University)
  
• Mahsa Shirmohammadi (IRIF, Université de Paris)
  

   

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

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_2026.pdf

jeu. 26 novembre

09:00 Welcome coffee

1 TBA

Orateur: Gleb Koshevoy (IITP, Moscow)

2 Higher-Bessel Kernel Polynomials, Multiset Eulerian Numbers and Beyond

We identify the Frobenius numerators of the higher-Bessel multiplication kernel with MacMahon’s multiset Eulerian polynomials. This gives a combinatorial description of polynomial coefficients arising from period integrals. This identification explains why the polynomials appeared in the product kernels are palindromic, have simple negative zeros, and have positive integer coefficients. The descent refinement specializes to the Dwork period and a common Landau-Ginzburg potential relates this period to the higher-Bessel function. We also identify the Newton polytope of the kernel’s Laurent model as the polar of the type - $A_{N-1}$ root polytope and give a complete-intersection presentation in $(P1)^N$. For $N = 5$ this has the configuration occurring in mirror Hulek–Verrill geometry. These explicit and motivate links with the Calabi–Yau geometry of Feynman integrals.
Based on a joint paper with Ilya Gaiur, Feihu Liu and Zihao Zhang.

Orateur: Vladimir Roubtsov (Université d'Angers)

11:40 Coffee Break

3 TBA

Orateur: Lucia Di Vizio (Université de Versailles Saint-Quentin)

12:50 Lunch

4 Galois Groups of D-finite Series modulo Primes

A D-finite series is a power series over the rational numbers that satisfies a linear differential equation with polynomial coefficients. Reducing a D-finite series modulo a prime number, one often obtains an algebraic power series in positive characteristic. For diagonals of multivariate rational functions, this was observed by Furstenberg in 1967; for hypergeometric functions, it can be deduced from results of Christol and was made precise in recent work of Vargas-Montoya. In these cases, one can consider the Galois group of the reduction over the field of rational functions.
In this talk, I will present many examples of D-finite series of various kinds for which we can (almost) compute these Galois groups for all but finitely many primes for which the reduction is defined: generating functions of binomial sums and of Apéry-like sequences, as well as some hypergeometric functions. Then, building on these observations, I will raise questions about the general behavior of these groups: Is there some uniformity in the Galois groups as the prime varies? Is there an object in characteristic zero governing their behavior, such as the differential Galois group of the minimal differential equation of the series? In the spirit of the Grothendieck p-curvature conjecture, we are looking for a local-global principle; however, we are concerned with only a single solution of a differential equation rather than a full basis of solutions.
This talk is based on joint work with X. Caruso, D. Vargas-Montoya and W. Zudilin.

Orateur: Florian Fürnsinn (Universität Wien)

5 TBA

Orateur: Stéphane Gaubert (INRIA)

15:40 Coffee Break

6 TBA

Orateur: Toshiki Nakashima (Sophia University, Tokyo)

ven. 27 novembre

09:00 Welcome coffee

7 TBA

Orateur: Travis Scrimshaw (Hokkaido University)

8 Mitigation of Quantum Simulations with BBGKY Hierarchies

Orateur: Theo Saporiti (Université Paris-Saclay)

11:40 Coffee Break

9 TBA

Orateur: Mahsa Shirmohammadi (IRIF, Université de Paris)

12:50 Lunch

10 TBA

Orateur: Darya Polyakova (Hamburg University)

11 Zamolodchikov Periodic Cluster Algebras

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. When the system is periodic, we call this Zamolodchikov periodicity. In this talk, we discuss the classification of all Zamolodchikov periodic cluster algebras, with connections to W-graphs, root systems, and maximal green sequences. We also discuss the introduction of the related Zamolodchikov fan, which has connections to other known geometric objects.

Orateur: Ariana Chin (UCLA)

15:40 Coffee Break

12 Geometric and Quasi-Thermodynamic Principles of Protein Dynamics

Predicting the three-dimensional folds of proteins is a celebrated problem at the crossroads of molecular dynamics and artificial intelligence. Our novel approach sets aside amino-acid sequence information and focuses instead on the geometry of protein backbones. An empirical density on SO(3) is determined from experimental protein structures, and a Boltzmann-like construction yields approximations to free energies directly from backbone geometry. This provides a new approach to predicting protein function from structure, with diverse applications across biophysics and biology. Two new physico-geometric quantities are introduced: Fragility, which measures the intrinsic susceptibility of a protein to perturbation, and Lability, which measures its response to a specific perturbation. A remarkable empirical law emerges: regions of high native backbone Fragility tend to exhibit greater Lability under structural perturbations, reminiscent of the fluctuation–dissipation theorem of statistical mechanics.

Orateur: Robert Penner (University of California & IHES)