Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of Abelian categories to derived categories of factorizations. Some immediate geometric consequences include a realization of the derived category of a projective hypersurface as matrix factorizations over a noncommutative algebra and recover of a theorem of Baranovsky and Pecharich.

Using the DAHA-Fourier transform of q-Hermite polynomials multiplied by level-one theta functions, we obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to Rogers dilogarithm and Nahm's conjecture is discussed. The q-Hermite polynomials are closely related to the Demazure level-one characters in the twisted case (Sanderson, Ion), which connects our formulas to tensor products of level-one integrable Kac-Moody modules, their coset theory and the level-rank duality. © 2013 Elsevier Inc.

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety *X* via derived algebraic geometry. We prove that if *X* is a Calabi–Yau variety of dimension *d* then this moduli stack has a -shifted Poisson structure. In the case , we construct a natural foliation of the moduli stack by 0-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii.

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero–Moser systems. In the special case of symmetric pair g=gl(n,2m),k=osp(n,2m)we establish a natural bijection between projective covers of spherically typical irreducible g-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero–Moser integrals Dn,m acting on the corresponding Laurent quasi-invariants An,m.

We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and étale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, iff is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or étale topology, then all symmetric powers Symn(f) are weak equivalences too. This gives left derived symmetric powers in the corresponding motivic homotopy categories of schemes over a base, which aggregate into a categorical λ-structures on these categories.

We define a new 4-dimensional symplectic cut and paste operations arising from the *generalized star relations* (ta0ta1ta2⋯ta2g+1)2g+1=tb1tb2gtb3, also known as the *trident relations*, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the *chain relations* to the families of words (tc1tc2⋯tc2g−1tc2gtc2g+12tc2gtc2g−1⋯tc2tc1)2n=1, (tc1tc2⋯tc2gtc2g+1)(2g+2)n=1and (tc1tc2⋯tc2g−1tc2g)2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.

The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least three incarnations — as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinite-dimensional unitary group, and as the set of doubly infinite totally positive sequences. An old deep result due to Albert Edrei and Dan Voiculescu provides an explicit description of the boundary; it can be realized as a region in an infinite-dimensional coordinate space. The paper contains a novel approach to the Edrei–Voiculescu theorem. It is based on a new explicit formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand–Tsetlin schemes of trapezoidal shape). The formula is obtained via the theory of symmetric functions, and new Schur-like symmetric functions play a key role in the derivation.

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.

We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices.

In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.

The tropical arithmetic operations on R are defined by a⊕b=min{a, b} and a⊗b=a+b. Let A be a tropical matrix and k a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^{m×k} and C∈R^{k×n} satisfying B⊗C=A. We show that no algorithm for TMF is likely to work in polynomial time for every fixed k, thus resolving a problem proposed by Barvinok in 1993.

Which polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer. Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients. The set of all atypical systems turns out to be a hypersurface in the space of all systems of k equations in n variables, whose monomials are contained in k given finite sets. This hypersurface B contains all systems that have a singular solution, this stratum is conventionally called the discriminant, and the codimension of its components has not been fully understood yet (e.g. dual defect toric varieties are not classified), so the purity of dimension of B looks somewhat surprising. We deduce it from a similar tropical purity fact. A generic system of equations in a component B_i of the hypersurface B differs from a typical system by the Euler characteristic of its set of solutions. Regarding the difference of these Euler characteristics as the multiplicity of B_i, we turn B into an effective divisor, whose equation we call the Euler discriminant by the following reasons. Firstly, it vanishes exactly at those systems that have a singular solution (possibly at infinity). Secondly, despite its topological definition, there is a simple linear-algebraic formula for it, and a positive formula for its Newton polytope. Thirdly, it interpolates many classical objects (sparse resultant, A-determinant, discriminant of deformation) and inherits many of their nice properties. This allows to specialize our results to generic polynomial maps: the bifurcation set of a dominant polynomial map, whose components are generic linear combinations of finitely many monomials, is always a hypersurface, and a generic atypical fiber of such a map differs from a typical one by its Euler characteristic.

We derive a formula for the Sn-equivariant Euler characteristic of the moduli space Mg,n of genus *g* curves with *n* marked points.

The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex for an arbitrary prestack , we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If denotes the Grothendieck construction of , which is a -graded category, we explicitly construct inverse quasi-isomorphisms and between and the Hochschild complex , as well as a concrete homotopy , which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an -structure on , which controlls the higher deformation theory of the prestack . This answers the open problem about the higher structure on the Gerstenhaber–Schack complex at once in the general prestack case.

The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U(1), U(2), .... This novel object is a commutative graded algebra R with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group U(∞). These processes were defined in a joint work with Alexei Borodin (2012) [5]. The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R.

We classify triangulated categories that are equivalent to finitely generated thick subcategories $T\subset D^b(cohC)$ for smooth projective curves C over an algebraically closed field.

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau–Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main result shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf.

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkähler. In the paper “Non-Hermitian Yang–Mills connections”, Kaledin and Verbitsky proved that the twistor space of a hyperkähler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.