In this paper, we prove the following differential analog of the Noether normalization lemma: for every dd-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map on to the dd-dimensional affine space. Equivalently, for every integral differential algebra AA over differential field of zero characteristic there exist differentially independent b1,…,bdb1,…,bd such that AA is differentially algebraic over subalgebra BB differentially generated by b1,…,bdb1,…,bd, and whenever p⊂Bp⊂B is a prime differential ideal, there exists a prime differential ideal q⊂Aq⊂A such that p=B∩qp=B∩q. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.

The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the Kolmogorov property and satisfies the central limit theorem, but our functional limit theorem is very different from the Donsker Invariance Principle. We show that the time integral of our process can be approximated by the sum of a linear Gaussian process and independent Gaussian fluctuations whose covariance matrix is computed explicitly. We interpret these results in terms of the Gaussian free field convergence for the random matrix models. The proof relies on a general form of the multidimensional central limit theorem under the sineprocess for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity 1/2.

In this paper we describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.

We define an algebraic set in 23-dimensional projective space whose ℚ-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight 3examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight 2 is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over ℚ.

The Gaudin algebra is the commutative subalgebra in U(g)^⊗N generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra g. This algebra depends on a collection of pairwise distinct complex numbers z1,…,zN. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible g-modules, for all values of the parameters z1,…,zN. We deduce from this result the Bethe Ansatz conjecture in the Feigin–Frenkel form that states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional g-modules are in 1-1 correspondence with monodromy-free LG-opers on the projective line with regular singularities at the points z1,…,zN,∞, and the prescribed residues at the singular points.

In a recent paper, the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula.

We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface X is always Jordan, and the birational automorphism group is Jordan unless X is birational to a product of an elliptic and a rational curve.

A manifold M is locally conformally Kähler (LCK), if it admits a Kähler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that S1 acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kähler cover is not isometric. We show that admits an automorphic Kähler potential, and hence (for dimℂ M > 2) the manifold M can be embedded to a Hopf manifold.

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

A locally conformally Khler (LCK) manifold is a complex manifold which admits a covering endowed with a Kähler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kähler. We also prove that a twistor space (of a compact four-manifold, a quaternion-Kähler manifold, or a Riemannian manifold) cannot admit an LCK metric, unless it is Kähler.

We study two rational Fano threefolds with an action of the icosahedral group 𝔄5. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are 𝔄5-Fano varieties that are 𝔄5-birationally superrigid. This gives two new embeddings of the group 𝔄5 into the space Cremona group.

Cactus group is the fundamental group of the real locus of the Deligne–Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra g. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case g=sl2 (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand–Tsetlin polytope.

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category that has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.

In this article, we calculate the ring of unstable (possibly nonadditive) operations from algebraic Morava K-theory K(n)^∗ to Chow groups with ℤ_(p) -coefficients. More precisely, we prove that it is a formal power series ring on generators c_i:K(n)^∗→CH^i⊗ℤ_(p) , which satisfy a Cartan-type formula.

Consider the space *M* = *O*(*p*, *q*)/*O*(*p*) × *O*(*q*) of positive *p*-dimensional subspaces in a pseudo-Euclidean space *V* of signature (*p*, *q*), where *p* > 0, *q* > 1 and (p,q)≠(1,2), with integral structure: V=Vℤ⊗ℤ. Let Γ be an arithmetic subgroup in G=O(Vℤ), and R⊂Vℤ a Γ-invariant set of vectors with negative square. Denote by *R*⊥ the set of all positive *p*-planes *W* ⊂ *V* such that the orthogonal complement *W*⊥ contains some *r* ∈ *R*. We prove that either *R*⊥ is dense in *M* or Γ acts on *R* with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of “negative” minimal rational curves) on a hyperkähler manifold *X*are bounded by a number which depends only on the deformation class of *X*. We also state and prove the density of orbits in a more general situation when *M* is the space of maximal compact subgroups in a simple real Lie group.

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.