Leclerc
Title : A cluster structure on the category O for shifted quantum affine algebras
Abstract : I will report on a joint work with C. Geiss and D. Hernandez. We prove that the Grothendieck ring of the category O for the shifted quantum affine algebras, introduced by Hernandez in arXiv:2010.06996, has the structure of a cluster algebra of infinite rank, with explicit initial seeds parametrized by reduced expressions of the longest element of the associated (finite) Weyl group. The initial cluster variables are constructed by means of the new Weyl group action of Frenkel and Hernandez (arXiv:2211.09779). In type A,D,E, we obtain a surprising connection with the Berenstein-Fomin-Zelevinsky cluster algebra structure on the open double Bruhat cell of the corresponding simple algebraic group.
Cerulli Irelli
Title: Quivers with self-duality and possible applications to cluster algebras
Abstract: A quiver with self-duality is a quiver isomorphic to its opposite. For example, the linearly-oriented quiver of type A is a quiver with self-duality. They were introduced by Derksen and Weyman with the name “symmetric quivers”. The category of representations of such quivers is endowed with a self-duality, and the underlying vector space of self-dual representations can be endowed with a symplectic or orthogonal form. We call them symplectic or orthogonal quiver representations, or symmetric representations for short. In the variety of symmetric representations the group acting is not anymore a product of general linear groups, but instead there are also the symplectic and the orthogonal group as factors. By a theorem of Magyar, Weyman and Zelevinsky two symmetric representations are isomorphic if and only if they are isomorphic as quiver representations. To my surprise I discovered that not much is known for the orbit closures. In collaboration with Magdalena Boos we prove that for symmetric quivers of type A, the closure of a symmetric orbit is precisely the intersection of the closure of the orbit for the usual action with the variety of symmetric representations. To prove this we show that the orbit closures are induced by isotropic subrepresentations. This result is used in the work of my Ph.D. student Azzurra Ciliberti to find a character associating to each indecomposable symmetric representation of a symmetric quiver of type A (with an odd number of vertices) a cluster variable of a cluster algebra of type B and C.
Ovsienko
Title: Shadows of numbers: supergeometry with a human face
Abstract: In this elementary and accessible to everyone talk I will explain an attempt to apply supersymmetry and supergeometry to arithmetic. The following general idea looks crazy. What if every integer sequence has another integer sequence that follows it like a shadow? I will demonstrate that this is indeed the case, though perhaps not for every integer sequence, but for many of them. The main examples are those of the Markov numbers and Somos sequences. In the second part of the talk, I will discuss the notions of supersymmetric continued fractions and the modular group, and arrive at yet a more crazy idea that every rational and every irrational has its own shadow.
The « shadow ideas » arose from the notion of cluster superalgebra (joint with Michael Shapiro).
Tabachnikov
Title: Integrable transformations on centroaffine polygons
Abstract: U. Pinkall interpreted the Korteweg-de Vries equation as a completely integrable evolution on centroaffine curves. Accordingly, symmetries (the Backlund transformation) of the KdV equation also can be realized as transformations of centroaffine curves. I shall discuss a discrete version of these transformations, where the curves are replaced by polygons. The centroaffine version of polygon recutting of V. Adler will be considered as well. The focus of the talk is on the geometrical aspects of the problem (and the cluster-algebraic aspects are up for grabs). This is a joint work with M. Arnold and D. Fuchs
Lando
Title: Weight systems related to Lie algebras
Abstract: V. A. Vassiliev’s theory of nite type knot invariants allows one to associate to such an invariant a function on chord diagrams, which are simple combinatorial objects, consisting of an oriented circle and a tuple of chords with pairwise distinct ends in it. Such functions are called weight systems . According to a Kontsevich theorem, such a correspondence is essentially one-to-one: each weight system determines certain knot invariant.
In particular, a weight system can be associated to any semi-simple Lie algebra. However, already in the simplest nontrivial case, the one for the Lie algebra sl(2), computation of the values of the corresponding weight system is a computationally complicated task. This weight system is of great importance, however, since it corresponds to a famous knot invariant known as the colored Jones polynomial.
The last year was a period of signi cant progress in understanding and computing Lie algebra weight systems, both for sl(2)- and gl(N)-weight system, for arbitrary N. New recurrence relations were deduced, which allow for a lot of explicit formulas. These methods are based on an idea, due to M. Kazarian, which suggests to extend the gl(N)-weight system to permutations.
Questions concerning possible integrability properties of the Lie algebra weight systems will be formulated.
The talk is based on work of M. Kazarian, the speaker, and the students P. Zakorko, Zhuoke Yang, and P. Zinova.
Khesin
Title: Hamiltonian geometry and the golden ratio in the Euler hydrodynamics
Abstract: The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present a Hamiltonian framework for the binormal equation in higher-dimensions and its explicit solutions that collapse in finite time. On the other hand, by going to lower dimensions, we observe a curious appearance of the golden ratio in the motion of point vortices in the plane. This is a joint work with C.Yang and H.Wang.
Felikson Tumarkin
Title: 3D Farey graph from ideal tetrahedra
Abstract: We construct a 3-dimensional analog of the Farey tesselation and show that it inherits many properties of the usual 2-dimensional Farey graph. In particular, we get a classification of SL_2-tilings over Eisenstein numbers in terms of pairs of paths on the graph. We also get a 3-dimensional counterpart of the Ptolemy relation. The talk is based on an ongoing work joint with Oleg Karpenkov and Khrystyna Serhiyenko.
Fomin
Title: Incidences and tilings
Abstract: We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a single “master theorem” that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.
This is joint work with Pavlo Pylyavskyy.
Geiss
Title: MSW-bangle functions and generic basis for surface cluster algebras
Di Francesco
Title: From Koornwinder operators to cluster algebra: Proof of the Macdonald-Q-system conjecture
Abstract: We present various constructions of commuting difference operators for the theory of Koornwinder polynomials. We show how a specialization/limiting procedure produces a functional representation for quantum Q-system cluster algebras associated to affine and twisted types A,B,C,D, also interpreted as discrete algebraic quantum integrable systems. The correspondence uses Koornwinder duality and a suitable Fourier-Whittaker transform allowing to interpret Koornwinder polynomial Pieri rules as relativistic Toda systems. (Based on joint work with Rinat Kedem).
Yakimov
Title: Poisson geometry and representation theory of cluster algebras
Abstract: One of the fundamental constructions in the area of cluster algebras is the Gekhtman-Shapiro-Vainshtein Poisson structure on an upper cluster algebra. We prove that their spectra have a Zariski open torus orbit of symplectic leaves and provide an explicit description of it. It is a far reaching generalization of the complement of the Richardson divisor of Schubert cells in Lie theory. Using Cayley-Hamilton structures, we show that, if a root of unity upper quantum cluster algebra is finitely generated, then the same holds for the corresponding classical algebra and that the quantum algebra is finitely generated module over a canonical central subalgebra that is isomorphic to the classical one. Using these two methods we give a description of the Azumaya locus of the quantum algebra at roots of unity. This is a joint work with Shengnan Huang, Thang Le, Greg Muller, Bach Nguyen and Kurt Trampel.
Burman
Title: Hurwitz numbers and ribbon gluing, real and complex case.
Abstract: Classical (or complex) Hurwitz numbers are integers solving several problems in combinatorics (the number of sequences of transpositions with a product of a specified cyclic structure), algebraic geometry (the number of rational functions with prescribed pole divisor and critical values), low-dimensional topology (the number of ways to obtain a given surface gluing ribbons to a collection of disks), and more. We discuss these problems together with their real (or “twisted”’) analogs.
The talk is based on a joint work with R.Fesler.
Fock
Title: Clusters, integrable systems and tame symbol
Abstract: A tame symbol is a bimultiplicative 2-cocycle on the group of nonvanishing functions a circle given by an explicit formula. The tame symbol is related to Heisenberg group, resultant, Witt ring, Gauss reciprocity and many other subjects. We will use the tame symbol to define a homology class (with values in the multiplicative group) of a Lagrangian subvariety of a cluster A-variety. Say that a Lagrangian subvariety is Bohr-Sommerfeld if this class is trivial. We will show in the example of dimension 2 that every Bohr-Sommerfeld curve gives a solution of a difference equation, which is a quantization of the equation of the curve with quantization parameter equal to 1. The solution is given by quantum dilogarithms of algebraic functions.
Baur
Title: Tiled surfaces, string algebras and laminations
Abstract: We consider tilings of surfaces. We show how they can be used as a geometric model for string algebras, a large class of algebras satisfying monomial relations. We consider laminations on tiled surfaces. They give rise to classes of locally gentle algebras and infinite dimensional modules. We study these laminations and their properties. This is joint work with R. Coelho Simoes and with B. Marsh.
Plamondon
Title: Some configuration spaces and the categorification of cluster algebras
Abstract: The starting point for this talk is the realization of certain configuration spaces as affine varieties defined by so-called “u-equations”. We will see how this construction can be interpreted and generalized using the representation theory of finite-dimensional algebras, using the tools of additive categorification of cluster algebras, and in particular the interpretation of F-polynomials and g-vectors.
This is a report on ongoing work with Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori and Hugh Thomas.
Williams
Title: Rigidity and Integrability in Monoidal Categorifications
Abstract: A familiar theme in cluster algebras is that of discrete integrable systems being realizable as sequences of cluster transformations. In this talk we will discuss this theme through the lens of tensor categories related to Coulomb branches of 4d gauge theories, introduced in joint work with Sabin Cautis. The rigidity of these categories, that is the existence of left and right duals, gives rise to a canonical discrete dynamical system on their Grothendieck ring. We explain how the geometry behind these duals guarantees that these systems are always integrable, and illustrate in the simplest examples that they are indeed cluster integrable systems (though this remains a conjecture in general)