The minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric ${\mathbb{E}}_2^4$ whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the condition $K^2-{\varkappa }^2\ne 0$ are called minimal Lorentz surfaces of general type. We prove that these surfaces admit special isothermal parameters, called canonical, and in terms of the canonical parameters the Gauss curvature $K$ and the normal curvature $\varkappa$ satisfy the following system of natural PDEs: $$\begin{array}{lll}\sqrt[4]{|K^2-{\varkappa }^2|}{\Delta }^h\ln |K^2-{\varkappa }^2|& =& 8K;\\ \sqrt[4]{|K^2-{\varkappa }^2|}{\Delta }^h\ln \left|\frac{K+\varkappa }{K-\varkappa }\right|& =& 4\varkappa ;\end{array}{K}^2-{\varkappa }^2\ne 0.$$

We find a Weierstrass representation with respect to isothermal parameters of any minimal Lorentz surface of general type. Further, we obtain a Weierstrass representation with respect to canonical parameters and describe all these surfaces in terms of four real functions. Using the canonical Weierstrass representation we solve explicitly system (1) expressing any solution to this system by means of four real functions of one variable. Finally, we give examples of minimal Lorentz surfaces of general type in ${\mathbb{E}}_2^4$ parametrized by canonical parameters.

The author is partially supported by the National Science Fund, Ministry of Education and Science of Bulgaria under contract KP-06-N82/6.

This talk addresses left-invariant geometric structures on the cotangent bundle of the $(2n+1)$-dimensional Heisenberg group. A complete classification of left-invariant pseudo-Riemannian metrics is available in the low-dimensional case $n=1$, while in higher dimensions the focus is on the Riemannian setting and selected classes of pseudo-Riemannian metrics.

Within this framework, the moduli space of metrics up to equivalence is described, together with distinguished structures such as pseudo-Kähler and pp-wave metrics. Their geometric features are examined, including curvature and related invariants. The complex structure is shown to be unique, the corresponding pseudo-Kähler metrics are Ricci-flat, and the uniqueness of an ad-invariant metric of neutral signature is established.

Left-invariant sub-Riemannian structures are also classified, related to compatible Riemannian metrics, and their geodesics are described.

Algebras with polynomial identities (or shortly PI-algebras) are natural generalizations of commutative algebras and possess many similar properties. Given a PI-algebra $A$ over a field $K$, one can associate to it the quotient algebra $F_n(A)=K⟨X_n⟩/T(A)$, where $K⟨X_n⟩$ denotes the free associative algebra generated by a set $X_n$ with $n$ elements over $K$ and $T(A)$ is the ideal of all polynomial identities of $A$. The algebra $F_n(A)$ has a natural structure of a $\mathrm{GL}(n,K)$-module and one important question in the theory of PI-algebras over a field $K$ of characteristic zero is to determine the $\mathrm{GL}(n,K)$-module structure of $F_n(A)$ for different algebras $A$. This question has been answered explicitly only for a few classes of PI-algebras. Furthermore, there are general results about the properties of the partitions $\lambda$ such that the irreducible $\mathrm{GL}(n,K)$-module with highest weight $\lambda$ appears with non-zero multiplicity in the decomposition of $F_n(A)$. In this talk, we are interested in the case when $A$ is a Lie nilpotent associative algebra, i.e., $A$ satisfies a long commutator identity $[x_1,\dots ,x_{p+1}]=0$ for some integer $p\ge 1$. We discuss different properties of the $\mathrm{GL}(n,K)$-module structure of $F_n(A)$ and we give a criterion when a PI-algebra $A$ is Lie nilpotent in terms of the $\mathrm{GL}(n,K)$-module structure of $F_n(A)$. We show that, in some sense, the class of Lie nilpotent associative algebras is the closest to commutative algebras among all classes of PI-algebras. In the end of the talk, we discuss also properties of the tensor product of Lie nilpotent associative algebras.

This talk is about certain non-periodic frieze patterns, which can be obtained from a categorification of a Grassmannian cluster algebra of infinite rank: the category of maximal Cohen-Macaulay modules over the so-called A-infinity curve singularity. This Frobenius category has a rich combinatorial structure and was studied in the context of triangulations of the infinity-gon by August, Cheung, Faber, Gratz, and Schroll. Extending the cluster character from work of Paquette and Yıldırım to this setting we obtain a new type of infinite friezes that can be related to Penrose tilings.

This is joint work with Özgür Esentepe.

The index with respect to cluster tilting subcategories in triangulated categories plays a key role in the categorification of cluster algebras. In recent years, the notion of index has been generalised in various directions, including defining it with respect to contravariantly finite rigid subcategories of triangulated categories, via the use of extriangulated categories.

In this joint project with Jørgensen and Shah, we drop the rigid assumption and prove the core properties of the index continue to hold. After introducing the topic, in this talk we will see how this index theory applies to Paquette-Yıldırım completed discrete cluster categories of type A with respect to fan triangulations, whose associated subcategories are often not rigid.

For an exact category I will present two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel–Quillen embedding. The first construction yields a pre-aisle and I will provide necessary and sufficient conditions when this is an aisle of a t-structure.

This is joint work with Marianne Lawson and Julia Sauter.

Torsion theories give rise to nice lattice structures in abelian categories. Pretorsion theories were recently introduced as a non-pointed analogue, allowing torsion-theoretic methods to be extended to broader categorical settings. In this talk, we investigate pretorsion classes in the module category of a finite-dimensional algebra and prove that they are naturally organized into a lattice under inclusion. We further analyze when this lattice is distributive and compare it with the classical lattice of torsion classes.

This is joint work with Federico Campanini and Francesca Fedele.

Surface combinatorics have been instrumental in describing algebraic structures such as module categories, derived categories, cluster categories, etc. We show how to construct string algebras and their module categories from tiled surfaces. The surface combinatorics can be applied to characterise reductions (using perpendicular subcategories) as we show. The constructed surfaces is not unique in general and it is an open question how to relate different surfaces with the same underlying algebra.

This is joint work with R. Coelho Simoes and work in progress with W. Chang and E. Hanson.

Caldero and Chapoton described a function which sends representations of an ADE quiver to elements of the associated cluster algebra. This transformative work paved the way for many more interesting studies, and in particular this CC map has been generalized to a variety of settings. We propose a CC map for the derived category of a gentle algebra. A main feature of our definition is how it interacts with the geometric model of the derived category, and in particular we show that our CC functions respect skein relations from (graded) intersections of curves.

This is based on ongoing joint work with Azzurra Ciliberti, Ilaria Di Dedda, Khrystyna Serhiyenko, Yadira Valdivieso-Diaz and Kayla Wright.

In this work, we give a way of expressing super cluster expansions associated with the Decorated Super Teichmüller space for triangulated polygons in terms of rank polynomials of a family of labeled posets. This allows us to use the theory of oriented posets to give explicit formulas for the super expansions in terms of $3\times 3$ matrices.

In a celebrated 1984 paper Gelfand and Zelevinskii proposed the notion of model, parametrizing irreducible unitary representations of a simple Lie group G. Later on, Gelfand models were inspiring the theory of geometric quantization, where representations of G are realized in the space of sections of unitary line bundles on a suitable symplectic quotient of G.

We generalize this construction to the supergeometric setting and provide a criterion for the realization of highest weight super representations, based on the super moment map.

This is a joint work with M.-K. Chuah.

Quantum symmetric pair coideal subalgebras are certain quantum analogues of fixed point Lie subalgebra of a semi-simple Lie algebra under an involution. Originally considered in relation to quantum symmetric spaces and special functions, they have been described as Lie theoretic objects by Letzter and Kolb, and in the last thirteen years shown to have a rich structure analogous to quantum groups, in particular allowing a construction of canonical bases, solutions of quantum reflection equations, superalgebra analogues, and a diagrammatic calculus. Despite this, their finite dimensional representations have not yet been classified in many types. We tackle this problem for a particular series of these algebras over a wide choice of fields, and for a given QSP subalgebra classify its finite dimensional representations in terms of “integral” highest weights of its Letzter-Cartan subalgebra.

Nilmanifolds are compact manifolds obtained as quotients of simply connected nilpotent Lie groups by discrete subgroups, and they serve as a rich source of examples and counterexamples in geometry. In this talk, we study nilmanifolds endowed with invariant complex structures in the case where the underlying Lie group is almost abelian, that is, its Lie algebra admits an abelian ideal of codimension one. In this setting, the Lie bracket is completely determined by a matrix. Using the Jordan normal form of this matrix and representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, we derive explicit expressions for the Betti and Hodge numbers of these nilmanifolds.

This is joint work with Adrián Andrada, María Laura Barberis, Sönke Rollenske, and Konstantin Wehler.

I will present a construction of non-algebraic complex manifolds acted on by an algebraic group by means of geometrical representation theory. This construction generalizes that of Hopf manifolds and more generally of LVMB-manifolds (acronym for López de Medrano-Verjovsky-Meersseman-Bosio).

In this talk I will report on joint work with Alessio Cipriani aimed at understanding Bridgeland stability spaces for the bounded derived category of constructible complexes on flag varieties. I will present a very first step towards this understanding, that is the study of the so-called wall and chamber structure of the category of perverse sheaves on projective spaces, the easiest flag variety we can think of. Such a study boils down to the investigation of modules for a special biserial algebra whose module category is equivalent to that of perverse sheaves on a projective space.

(joint work with Osamu Iyama and Emre Sen)

For the path algebra $H$ of a Dynkin quiver $Q$, the fundamental domain ${\mathcal{F}}_n$ of $n$-cluster category ${\mathcal{C}}_n$ is known to be $(3n-2)$-representation finite (Sen 2023), in particular global dimension of mod-${\mathcal{F}}_n$ is $(3n-2)$. In this work, we show that ${\mathcal{F}}_n$ and ${\mathcal{C}}_n$ are related as: the $(3n-1)$-preprojective category of ${\mathcal{F}}_n$ is equivalent to the category ${\mathcal{C}}_n$. In fact, for any acyclic quiver $Q$, we prove the same result, that is, the $(3n-1)$-preprojective category ${\Pi }_{(3n-1)}$ of the fundamental domain ${\mathcal{F}}_n$ is equivalent to the $n$-cluster category ${\mathcal{C}}_n$.

This is a case of a functorial/categorical generalization of the notion of preprojective algebras.