TBA

• Gleb Koshevoy (IITP, Moscow)

Room: Centre de conférences Marilyn et James Simons

Higher-Bessel Kernel Polynomials, Multiset Eulerian Numbers and Beyond

• Vladimir Roubtsov (Université d'Angers)

Room: Centre de conférences Marilyn et James Simons 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.

TBA

• Lucia Di Vizio (Université de Versailles Saint-Quentin)

Room: Centre de conférences Marilyn et James Simons

Galois Groups of D-finite Series modulo Primes

• Florian Fürnsinn (Universität Wien)

Room: Centre de conférences Marilyn et James Simons 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.

TBA

• Stéphane Gaubert (INRIA)

Room: Centre de conférences Marilyn et James Simons

Geometric Crystal on Unipotent Variety and Crystal Base of Modified Quantum Algebra.

• Toshiki Nakashima (Sophia University, Tokyo)

Room: Centre de conférences Marilyn et James Simons Let $G$ be a simply connected, simple algebraic group over $\mathbb{C}$, and let $\mathcal{U}(G)$ be its unipotent variety, namely, the set of all unipotent elements. We introduce a unipotent crystal structure on $\mathcal{U}(G)$ and show that it gives rise to a geometric crystal structure on $\mathcal{U}(G)$. In the case of ${\rm SL}_2$, we define certain positive structure and a decoration function on $\mathcal{U}({\rm SL}_2)$ and show that the resulting tropicalized crystal is isomorphic to $B(\infty) \otimes T_\lambda \otimes B(-\infty)$, which is the crystal base of the $\lambda$-component $U_q(\mathfrak{sl}_2)a_\lambda$ of the modified quantum algebra $\widetilde{U}_q(\mathfrak{sl}_2) = \bigoplus_{\lambda} U_q(\mathfrak{sl}_2)a_\lambda$. Here $B(\pm\infty)$ is the crystal base of the nilpotent half subalgebra $U_q^{\mp}$ of the quantum algebra $U_q$.

TBA

• Travis Scrimshaw (Hokkaido University)

Room: Centre de conférences Marilyn et James Simons

Mitigation of Quantum Simulations with BBGKY Hierarchies

• Theo Saporiti (Université Paris-Saclay)

Room: Centre de conférences Marilyn et James Simons

TBA

• Mahsa Shirmohammadi (IRIF, Université de Paris)

Room: Centre de conférences Marilyn et James Simons

Poset Dimensions of Higher Bruhat Orders

• Darya Polyakova (Hamburg University)

Room: Centre de conférences Marilyn et James Simons Higher Bruhat orders $B(n,k)$ are certain generalizations of the weak order on the symmetric group, introduced originally by Manin-Schekhtman. The dimension of a poset $P$ is the smallest $d$ such that $P$ admits an order embedding into $R^d$. While the dimension computation for $B(n,1)$, the weak order case, is classical, the dimensions of general $B(n,k)$ were not previously studied. For $B(n,k)$ with inclusion order and $k$ fixed, we prove that the order of dimension growth is $n^k$: for lower bounds, we construct standard examples, and for upper bounds, we construct explicit coordinates, and further utilize an unexpected connection to Turan systems to reduce the number of coordinates. For $B(n,k)$ with the flip order, we prove that the order of dimension growth is at least $n^{2k+1}$, by constructing large standard examples of generalized Ziegler gaps. This is the quantitative incarnation of the principle that the flip order is way more complicated than the inclusion order. All the relevant definitions will be given in the talk.

Zamolodchikov Periodic Cluster Algebras

• Ariana Chin (UCLA)

Room: Centre de conférences Marilyn et James Simons 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.

Geometric and Quasi-Thermodynamic Principles of Protein Dynamics

• Robert Penner (University of California & IHES)

Room: Centre de conférences Marilyn et James Simons 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.