We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth ℤZ-graded gentle algebra *A* we associate a triple (ΣA,ΛA;ηA), where ΣA is an oriented smooth surface with non-empty boundary, ΛA is a set of stops on ∂ΣA and ηA is a line field on ΣA, such that the derived category of perfect dg-modules of *A* is equivalent to the partially wrapped Fukaya category of (ΣA,ΛA;ηA). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of ΣA on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number I of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number 2I of coefficients depending on the quasimomentum and show that the number I is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant I and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians.Moreover, similar results for Schrödinger operators with periodic potentials are obtained.

We propose a natural definition of a category of matrix factorizations for nonaffine Landau–Ginzburg models. For any LG-model we construct a fully faithful functor from the category of matrix factorizations defined in this way to the triangulated category of singularities of the corresponding fiber. We also show that this functor is an equivalence if the total space of the LG-model is smooth.

This work was partially supported by RFBR grants 10-01-93113, 11-01-00336, 11-01-00568, NSh grant 4713.2010.1,by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, and by Simons Center for Geometry and Physics.

In this paper we study the moduli stack M_{1,n} of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In Polishchuk (A modular compactification of M_{1,n} from A∞-structures, arXiv:1408.0611) this stack was realized as the quotient of an explicit scheme U^{ns}_{1,n} affine of finite type over ℙn−1, by the action of 𝔾m^n . Our main result is an explicit description of the corresponding GIT semistable loci in U^{ns}_{1,n}. This allows us to identify some of the GIT quotients with some of the modular compactifications of M_{1,n} defined in Smyth (Invent Math 192:459–503, 2013; Compos Math 147(3):877–913, 2011).

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.

We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued functions, and the computation of the monodromy group of a general system of equations, which may be of independent interest.

We consider the class of singular double coverings X → P3 ramified in the degeneration locus D of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface D, one can associate an Enriques surface S which is the factor of the blowup of D by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface S is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of X.

We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards.

We provide descriptions of the derived categories of degree d hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem of Orlov. Using a local generator and Morita theory, we re-interpret the resulting matrix factorization category as a derivedequivalent sheaf of dg-algebras on the base. Then, applying homological perturbation methods, we obtain a sheaf of A∞-algebras which gives a new description of homo-logical projective duals for (relative) d-Veronese embeddings, recovering the sheaf of Clifford algebras obtained by Kuznetsov in the case when d = 2.

Let *M* be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant *𝐼*1(*𝑀*) using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator Δ*𝑔* of a Riemannian metric *g*. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use *k*-th eigenvalues of Δ*𝑔* to define the invariants *𝐼**𝑘*(*𝑀*) indexed by positive integers *k*. In the present paper the values of these invariants on surfaces are investigated. We show that *𝐼**𝑘*(*𝑀*)=*𝐼**𝑘*(𝕊2) unless *M* is a non-orientable surface of even genus. For orientable surfaces and *𝑘*=1 this was earlier shown by Petrides. In fact Friedlander and Nadirashvili suggested that *𝐼*1(*𝑀*)=*𝐼*1(𝕊2) for any surface *M* different from Rℙ2. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has *𝐼**𝑘*(*𝑀*)>*𝐼**𝑘*(𝕊2). We also discuss the connection between the Friedlander–Nadirashvili invariants and the theory of cobordisms, and conjecture that *𝐼**𝑘*(*𝑀*) is a cobordism invariant.

We study analytic surfaces in 3-dimensional Euclidean space containing two circular arcs through each point. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We reduce finding all such surfaces to the algebraic problem of finding all Pythagorean 6-tuples of polynomials. The reduction is based on the Schicho parametrization of surfaces containing two conics through each point and a new approach using quaternionic rational parametrization.

The earlier work of the first and the third named authors introduced the algebra A_q,t and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert schemes of points on the plane. In this presentation, the fixed points of torus action correspond to generalized Macdonald polynomials and the the matrix elements of the operators have explicit combinatorial presentation.

We develop the topological polylogarithm which provides an integral ver- sion of Nori’s Eisenstein cohomology classes for GLn (Z) and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of spe- cial values of L-functions of totally real fields and a construction of the associated p-adic L-function. Using a result of Graf, we also apply this to prove some integrality and p-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.

Given a tropical variety X and two non-negative integers p and q we define a homology group Hp,q (X) which is a finite-dimensional vector space over Q. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then dim Hp,q (X) coincides with the Hodge number h p,q of a general member of the family.

Let G be a semisimple simply connected algebraic Lie group over complex numbers. Following Gerasimov, Lebedev and Oblezin, we use the q-Toda integrable system obtained by the quantum group version of the Kostant-Whittaker reduction to define the notion of q-Whittaker functions. This is a family of invariant polynomials on the maximal torus T in G depending on a dominant weight of G, whose coefficients are rational functions in the variable q. For a conjecturally the same (but a priori different) definition of the q-Toda system these functions were studied by I.Cherednik. For G=SL(N) these functions were extensively studied by Gerasivom, Lebedev and Oblezin. We show that when G is simply laced, the Whittaker function is equal to the character of the global Weyl module. When G is not simply laced a twisted version of the above result holds. Our proofs are algebro-geometric.