Invited Talks

• Mikhail Bershtein
• Xiang-Ke Chang
• Cesar Cuenca
• Evgeny Feigin
• Bin Gui
• Babak Haghighat
• Rinat Kedem
• Karol Kozlowski
• Jianrong Li
• Oleg Lisovyi
• Qing Ping Liu
• Kang Lu
• Andrei Mironov
• Hajime Nagoya
• Satoshi Nawata
• Gleb Nenashev
• Tomasz Przezdziecki
• Leonid Rybnikov
• Andrei Smirnov
• Changjian Su
• Trung Vu
• Oksana Yakimova

Posters

• Jiakang Bao
• Nikita Belousov
• Siqi Chen
• Andrei Grigorev
• Yuan Miao
• Maria Onufrienko
• Rahul Singh
• Jinfeng Song
• Pengyu Sun
• Хiaolu Yue
• Siyao Yin
• Weinan Zhang

Invited Talks

• Mikhail Bershtein (SISSA, Italy)
  Title: Cyclically shifted $U_q({\widehat{𝔰𝔩}}_N)$ algebra and cluster realizations (PDF)
  Abstract: Cyclically shifted $U_q({\widehat{𝔰𝔩}}_N)$ algebras were introduced by Bourgine and Jeong in the context of gauge theories with surface defects. It turns out that they can also be used to study the standard, non-shifted algebras; in particular, they lead to explicit formulas for $q$–Heisenberg realizations, $R$–matrices, and quantum evolution operators. The main source of these formulas is a cluster realization of the algebra. This is based on joint work with Bourgine and Shiraishi.
• Xiang-Ke Chang (Chinese Academy of Sciences, Beijing, China)
  Title: Hyperelliptic curves, continued fractions and Somos–Gale–Robinson recurrences
  Abstract: Somos-4 and Somos-5 are bilinear recurrence relations that can obtained as reductions of the discrete KP equation. They exhibit interesting integrality, behind which it is the Laurent phenomenon appearing as a key property of cluster variables in Fomin and Zelevinsky’s cluster algebras. In this talk, we present how to derive explicit solutions of Somos-4 and Somos-5 in terms of Hankel determinants based on elliptic curves and continued fractions, from which the Laurent properties straightforwardly follow. Moreover, we also talk about some recent results on a unified approach for solving initial value problems for a class of three-term Gale–Robinson recurrences, including Somos-4 and Somos-5, based on their Lax pairs.
• Cesar Cuenca (Ohio State University, USA)
  Title: $N$-particle ensembles at high temperature via Dunkl and Cherednik operators (PDF)
  Abstract: Following a discussion of the continuous Gaussian beta ensemble and the classical Law of Large Numbers (LLN), we switch to the setting of discrete-space particle systems. By using Fourier transforms based on Jack symmetric polynomials, we study d iscrete $N$*–particle ensembles in the regime where the inverse temperature parameter tends to zero, simultaneously as the number of particles in the system tends to infinity. We prove the LLN and characterize the limiting measure in terms of a moment problem. For fixed-time distributions of the discrete beta-Dyson Brownian motion, we calculate the densities of the limiting measures and express them in terms of the zeroes of certain entire functions or the eigenvalues of certain Jacobi operators. This talk is based on joint works with Florent Benaych-Georges, Vadim Gorin and Maciej Dolega.
• Evgeny Feigin (Tel Aviv University, Israel)
  Title: Type $A$ algebraic coherence conjecture
  Abstract: The coherence conjecture of Pappas-Rapoport, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to line bundles on spherical Schubert varieties in affine Grassmannians and to line bundles on unions of Schubert varieties in affine flag varieties. We will formulate an algebraic version of the geometric construction relating certain Demazure modules on one side to sums of Demazure modules on the other side. Our construction works only in type $A$, but it is applicable to a much wider class of representations than those popping up in the geometric coherence conjecture. It also leads to a more general conjecture for representations of finite ${𝔰𝔩}_n$ , which we will discuss in detail.
• Bin Gui (YMSC, Tsinghua University, China)
  Title: Conformal blocks and associative algebras in logarithmic conformal field theory (PDF)
  Abstract: A main theme in the development of the theory of vertex operator algebras (VOAs) is the factorization property. If $V$ is a $C_2$-cofinite rational VOA (which corresponds to the chiral algebra of a rational CFT in physics), the factorization property — saying roughly that higher genus conformal blocks can be decomposed into lower genus ones — was recently completely proved. Its low genus special cases (such as Zhu's modular invariance theorem in 1996 and Huang's associativity theorem for intertwining operators in 2005) are crucial to the understanding of the representation theory of $V$.
  This talk focuses on $C_2$-cofinite VOAs that are not necessarily rational. Such VOAs correspond to chiral algebras of finite-type logarithmic CFTs. Since their representation categories are not necessarily semisimple, the study of their conformal blocks differ significantly from the rational case. In this talk, I will present my recent joint work with Hao Zhang on the complete proof of the factorization property for conformal blocks of such VOAs. We will also discuss the natural associative algebras in log CFT that play the role of Zhu's algebra in rational CFT. In particular, we show that the $0$-th order Hochschild cohomology of that algebra is isomorphic to the space of torus conformal blocks. This is based on our works arXiv:2503.23995 and arXiv:2508.04532.
• Babak Haghighat (YMSC, Tsinghua University and BIMSA, China)
  Title: Quantum flat connections, KZ equations, and integrability (PDF)
  Abstract: $N=2$ supersymmetric Yang-Mills theories are described in terms of a Hitchin system over a Riemann surface $C$. Focusing on strongly coupled Argyres-Douglas theories, we show that the corresponding flat bundle over $C$ can be quantized such that the resulting quantum flat connection is integrable. For ${𝔰𝔩}_2$, the quantum connection takes values in ${𝔤𝔩}_2(A)$ where $A$ is an associative algebra which we explicitly describe for the cases of Painlevé I, II and IV. Moreover, we find that the quantum connection is equivalent to irregular versions of Knizhnik-Zamolodchikov (KZ) connections. Utilizing a suitable gauge transformation, one can show that the corresponding KZ equations give rise to BPZ equations.
• Rinat Kedem (University of Illinois at Urbana-Champaign, USA)
  Title: Affine Laumon functions and $Q$–system quivers (PDF)
  Abstract: We discuss some new constructions of Toda and affine Laumon functions using affine versions of the $Q$ system quantum cluster algebras. This is based on work with Bershtein, Bourgine, Di Francesco, Pasquier and Shiraishi.
• Karol Kozlowski (École Normale supérieure de Lyon, France)
  Title: Free energy of the classical Toda chain in a generalised Gibbs ensemble
  Abstract: Classical integrable systems exhibit a tower of conserved quatities having local densities built out of traces of powers of the model's Lax matrix. It is argued that this local structure, absent in general models, leads to peculiar thermalisation properties of integrable systems. In particular, their equilibrum properies are expected to be grasped by so-called Generalised Gibbs measures. The study of Generalised Gibbs ensembles' partition functions was initiated by Spohn. He focused on the $N$–particle Toda chain and managed to describe the $N\to +\infty$ limiting distribution of the Eigenvalues of the model's Lax matrix under a Generalised Gibbs distribution. He was also able to conjecture an expression for the associated free energy. A thorougher description of the Gibbs measure, in the form of a large deviation principle with an explicit rate function, was later conjectured by Doyon and, independently, Spohn.
  In this talk, after reviewing the various motivations for the study of the problem, I will explain how one can establish, on rigorous grounds, the explicit form of the Generalised Gibbs ensemble Toda chain rate function and free energy by using the separated variables representation of the model's partition function. This result constitutes the first step towards studying the thermodynamic limit of the model's dynamical correlation functions in such a setting. This is a joint work with T. Grava, A. Guionnet and A. Little.
• Jianrong Li (University of Vienna, Austria)
  Title: Tropical symmetries of cluster algebras (PDF)
  Abstract: In this talk, I will present joint work with James Drummond and Ömer Gürdoğan on tropicalizations of quasi-automorphisms of cluster algebras.
  We study tropicalizations of quasi-automorphisms of cluster algebras and show that their induced action on $g$–vectors can be realized by tropicalizing their action on the homogeneous $X$-variables ($\hat{y}$–variables) of a chosen initial cluster. This perspective allows us to interpret the action on $g$–vectors as a change of coordinates in the tropical setting.
  Focusing on Grassmannian cluster algebras, we analyze tropicalizations of quasi-automorphisms in detail. In particular, we derive tropical analogues of the braid group action and the twist map, both on $g$–vectors and on tableaux. We also introduce the notions of unstable and stable fixed points for quasi-automorphisms.
  As an application, we show that the number of prime non-real tableaux with a fixed number of columns in $\mathrm{SSYT}(3,[9])$ and $\mathrm{SSYT}(4,[8])$, arising from the braid group action on stable fixed points, is governed by Euler’s totient function. Furthermore, we apply our results to scattering amplitudes in physics, providing a new interpretation of the square root appearing in the four-mass box integral in terms of stable fixed points of quasi-automorphisms of the Grassmannian cluster algebra $C[\mathrm{Gr}(\mathrm{4,8})]$.
• Oleg Lisovyi (Université de Tours, France)
  Title: Connection problem for Painlevé II-tau function (PDF)
  Abstract: I will start by describing the monodromy manifold associated with the general (inhomogeneous) Painlevé-II equation and recalling Kapaev’s description of the asymptotics of generic Painlevé-II transcendents in terms of Stokes data. I will then present a conjectural solution of the corresponding connection problem for the generic Painlevé-II tau function and discuss possible approaches to its proof.
• Qing Ping Liu (China University of Mining and Technology, Beijing, China)
  Title: Bäcklund transformation approach to the modified Camassa-Holm equation (PDF)
  Abstract: The Modified Camassa-Holm (MCH) equation, an integrable equation with peakon solutions, has been the subject of extensive research from both the community of partial differential equations and the community of integrable systems. In this presentation, a Bäcklund transformation will be constructed, which involves the alteration of both dependent and independent variables for the MCH equation. In addition, the corresponding nonlinear superposition formula and Darboux transformation for this equation will be discussed. The applications of these results will be considered.
• Kang Lu (SUSTech, Shenzhen, China)
  Title: Drinfeld presentations of twisted Yangians and their applications (PDF)
  Abstract: In recent joint work with Weiqiang Wang and Weinan Zhang, we established Drinfeld-type current presentations for twisted Yangians of type AI and beyond. These presentations are realized via Gauss decomposition and the degeneration of affine iquantum groups. In this talk, I will discuss recent progress in the study of these twisted Yangians (also known as iYangians) and explore their applications. Key topics will include their coideal structure, shifted generalizations, and connections to finite $W$–algebras of classical types, as well as affine Grassmannian islices (time permitting). This talk is based on joint works with Yung-Ning Peng, Lukas Tappeiner, Lewis Topley, Weiqiang Wang, Alex Weekes, and Weinan Zhang.
• Andrei Mironov (Sobolev Institute of Mathematics and Novosibirsk State University, Russia)
  Title: Integrable geodesic flows on cones over Riemannian manifolds (PDF)
  Abstract: We study the behavior of geodesics on cones over arbitrary ${ℂ}^3$–smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for cone generatrices. We also prove that the geodesic flow restricted to the open dense subset of the cotangent bundle corresponding to all non-radial trajectories is completely integrable. This investigation is inspired by our recent results on billiards inside cones over manifolds where similar results hold true. The talk is based on a joint work with Siyao Yin.
• Hajime Nagoya (Kanazawa University, Japan)
  Title: On quantum Sasano systems of type $D_{2n+2}^{(1)}$ (PDF)
  Abstract: In this talk, we propose a quantum Hamiltonian for the higher-order Painlevé system of type $D_{2n+2}^{(1)}$, also known as the Sasano system, which reduces to the sixth Painlevé equation for $n=1$. We establish the invariance of this quantum Hamiltonian under the action of the affine Weyl group $W(D_{2n+2}^{(1)})$.
  Furthermore, we investigate the time-dependent, non-stationary Schrödinger equation associated with this quantum system. We establish the realization of the Bäcklund transformations on the wave functions utilizing fractional calculus, incorporating spatial gauge transformations and Riemann-Liouville fractional integrals. We identify the exact parameter conditions required for the existence of polynomial solutions. Finally, we demonstrate that shift operators yield an infinite number of solutions to the quantum Sasano system.
• Satoshi Nawata (Fudan University, Shanghai, China)
  Title: Branes and representations of DAHA $C^{\vee }{C}_1$: affine braid group action on category
  Abstract: We study the representation theory of the spherical double affine Hecke algebra (DAHA) of $C^{\vee }{C}_1$, using brane quantization. By showing a one-to-one correspondence between Lagrangian $A$–branes with compact support and finite-dimensional representations of the spherical DAHA, we provide evidence of derived equivalence between the $A$–brane category of $\mathrm{SL}(2,ℂ)$–character variety of a four-punctured sphere and the representation category of DAHA of $C^{\vee }{C}_1$. The $D_4$ root system plays an essential role in understanding both the geometry and representation theory. In particular, this $A$–model approach reveals the action of an affine braid group of type $D_4$ on the category. As a by-product, our geometric investigation offers detailed information about the low-energy effective dynamics of the $\mathrm{SU}(2)$ $N_f=4$ Seiberg-Witten theory.
• Gleb Nenashev (Saint Petersburg State University, Russia)
  Title: Algebras generated by the Bott-Chern forms on flag varieties and graphs (PDF)
  Abstract: We introduce the algebra generated by the Bott-Chern forms of an arbitrary flag variety. For the case of complete flag varieties these algebras were introduced by V.I.Arnold and later they were generalized to the algebras which are now known as graphical Zonotopal algebras. We also define a promising family of algebras that generalizes both classes. These algebras are indexed by pairs, a flag variety and a graph. Some open questions will be presented.
• Tomasz Przezdziecki (University of Vienna, Austria)
  Title: $q$–character theory for quantum symmetric pairs (PDF)
  Abstract: It is well known that quantum affine algebras admit three distinct presentations (Kac-Moody, new Drinfeld and RTT). Relatively recently, the same has been shown to hold for a broad family of quantum affine symmetric pairs. In particular, a Drinfeld-type presentation, due to (among others) Lu and Wang, is a new and exciting development. The focus of my talk will be the relationship between the usual Drinfeld presentation of quantum affine algebras and the Lu-Wang presentation of their coideal subalgebras. Remarkably, both Drinfeld presentations exhibit large commutative subalgebras, which are of particular interest to representation theory and integrable systems. More specifically, I will present several results concerning the properties of the generators of these commutative subalgebras, including their behaviour under inclusion and coproduct, as well as their spectra on finite-dimensional representations. These results will then be used to construct an analogue of $q$-character theory for quantum symmetric pairs. Finally, I will discuss $q$-character formulae for evaluation modules of type AI quantum symmetric pairs.
• Leonid Rybnikov (University of Montreal, Canada)
  Title: Bethe subalgebras in Yangians and wonderful models for toric arrangements
  Abstract: Bethe subalgebras in Yangians are commutative subalgebras responsible for the higher Hamiltonians of the XXX Heisenberg spin chain and its generalizations (related to any semisimple Lie algebra $𝔤$). A natural question is how these commutative subalgebras depend on their parameters, and what their degenerations are. The parameter space of these Bethe subalgebras is the complement of a toric arrangement corresponding to the root system of $𝔤$. I will explain how this family extends regularly to the minimal wonderful model of this arrangement, in the sense of De Concini and Gaiffi. The subalgebras corresponding to boundary points of the compactification can be explicitly described in terms of Bethe subalgebras in smaller Yangians and shift-of-argument subalgebras (or, quantized Mishchenko-Fomenko subalgebras) in universal enveloping algebras. This is a joint work with Aleksei Ilin.
• Andrei Smirnov (University of North Carolina, USA)
  Title: Quantum K-theory of Grassmannians and 5-vertex models
  Abstract: In this talk, I discuss the equivalence between the Bethe algebra of the trigonometric 5-vertex model and the quantum K-theory of Grassmannians. The identification is based on the analysis of solutions to the quantum $q$–difference equations for these spaces and their asymptotics at $q=1$.
• Changjian Su (YMSC, Tsinghua University, China)
  Title: Geometric realizations of quantum symmetric pairs
  Abstract: Geometric realizations of quantum algebras serve as important tools for studying their representations. In this talk, I will survey some recent developments in the geometric study of quantum symmetric pairs, from both the Higgs branch and the Coulomb branch perspectives.
• Trung Vu (YMSC, Tsinghua University, China)
  Title: Rank recursion for $q$-Whittaker and Macdonald operator (PDF)
  Abstract: In this talk, we introduce and prove a set of rank recurrence relations for $q$-Whittaker and Macdonald operators. We also show an explicit expression for the $k$-th power $q$-Whittaker operator in terms of $q$-deformed binomial probability distribution. These rank recurrence relations once found can be used to explore a family of expectations called "$q$-moments" of the $q$-deformed Totally Asymmetric Simple Exclusion Process ($q$-TASEP).
• Oksana Yakimova (University of Jena, Germany)
  Title: Near-derivations and their applications to Lie algebras
  Abstract: We extend E.B. Vinberg’s theory of quasi-derivations of algebras to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. While basic results apply to arbitrary algebras, our main focus lies on the Poisson algebra $(S(𝔮),\{,\})$ associated to a Lie algebra $𝔮$. It will be shown that a near-derivation $D$ of $(S(𝔮),\{,\})$ naturally gives rise to a pencil of compatible Poisson structures on the dual of $𝔮$. Moreover, using $D$ one may naturally construct a Poisson-commutative subalgebra of $S(𝔮)$.
  Special emphasis is placed on near-derivations arising from $𝔮$ itself, which lead to both classical and novel families of compatible Poisson brackets. At the end of the talk, we will compare properties of near-derivations of $𝔮$ with Nijenhuis operators in $𝔤𝔩(𝔮)$, highlighting parallels between these two frameworks.
  This is a joint work with D. Panyushev

Posters

• Jiakang Bao (University of Tokyo, Japan)
  Title: Quiver (BPS) algebras and crystal representations
  Abstract: I will discuss some recent progress on the quiver algebras that encode the information of BPS counting in terms of the combinatorial modules (aka crystal representations). In particular, this includes theories arised from toric Calabi-Yau threefolds and fourfolds. I will mention their properties such as the behaviour under Seiberg duality/mutations and connections to integrable systems.
• Nikita Belousov (BIMSA, China)
  Title: Eigenfunctions of BC Toda chain
  Abstract: BC Toda chain is an integrable system of particles on a line with reflection at the boundary. In recent work with S. Derkachov and S. Khoroshkin, we derived explicit formulas for eigenfunctions of the corresponding quantum system. I will present our results and explain a diagram technique used to prove several key properties of these eigenfunctions. Based on [arXiv:2603.16380, arXiv:2603.16387].
• Siqi Chen (Qilu University of Technology, Jinan, China)
  Title: Painlevé transcendents and recurrence relations for orthogonal polynomials with discontinuous Gaussian-type weights
  Abstract: We study orthogonal polynomials associated with the weight $w(x;s)={\mathrm{e}}^{-s{x}^2}|x-s{|}^{\alpha }(A+B\theta (x-s))$, $x\in ℝ$, where $s,\alpha >0$, $A\ge 0$, and $A+B\ge 0$. Using the ladder operator method, we derive key equations for the three-term recurrence coefficients ${\alpha }_n(s)$ and ${\beta }_n(s)$, including first-order (Toda evolution) and second-order differential equations. We also show that these coefficients satisfy the discrete Painlevé equation $\mathrm{d}-\mathrm{P}(A_2^{(1)}/E_6^{(1)})$. For the special case $A=0,B=1$, we obtain the asymptotic behavior of ${\alpha }_n(s)$ and ${\beta }_n(s)$, as $n\to \infty$, and construct the associated Heun equations and monic orthogonal polynomials for the weight $w(x)=x^{\alpha }{\mathrm{e}}^{-s{x}^2-2s^{2}x}$, where $x,\alpha ,s>0$.
• Andrei Grigorev (NRU HSE, Moscow, Russia)
  Title: Monodromy-free Shrodinger operators and master functions for affine ${𝔰𝔩}_2$.
  Abstract: A master function is a multivariable function constructed from a Kac–Moody Lie algebra and a set of integral dominant weights for this algebra. Master functions of type $A$ help relate the Bethe ansatz for Gaudin model to monodromy-free differential operators and facilitate the study of both sides of this correspondence. In our work, we clarify the relation between master functions of type $A_1^{(1)}$ and certain families of monodromy-free Schrödinger operators. Poster is to be based on joint work with E. Mukhin.
• Yuan Miao (Kavli IPMU, University of Tokyo, Japan)
  Title: Hidden Onsager symmetry in XXZ model at root of unity ${𝔰𝔩}_2$.
  Abstract: Symmetries play an important role in quantum many-body systems. We report a rigorous proof for a hidden Onsager algebraic symmetry (i.e. infinite-dimensional Lie algebra) of the 6-vertex model (XXZ model) at root of unity values of anisotropy. We use Baxter's ${\tau }_2$ transfer matrix as the generating functions for Onsager generators and prove the Dolan-Grady relations making use of Yang-Baxter-like relations. As a by-product, we find a lattice construction of topological defect lines of $\mathrm{TY}({\mathbb{Z}}_N)$ category from the ${\tau }_2$ transfer matrix as an automorphism of the Onsager algebra.
• Maria Onufrienko (Lomonosov Moscow State University, Moscow, Russia)
  Title: Semiglobal classification of corank-1 singularities in integrable Hamiltonian systems with three degrees of freedom
  Abstract: We study real-analytic integrable systems with three degrees of freedom [2] near a compact two-dimensional singular orbit of corank 1. Under generic non-degeneracy conditions the local momentum map is reduced to one of the standard normal forms depending on the resonance order d and codimension <3. We establish a semiglobal classification [1,2] of the corresponding singular Liouville fibers: there are 9 supercritical cases (the orbit coincides with the fiber), 10 subcritical cases with a unique semiglobal type, and the hyperbolic swallowtail yields exactly two topologically distinct semiglobal singularities. The number of regular Liouville tori in the chambers of the bifurcation diagram is computed.
  1. [1] E.A. Kudryavtseva, M.V. Onufrienko, Classification of singularities of smooth functions with a finite cyclic symmetry group, Russ. J. Math. Phys. 30:1 (2023), 76–94
  2. [2] A.Z. Ali, V.A. Kibkalo, E.A. Kudryavtseva, M.V. Onufrienko, Bifurcations in integrable Hamiltonian systems near corank-one singularities, Differ. Eq. 60:10 (2024), 1311–1368.
• Rahul Singh (YMSC, Tsinghua University, Beijing, China)
  Title: $q$–opers and Bethe Ansatz for open spin chains
  Abstract: We initiate a geometric study of open XXZ spin chains using the framework of $q$–opers. By introducing reflection-invariant $q$–opers, we derive generalized QQ-systems and Bethe Ansatz equations associated with integrable models with open boundary conditions. In the $\mathrm{GL}(2)$ case, our construction reproduces the known Bethe equations for open XXZ chains with diagonal and generic boundary conditions. For higher rank, however, the construction leads to a new family of generalized open-chain Bethe equations.
• Jinfeng Song (The Hong Kong University of Science and Technology, Hong Kong, China)
  Title: Braid group symmetries on Poisson homogeneous spaces
  Abstract: The celebrated De Concini–Kac forms of quantum groups provide a quantization of the dual Poisson–Lie group associated to a complex semisimple Lie group. The i-quantum groups arising from quantum symmetric pairs are vast generalizations of quantum groups and have seen burgeoning development in recent years. In this poster, I will introduce De Concini–Kac forms for i-quantum groups and explain that they yield quantizations of certain Poisson homogeneous spaces. I will also show that these integral forms admit braid group symmetries, and that by taking semiclassical limits, one obtains braid group actions on the corresponding Poisson homogeneous spaces. These symmetries encompass many examples of interest in mathematical physics.
• Pengyu Sun (Shanghai University, Shanghai, China)
  Title: Soliton solutions of the cross-ratio equation on the half-plane
  Abstract: We construct soliton solutions of the cross-ratio equation defined on the half-plane using the dressing method. By assigning integrable boundary conditions, we derive explicit soliton configurations that satisfy the integrability structure of the equation. Furthermore, we establish a direct relationship between the dressing technique and the Riemann-Hilbert formalism.
• Хiaolu Yue (City University of Hong Kong, Hong Kong, China)
  Title: $(1,𝑚)$-type biorthogonal polynomials and discrete Painlevé-type equations
  Abstract: This paper is devoted to investigating the link between $(1,m)$–type biorthogonal polynomials (bi-OPs) and discrete Painlevé-type equations. On one hand, a structure relation is established for a family of bi-OPs with the semiclassical weight function $\omega (x,t_1,t_2,\alpha )=x^{\alpha }{e}^{-t_1{x}^m-t_2{x}^{2m}}$ on $x\in (0,\infty )$, where $t_1\in ℝ$, $t_2>0$, $\alpha >-1$ and $m$ is a fixed positive integer, through the biorthogonality condition. A family of generalized discrete Painlevé-II (${\text{d-P}}_{\text{II}}$) is subsequently derived by utilizing the compatibility condition between the recurrence relation and the structure relation. On the other hand, we also investigate a symmetric weight function, i.e. $\omega (x,t_1,t_2,\alpha )=|x{|}^{2\alpha +1}{e}^{-t_1{x}^{2m}-t_2{x}^{4m}}$ on $x\in ℝ$, where $t_1\in ℝ$, $t_2>0$, $\alpha >-1$ and$n$ is a fixed positive odd integer, leading to a family of generalized asymmetric discrete Painlevé-I (${\text{d-P}}_{\text{I}}$). Furthermore, a unified Miura transformation between the generalized ${\text{d-P}}_{\text{II}}$ and generalized asymmetric ${\text{d-P}}_{\text{I}}$ is discovered.
• Siyao Yin (Sobolev Institute of Mathematics, Novosibirsk, Russia)
  Title: Integrable Birkhoff billiards inside cones
  Abstract: We study Birkhoff billiards inside cones in ${ℝ}^n$. We show that every trajectory inside a cone over a $C^3$ strictly convex closed hypersurface embedded in ${ℝ}^{n-1}$ with non-degenerate second fundamental form undergoes only finitely many reflections. Using this result, we prove that the system is both superintegrable and completely integrable. To our knowledge, this provides the first example of an integrable billiard in which the billiard table is neither a quadric nor composed of pieces of quadrics. This poster is based on joint work with Andrey E. Mironov.
• Weinan Zhang (The Hong Kong University, Hong Kong, China)
  Title: Quantum symmetric pairs at roots of unity
  Abstract: The representation theory of quantum groups at roots of unity was developed by De Concini-Kac-Procesi in series of works, and this has applications and connections in modular representations, Poisson geometry, and geometric representation theory. In this talk, we generalize the approach of De Concini-Kac-Procesi to quantum symmetric pairs. Let $\theta$ be an involution on a Lie algebra $𝔤$. We construct a Frobenius center for the associated iquantum group at odd roots of unity and show that the irreducible representations of iquantum group are parametrized by $\theta$–twisted conjugacy classes of the Lie group $G$. We determine the degree of iquantum group and study its Azumaya locus. This is based on a joint work with Jinfeng Song.