**Berenstein, Arkady (Oregon),
Noncommutative surfaces, clusters, and their symmetries
**

The aim of my talk (based on joint work in progress with Min Huang and Vladimir Retakh) is to introduce and study certain noncommutative algebras A for any marked surface. These algebras admit noncommutative clusters, i.e., embeddings of a given group G which is either free or one-relator (we call it triangle group) into the multiplicative monoid A^{¡ß}. The clusters are parametrized by triangulations of the surface and exhibit a noncommutative Laurent Phenomenon, which asserts that generators of the algebra can be written as sums of the images of elements of G for any noncommutative cluster. If the surface is unpunctured, then our algebra A can be specialized to the ordinary quantum cluster algebra, and the noncommutative Laurent Phenomenon becomes the (positive) quantum one.

It turns out that there is a natural action of a certain braid-like group Br_{A} by automorphisms of G on each cluster in a compatible way (this is, indeed, the braid group Br_{n} if the surface is an unpunctured disk with n+2 marked boundary points). If surface is punctured, the algebra A admits a family of commuting automorphisms which will give new clusters and new "tagged" noncommutative Laurent Phenomena.

There are important elements in A assigned to each marked point, which we refer to as noncommutative angles (or h-lengths). They belong to the group algebra of each cluster group and are invariant under all noncommutative cluster mutations. This eventually gives rise to noncommutative integrable systems on unpunctured cylinders and other surfaces which, in particular, recover the ones introduced by Kontsevich in 2011 together with their Laurentness and positivity.

**
Bobrova, Irina (Max-Planck-Institute for Mathematics in the Sciences),
Non-Abelian ODEs and O¦¤Es
**

In this series of lectures, we will discuss some methods for the deriving and classification of such equations as well as investigation their integrability.

**Carillo, Sandra
(Roma), Backlund transformations and non-Abelian nonlinear evolution equations
**

Backlund transformations are well known to represent a powerful tool in investigating nonlinear differential equations. In particular, we are concerned about so-called soliton equations since they admit soliton type solutions. The aim of the present study is twofold since, on one side, we consider the connections which can be established and the induced structural properties; on the other side, we consider Backlund transformations as a tool to construct solutions, admitted by nonlinear evolution equations. Hence, first of all, we consider the links which can be established among different nonlinear evolution equations via Backlund transformations. Accordingly, a net of connections among different nonlinear evolution equations is depicted in a Backlund Chart, as we term such a net of links. The attention is focussed on third order, nonlinear evolution questions in particular, the comparison between the commutative (Abelian) and the non-commutative cases is analyzed. Notably, a richer structure can be observed when the commutativity condition is removed. Then, via Backlund transformations, solutions of matrix modified KdV equation can be constructed. Finally, some new results as well as some problems, currently under investigation, concerning fifth order nonlinear evolution equations are mentioned.

Most of the presented results are part of a joint research project with Cornelia Schiebold, Sundsvall University, Sweden which involves also, in alphabetical order, M. Lo Schiavo, Rome, E. Porten, Sundsvall, and F. Zullo, Brescia.

** Gilson, Claire
(Glasgow), Non-commutative Pfaffians
**

Solutions to a number of integrable systems an be expressed in the form of Pfaffians. In this talk we shall investigate forms for non-commutative Pfaffians via the quasi-determinant formalism. We shall explore the possibility of constructing new integrable systems employing these non-commutative Pfaffians. Among the equations we shall explore are the BKP equation, the Novikov-Veselov equation and the Hirota-Ohta coupled soliton equations.

Generalization of soliton theory and integrable systems to their noncommutative counterparts is an interesting topic. Some classical integrable systems have been generalized to their noncommutative versions and their integrability has been investigated. Moreover, as is known that integrable systems are closely related to other topcis such as orthogonal polynomials and combinatorics. Their noncommutative generalization is of great research interest too. KP equation is one of the most fundamental among many soliton equations. Its generalizations and extensions have been paid much attention to. In this talk, I will talk about how to construct the noncommutative extended KP equation and the noncommutative extended modified KP equation by using variation of parameter. As a consequence, two types of quasideterminant solutions are presented for the two noncommutative extended integrable systems respectively. In addition, Miura transformations between them are established successfully as well.

** Roubtsov, Volodya (Angers),
¡ÔPainleve equations -- different facets of non-commutativity¡Õ
**

I propose an overview my results on different ¡Ônon-commutative aspects¡Õ Painleve equations and some related systems. I shall describe various non-commutative models associated with different Painleve and corresponding toolbox and resulted properties and applications. Our methodology includes Gelfand-Retakh quasidetrminant technics for Painleve II and IV, isomonodrtomy representations for the matrix Takasaki Hamiltonian Calogero-Painleve systems and some analogues of Ruijsenaars duality.

In 2009, Etingof--Nikshych--Ostrik formulated nilpotency and solvability for fusion categories. If we define nilpotency and solvability for finite quantum groups via this language, nilpotency does not imply solvability in general. In 2016, Cohen--Westreich proposed definitions of nilpotency and solvability via integrals. Their definitions are satisfactory in that nilpotency implies solvability and the analogue of Burnside's $p^aq^b$ theorem holds.

In this talk, we introduce Cohen--Westreich's definitions for solvability and nilpotency and give examples of nilpotent finite quantum groups. We also give a direct computation of $R$-matrices for some finite quantum groups if time allows. This talk is based on the joint work with Gerard Glowacki (Nagoya University) and Masamune Hattori (Nagoya University).

[1] M. S., Discrete field theory: symmetries and conservation laws, Math. Phys. Anal. Geom. 26:19 (2023).

**Sharygin, Georgy
(Lomonosov Moscow State University), Argument shift method in the algebra Ugl(n)
**

I will describe a construction that allows one to obtain commutative subalgebras in Ugl(n). This method is analogous to the classical "shift of the argument" construction on symmetric algebras.

Mishchenko-Fomenko theorem, often referred to as the argument shift method, constructs large Lie-Poisson commutative subalgebras in the symmetric algebra of a Lie algebra. Commutative lifts of these algebras in the universal enveloping algebra are called quantum argument shift algebras and established in many cases. We will explain that argument shift operators themselves can be quantised and they generate the quantum argument shift algebras for general linear Lie algebras. This is joint work with Georgy Sharygin (Moscow State University). Georgy Sharygin will talk about quantum derivations and I will focus on the theorem and its application.

We investigate a real symmetric $\Phi^{4}$ matrix model as a Grosse and Wulkenhaar type noncommutative scalar field theory. Our study has revealed that the partition function of the real symmetric $\Phi^{4}$ matrix model corresponds to a zero-energy solution of the Schrodinger-type equation with Calogero-Moser Hamiltonian. Additionally, we find that the partition function satisfies a system of partial differential equations generated by Virasoro algebra (Witt algebra).

In a noncommutative scalar $\Phi^{4}$ theory, a field is discretized and a matrix appears. Therefore, a noncommutative scalar $\Phi^{4}$ theory can be replaced by a matrix model. There is one important problem with a noncommutative scalar $\Phi^{4}$ theory. The UV/IR mixing problem occurs and renormalization is impossible without ingenuity when considering a noncommutative field theory. To avoid the UV/IR mixing problem, Grosse and Wulkenhaar adjusted the action of a scalar $\Phi^{4}$ theory on the Moyal space by adding a harmonic oscillator potential. Incidentally, where the Moyal space is a kind of noncommutative space. This model is called Grosse-Wulkenhaar $\Phi^{4}$ model ($\Phi^{4}$ matrix model).

This talk is based on collaborative work with Professor Harald Grosse, Professor Akifumi Sako, and Professor Raimar Wulkenhaar.

This is the joint work with Akifumi Sako (Tokyo University of Science). In this talk, we will explain that the noncommutative $G_{2,4}\left(\mathbb{C}\right)$ as deformation quantization is constructed by using the construction method proposed by Hara-Sako. Here $G_{2,4}(\mathbb{C})$ is the simplest complex Grassmannian which is not $\mathbb{C}P^{N}$. It is known that the construction method by Hara-Sako is the tool to give a deformation quantization with separation of variables for a locally symmetric K\"{a}hler manifold. Since complex Grassmannians are locally symmetric K\"{a}hler manifolds, we can construct the deformation quantization for $G_{2,4}(\mathbb{C})$ by using this method. To give a deformation quantization via the construction method by Hara-Sako, It is necessary to determine a star product by solving the recurrence relations which have the coefficients of a star product as the solutions. We show that for $G_{2,4}(\mathbb{C})$ case, its recurrence relations can be solved by using Fock representation. From the obtained general terms of the recurrence relations, we also give the explicit star product with separaion of variables on $G_{2,4}(\mathbb{C})$.

**Li, Shangshuai
(Shanghai University), Solutions to the SU(N) self-dual Yang-Mills equation
**

In this talk, we aim to derive solutions for the SU(N) self-dual Yang-Mills (SDYM) equation by Cauchy matrix approach. A set of non-commutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

**Amano, Markus
(Yamagata University), Non-Linear Deformed Holographic Abelian-Higgs Vortices
**

In order to study strongly coupled $U(1)$ vortex-vortex interactions, employed a (3+1)D U(1) phi^4 scalar field in an AdS/CFT framework.

By introducing a U(1) symmetry-breaking phase, we can admit line vortices.

In this phenomenological study, we investigate the binding energy of a pair of point vortices in a dual (2+1)D theory which correspond to line vorticies in the bulk.

We find that the binding dynamics of a pair of vortices, depending on the temperature and distance of separation, are intricate.