Pentagram Nets

• Anton IZOSIMOV (University of Glasgow)

Room: Centre de conférences Marilyn et James Simons In the definition of the pentagram map, both space and time variables are discrete, but play unequal roles. Unifying time and space leads to the notion of pentagram nets. A pentagram net is an assignment of points to the edges of a regular triangular lattice such that the points assigned to the edges of every elementary lattice rhombus are collinear. We show that pentagram nets are critical points of a simple area-like functional, yielding a variational description of the pentagram map and, consequently, an invariant presymplectic structure. In contrast to the known Poisson bracket for the pentagram map, arising from the trigonometric r-matrix, the new structure is related to the rational r-matrix. Moreover, we show that pentagram nets are integrable in a natural sense. The existence of integrals in this setting is a consequence of a high-school-level theorem about triangles. Restricting to the pentagram map, we recover the known integrals. Together with our new presymplectic structure, this yields integrability of the pentagram map directly on the space of planar polygons, without passing to twisted polygons or taking a quotient by the projective group. Time permitting, we may also discuss multidimensional pentagram nets and how they lead to dented pentagram maps, Grassmannian pentagram maps, and some new integrable pentagram maps; connections to Glick’s results on the limit point of the pentagram map; connections to Veselov’s work on solitons for matrix KdV; and connections to the Adler–Bobenko–Suris theory of 3D consistency.

The dSKP recurrence: combinatorial aspects and geometric systems

• Béatrice DE TILIÈRE (Université Paris Dauphine - PSL)

Room: Centre de conférences Marilyn et James Simons he subject of this talk is the discrete Schwarzian Kadomtsev–Petviashvili (dSKP) recurrence. We will first derive an explicit expression for its solution in terms of the initial data. More precisely, we will show that the solution is the ratio of two partition functions associated with a model of oriented dimers. Each of these partition functions involves cancellations, and we will derive an alternative expression without cancellations, involving trees and forests. Beyond its combinatorial interest, this expression is used to establish results on the singularities of the recurrence. We will then explain how the dSKP equation arises in a number of discrete geometric systems, including discrete holomorphic functions, polygon recutting, and the pentagram map. We will discuss the implications of our results for these systems. This is joint work with Niklas Affolter (TU Wien) and Paul Melotti (Université Paris-Saclay).

Collapsing of Polygons, Beyond Convexity and Pentagrams

• Jean-Baptiste STIEGLER (Université Paris-Saclay)

Room: Centre de conférences Marilyn et James Simons One of the first results from Schwartz about the pentagram map is the collapsing of convex polygons. Much later, Glick found a formula for the coordinates of the collapse point. This collapsing behavior reappears in other dyncamics, and the goal of this talk is to understand why. To do so, we'll consider the general setting of polygonal dynamics, defined for polygons in any projective spaces. We give a coordinate system on the moduli space, using generalized cross-ratios. If the dynamics admits a scaling symmetry this yields a formula for the generalized Glick operator. To do this, we use the formalism of infinitesimal monodromies, developped by Aboud and Izosimov. As a by-product, we automatically get preserved quantities (the monodromy invariants), which paves the way for integrability. Then we prove the collapsing phenomenon under a periodicity assumption on the moduli space, and present ideas to prove it in the quasi-periodic case.

Noncommutative Nonunitary Friezes

• Anna FELIKSON (Durham University)

Room: Centre de conférences Marilyn et James Simons Given a marked surface S, a frieze on S over a ring R is an assignment of elements of R to every arc of S, such that Berenstein - Retakh's exchange relations are satisfied. When R is commutative this simplifies to a usual frieze on a surface i.e. a homomorphism to nonzero elements of R; otherwise we call a frieze noncommutative. A frieze is called unitary when there exists a triangulation in which every arc is assigned with a unit of R. We investigate unitarity of integral friezes over matrices. This is an ongoing work joint with Zack Greenberg and Pavel Tumarkin.

The Inverse Spectral Map in the Pentagram Case

• Max GLICK (Google)

Room: Centre de conférences Marilyn et James Simons The dimer model on the torus is one of the various ways to include the pentagram map within a more general integrability framework. A central object in that theory is the spectral map which associates to each input configuration an algebraic curve and a divisor there-on. Following Goncharov-Kenyon 2013 and George-Goncharov-Kenyon 2023, I will describe the spectral map and its inverse in very concrete terms in the pentagram case.

Cluster Structures via Birational Poisson Maps

• Misha GEKHTMAN (University of Notre Dame)

Room: Centre de conférences Marilyn et James Simons Based on a joint works with M. Shapiro and A. Vainshtein and D. Voloshyn I will present an overview of recent results showing how birational Poisson maps intertwining Poisson-homogeneous structures on the same Lie group can be used to construct exotic cluster structures starting from “standard” ones.

Skew Pentagram Maps

• Max WEINREICH (CUNY Baruch College)

Room: Centre de conférences Marilyn et James Simons Skew pentagram maps act on polygons by intersecting diagonals of different lengths. They were introduced by Khesin-Soloviev in 2015. In this paper, we show that certain skew pentagram maps have exponential degree growth and no preserved fibration. We show that the dynamical degree of any equal-length pentagram map is 1, but that there are infinitely many skew pentagram maps with dynamical degree 4.

Area-Normalized Pentagram Map

• Michał ZWIERZYŃSKI (Warsaw University of Technology)

Room: Centre de conférences Marilyn et James Simons We study the Euclidean asymptotics of the area-normalized pentagram map through the spectral properties of Glick’s operator. For projectively periodic orbits, we establish general criteria leading either to flattening onto a spectral line or to bounded elliptic oscillations. For pentagons, we obtain a complete projective spectral classification in terms of a single invariant and recover, with explicit spectral directions and rates, Schwartz’s long-and-thin theorem for strictly convex pentagons. We show that closely related, though richer, spectral phenomena occur for hexagons as well. We also compute the Darboux--Schwartz return spectrum for Poncelet polygons using a Jacobi-function normal form and obtain corresponding asymptotic results. Finally, exact higher-order projective returns illustrate how the same spectral mechanisms extend to larger polygons.

T-systems and Toda Hamiltonians in type A

• Rinat KEDEM (University of Illinois Urbana-Champaign)

Room: Centre de conférences Marilyn et James Simons Glick’s cluster algebra for the pentagram map can be considered as the $A_\infty$ T-system cluster algebra wrapped on a special torus. A closely related integrable system is obtained by considering the T-system on a cylinder of radius 2. Its quantization is the relativistic quantum Toda chain, whose Hamiltonians commute with the time evolution operator for this cluster algebra, a generalized mutation sequence. I will explain how a different cluster algebra quiver, representing the T-system relations, can be quantized and the cluster variables used to construct the eigenfunctions of the Toda Hamiltonians. [Joint with P. Di Francesco]

Tau function and all that.

• Vladimir FOCK (IRMA, Strasbourg)

Room: Centre de conférences Marilyn et James Simons As a contribution to the rock soup I would like to show that the pentagram maps fits into the standard approach to integrable systems using Sato tau function. In particular I'll show that integrable equations like KP and pentagram map and maybe even the GL(1) automorphic forms from number theory can be viewed as ingredients of the same soup.

Prediction Normalization and Lasso Complexity, Geometry and Extremal Combinatorics

• Vladimir DRAGOVIĆ (UT Dallas)

Room: Centre de conférences Marilyn et James Simons We bridge statistics, convex and discrete geometry, extremal combinatorics, spectral graph theory, and coding theory in our study of complexity of the lasso regularization path under predictor normalization. We show that normalization imposes a new global restriction on the sign patterns encountered by the path: any two full-support patterns must agree in at least two coordinates. This reduces the highest-dimensional complexity of such paths in $\mathbb R^p$ to Kleitman's binary diameter problem of calculating $h_{p,r}$, the number of binary $p$-words that match at least at $r$ places, for $r=2$, and leads us to explicit finite dimensional upper bounds. We obtain upper and lower bounds for the total complexity of such paths and we show that the normalized lasso complexity remains exponential, while exhibiting combinatorial restrictions absent in the unrestricted problem. Furthermore, by using the concept of mutual coherence from statistics, we identify the data for which the lasso paths have $h_{p,r}$ as upper bounds for their highest dimension complexity for any given $r>2$, while $r=1$ case is identified with the unrestricted data studied by Mairal and Yu. This is based on a joint work with Borislav Gajić.

"Quantum'' Rational Numbers, q-Deformed Coxeter Friezes, and the Cross-Ratio

• Valentin OVSIENKO (CNRS - Université de Reims Champagne-Ardenne)

Room: Centre de conférences Marilyn et James Simons The classical cross-ratio "crosses" both classical and modern mathematics, linking geometry with combinatorics, dynamics, and mathematical physics. It provides natural coordinates on many moduli spaces. In this talk, I will introduce a new q-deformed, or ``quantum'' cross-ratio: a PSL(2,Z)-invariant of ordered quadruples of distinct points on the rational projective line. The construction is based on the theory of q-deformed rational numbers developed jointly with Sophie Morier-Genoud. I will discuss several properties of this new invariant, with its close connection to q-deformed Coxeter frieze patterns taking centre stage.

Movable n4 Configurations, Commuting Pentagram Maps, and Poncelet Grid

• Sergei TABACHNIKOV (Penn State University)

Room: Centre de conférences Marilyn et James Simons An n4 configuration is a collection of points and lines in the plane such that exactly 4 points lie on each line and exactly 4 lines pass through each point. The first such example, a 214 configuration, was found by F. Klein in 1878, but it was in the complex plane, and no projective transformation would make it real. The first real example of 214 configuration was discovered by B. Grünbaum and J. Rigby only in 1990. Their construction was based on the symmetries of a regular heptagon, and it was unknown whether it was rigid (modulo projective transformations). I shall present constructions of movable n4 configurations based on Poncelet polygons and relate this subject with commuting deeper diagonal pentagrams maps, the Poncelet grid, and integrable billiards. This is joint work with L. Berman, G. Gévay, and J. Richer-Gebert (recently published in Focus of Math., Sigma).