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We prove that Schubert varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under suitable conditions. This is used to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety.

We prove that the Schubert structure constants of the quantum K-theory rings of symplectic Grassmannians of lines have signs that alternate with codimension and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings, including the Seidel representation and a Chevalley formula.

We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components., Comment: 54 pages, 1 table

In this paper we study general hyperplane sections of adjoint and coadjoint varieties. We show that these are the only sections of homogeneous varieties such that a maximal torus of the automorphism group of the ambient variety stabilizes them. We then study their geometry, provide formulas for their classical cohomology rings in terms of Schubert classes and compute the quantum Chevalley formula. This allows us to obtain results about the semi-simplicity of the (small) quantum cohomology, analogous to those holding for (co)adjoint varieties., Comment: 57 pages

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$ that occur in the product of two Schubert classes in the quantum cohomology or quantum $K$-theory ring of a cominuscule flag variety form an integer interval. Our proofs are based on several new results, including an explicit description of the most general non-empty intersection of two Schubert varieties in an arbitrary flag manifold, and a computation of the cohomology groups of any negative line bundle restricted to a Richardson variety in a cominuscule flag variety. We also give a type-uniform proof of the quantum-to-classical theorem, which asserts that the (3-point, genus 0) Gromov-Witten invariants of any cominuscule flag variety are classical triple-intersection numbers on an associated flag variety. Finally, we prove several new results about the geometry and combinatorics related to this theorem.

This paper is devoted to the study of the quantum cohomology of coadjoint varieties of simple algebraic groups across all Dynkin types. We determine the non-semisimple factors of the small quantum cohomology ring and relate them to ADE-singularities. Moreover, we show that the big quantum cohomology of a coadjoint variety is always generically semisimple even though in most cases the small quantum cohomology is not., Comment: v2: introduction is made more precise, sections 2-4 are reorganised and clarified, section 9 added, results remain unchanged

In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas., Comment: 23 pages

We propose a novel solution to challenging sparse-reward, continuous control problems that require hierarchical planning at multiple levels of abstraction. Our solution, dubbed AlphaNPI-X, involves three separate stages of learning. First, we use off-policy reinforcement learning algorithms with experience replay to learn a set of atomic goal-conditioned policies, which can be easily repurposed for many tasks. Second, we learn self-models describing the effect of the atomic policies on the environment. Third, the self-models are harnessed to learn recursive compositional programs with multiple levels of abstraction. The key insight is that the self-models enable planning by imagination, obviating the need for interaction with the world when learning higher-level compositional programs. To accomplish the third stage of learning, we extend the AlphaNPI algorithm, which applies AlphaZero to learn recursive neural programmer-interpreters. We empirically show that AlphaNPI-X can effectively learn to tackle challenging sparse manipulation tasks, such as stacking multiple blocks, where powerful model-free baselines fail.

Bidding in real-time auctions can be a difficult stochastic control task; especially if underdelivery incurs strong penalties and the market is very uncertain. Most current works and implementations focus on optimally delivering a campaign given a reasonable forecast of the market. Practical implementations have a feedback loop to adjust and be robust to forecasting errors, but no implementation, to the best of our knowledge, uses a model of market risk and actively anticipates market shifts. Solving such stochastic control problems in practice is actually very challenging. This paper proposes an approximate solution based on a Recurrent Neural Network (RNN) architecture that is both effective and practical for implementation in a production environment. The RNN bidder provisions everything it needs to avoid missing its goal. It also deliberately falls short of its goal when buying the missing impressions would cost more than the penalty for not reaching it.

The exploration-exploitation trade-off is at the heart of reinforcement learning (RL). However, most continuous control benchmarks used in recent RL research only require local exploration. This led to the development of algorithms that have basic exploration capabilities, and behave poorly in benchmarks that require more versatile exploration. For instance, as demonstrated in our empirical study, state-of-the-art RL algorithms such as DDPG and TD3 are unable to steer a point mass in even small 2D mazes. In this paper, we propose a new algorithm called "Plan, Backplay, Chain Skills" (PBCS) that combines motion planning and reinforcement learning to solve hard exploration environments. In a first phase, a motion planning algorithm is used to find a single good trajectory, then an RL algorithm is trained using a curriculum derived from the trajectory, by combining a variant of the Backplay algorithm and skill chaining. We show that this method outperforms state-of-the-art RL algorithms in 2D maze environments of various sizes, and is able to improve on the trajectory obtained by the motion planning phase.

In environments with continuous state and action spaces, state-of-the-art actor-critic reinforcement learning algorithms can solve very complex problems, yet can also fail in environments that seem trivial, but the reason for such failures is still poorly understood. In this paper, we contribute a formal explanation of these failures in the particular case of sparse reward and deterministic environments. First, using a very elementary control problem, we illustrate that the learning process can get stuck into a fixed point corresponding to a poor solution. Then, generalizing from the studied example, we provide a detailed analysis of the underlying mechanisms which results in a new understanding of one of the convergence regimes of these algorithms. The resulting perspective casts a new light on already existing solutions to the issues we have highlighted, and suggests other potential approaches., Comment: 19 pages, submitted to ICLR 2020

In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties., Comment: 23 pages. Comments are welcome

We propose a novel reinforcement learning algorithm, AlphaNPI, that incorporates the strengths of Neural Programmer-Interpreters (NPI) and AlphaZero. NPI contributes structural biases in the form of modularity, hierarchy and recursion, which are helpful to reduce sample complexity, improve generalization and increase interpretability. AlphaZero contributes powerful neural network guided search algorithms, which we augment with recursion. AlphaNPI only assumes a hierarchical program specification with sparse rewards: 1 when the program execution satisfies the specification, and 0 otherwise. Using this specification, AlphaNPI is able to train NPI models effectively with RL for the first time, completely eliminating the need for strong supervision in the form of execution traces. The experiments show that AlphaNPI can sort as well as previous strongly supervised NPI variants. The AlphaNPI agent is also trained on a Tower of Hanoi puzzle with two disks and is shown to generalize to puzzles with an arbitrary number of disk

In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods., Comment: 32 pages

In this paper, we provide an overview of first-order and second-order variants of the gradient descent method that are commonly used in machine learning. We propose a general framework in which 6 of these variants can be interpreted as different instances of the same approach. They are the vanilla gradient descent, the classical and generalized Gauss-Newton methods, the natural gradient descent method, the gradient covariance matrix approach, and Newton's method. Besides interpreting these methods within a single framework, we explain their specificities and show under which conditions some of them coincide., Comment: 13 pages

Deep neuroevolution, that is evolutionary policy search methods based on deep neural networks, have recently emerged as a competitor to deep reinforcement learning algorithms due to their better parallelization capabilities. However, these methods still suffer from a far worse sample efficiency. In this paper we investigate whether a mechanism known as "importance mixing" can significantly improve their sample efficiency. We provide a didactic presentation of importance mixing and we explain how it can be extended to reuse more samples. Then, from an empirical comparison based on a simple benchmark, we show that, though it actually provides better sample efficiency, it is still far from the sample efficiency of deep reinforcement learning, though it is more stable.

We study the geometry of non-homogeneous horospherical varieties. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study in detail the case of the horospherical variety associated to the exceptional group $G_2$, and construct a full rectangular Lefschetz exceptional collection in the derived category., Comment: 71 pages

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of $B$-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider $B$-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit $B$-stable canonical divisor on any spherical variety., Comment: 45 pages

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\text{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov., Comment: Appendix by Alexander Kuznetsov. The paper incorporates the results of the earlier preprint arXiv:1510.07903. Comments are welcome

We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws., Comment: 21 pages

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A., Comment: version 2: 28 pages; few examples, and a simplified Chevalley formula in the minuscule case, are added

We give another proof of the generic semisimplicity of the big quantum cohomology of the symplectic isotropic Grassmannians IG(2,2n)., Comment: The results of this paper have been revised and included into arXiv:1705.01819. Please refer to that preprint instead

We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height $2$ in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type $A$ or the element has rank $2$., Comment: 23 pages. Replaces arXiv:1412.5654. New results added

We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also give examples of Fano varieties having a non semisimple small quantum cohomology but a semisimple big quantum cohomology., Comment: 27 pages

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3 point, genus zero) K-theoretic Gromov-Witten invariants of X are determined by the projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety., Comment: 13 pages

For a reductive group G, the products of projective rational varieties homogeneous under G that are spherical for G have been classified by Stembridge. We consider the B-orbit closures in these spherical varieties and prove that under some mild restrictions they are normal, Cohen-Macaulay and have a rational resolution., Comment: 16 pages

In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces, Comment: 5 pages. Just some corrections in the acknowledgements

Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation., Comment: 17 pages

Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees., Comment: 6 pages

Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups., Comment: 40 pages

Using the theory of spherical varieties and especially Frobenius splitting results for symmetric varieties, we give a type independent very short proof of Wahl's conjecture for cominuscule homogeneous spaces for all primes different from 2., Comment: 7 pages

Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$ morphisms from $C$ to $X$ is irreducible, giving an explicit lower bound for $d$ which is optimal in many cases.

The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected., Comment: 16 pages; proofs slightly improved; explicit multiplications in QK(Cayley plane) from v1 no longer necessary

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of $X$. Finally we prove transplanting theorems {\`a} la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces., Comment: 20 pages