Fix a semisimple Lie algebra 𝔤. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for 𝔤-representations. These algebras depend on a parameter in the Deligne–Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. We study the monodromy of these eigenvectors as the parameter varies within the real locus, giving an action of the fundamental group of this moduli space called the *cactus* *group*. We prove that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of 𝔤-crystals (as conjectured by Etingof), and we prove that the coboundary category of normal 𝔤-crystals can be reconstructed using the coverings of the moduli spaces. To prove the conjecture, we construct a crystal structure on the set of eigenvectors for the shift of argument algebras, another family of commutative algebras acting on any irreducible 𝔤-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on 𝔤-crystals.

Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for *W* depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.

Given an irreducible affine algebraic variety X of dimension n≥2, we let SAut(X) denote the special automorphism group of X, that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus Xreg, then it is infinitely transitive on Xreg. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x∈Xreg the tangent space TxX is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class Γ_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by Γ_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_{coh}(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces, toric manifolds, certain toric complete intersections and Grassmannians.

It is now well-known that applications of the operad theory in general (and, in particular, to verifications of the Koszul property) are really difficult in particular computations. There was no known ``arithmetic'' of operations similar to the arithmetic of integers or polynomials (by an ``arithmetic'' we mean the usual notion of divisibility). The good analogue of multiplication for the operadic data is the composition of operations. But the action of the symmetric groups on the entries of operations contradicts with any possible functorial definition of divisibility. This paper contains a solution to this problem using the notion of * Shuffle operads. * The key idea is to forget about a certain part of the action of the symmetric group. In spite of being a very simple idea, it allowed us to introduce a theory of monomials, their divisibility and compatible orderings of monomials for operads. Summarizing these notions, we came up with the notion of Grobner bases for operads. Grobner bases is a remarkable technical tool initiated in the commutative algebra setting by Buchberger which allows one to solve systems of equations with many unknowns. The theory of Grobner bases for operads made it possible to provide a unified proof of the existing computational results in the field as well as to prove some new results. It is clear that there are many topics that can be successfully approached by these new methods.

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy

We apply some of the ideas of Margulis's Ph.D. dissertation to Teichmüller space. Let X be a point in Teichmüller space, and let B R.X/ be the ball of radius R centered at X (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B R.X/, and also for the cardinality of the intersection of B R.X/ with an orbit of the mapping class group.

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in SO(3, b_2-3). A Teichmüller space of $M$ is a space of complex structures on $M$ up to isotopies. We define a birational Teichmüller space by identifying certain points corresponding to bimeromorphically equivalent manifolds. We show that the period map gives the isomorphism between connected components of the birational Teichmüller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\times SO(b_2 -3, 1)$. We use this result to obtain a Torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group. When $M$ is a Hilbert scheme of $n$ points on a K3 surface, with $n-1$ a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkähler manifolds, the Hodge-theoretic Torelli theorem is known to be false).

We define and study the stack U^{ns,a}_{g,g} of (possibly singular) projective curves of arithmetic genus g with g smooth marked points forming an ample non-special divisor. We define an explicit closed embedding of a natural 𝔾gm-torsor over U^{ns,a}_{g,g} into an affine space and give explicit equations of the universal curve (away from characteristics 2 and 3). This construction can be viewed as a generalization of the Weierstrass cubic and the j-invariant of an elliptic curve to the case g>1. Our main result is that in characteristics different from 2 and 3 our moduli space of non-special curves is isomorphic to the moduli space of minimal A-infinity structures on a certain finite-dimensional graded associative algebra Eg (introduced in arXiv:1208.6332). We show how to compute explicitly the A-infinity structure associated with a curve (C,p1,...,pg) in terms of certain canonical generators of the algebra of functions on C−{p1,...,pg} and canonical formal parameters at the marked points. We study the GIT quotients associated with our representation of U^{ns,a}_{g,g} as the quotient of an affine scheme by 𝔾m^g and show that some of the corresponding stack quotients give modular compactifications of M_{g,g} in the sense of arXiv:0902.3690. We also consider an analogous picture for curves of arithmetic genus 0 with n marked points which gives a new presentation of the moduli space of ψ-stable curves (also known as Boggi-stable curves) and its interpretation in terms of A∞-structures.

We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields. The limiting process has previously arisen via the global scaling limit of spectra for submatrices of Wigner Hermitian random matrices.

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group (in this paper for simplicity we consider only untwisted and simply connected case). The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G_) of finitedimensional representations of the Langlands dual group G_ with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG = G(K)/G(O) of G (here K = C((t)) and O = C[[t]]). As a byproduct one gets a description of the irreducible G(O)-equivariant intersection cohomology sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite dimensional representations of G_. The purpose of this paper is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff (we shall refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian). More precisely, in this paper we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff (O)-orbit in GrGaff . We present a conjecture that computes the IC sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group G_aff and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian etc.) will be addressed in another publication.

Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible ``wild'' behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a topological classification of Morse-Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n-1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q,t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.