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Working paper

Quantum games represent the really 21st century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox) the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organising feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather nontrivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

Added: Oct 30, 2020

Working paper

We classify quasi-simple finite groups of essential dimension 3.

Added: Aug 28, 2017

Working paper

We prove that Q-Fano threefolds of Fano index ≥8 are rational.

Added: Jun 8, 2019

Working paper

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.

Added: Aug 19, 2020

Working paper

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

Added: Nov 19, 2019

Working paper

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus g∈{7,8,9,10} over an arbitrary field 𝗄 of zero characteristic. In the case of dimension n≥4 we prove that these varieties are 𝗄-rational if and only if they have a 𝗄-point except for the case of genus 9, where we assume n≥5. Furthermore, we prove that Mukai varieties of genus g∈{7,8,9,10} and dimension n≥5 contain cylinders if they have a 𝗄-point. Finally, we prove that the embedding X↪Gr(3,7) for prime Fano threefolds of genus 12 is defined canonically over any field and use this to give a new proof of the criterion of rationality.

Added: Aug 19, 2020

Working paper

We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of N,M-torus links and the rational q,t-Catalan power series for general positive N,M.

Added: Sep 3, 2019

Working paper

Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus RC consists of implications between formulas built up from propositional variables and constant `true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles.
We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity Π0n, for all n>0. We note that such operators, in a precise sense, cannot be represented in the full language of modal logic.
We formulate a formal system RC∇ extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal <ϵ0. On the other hand, we provide normal forms for its variable-free fragment. Thereby, the variable-free fragment is shown to be decidable and complete w.r.t. its natural arithmetical semantics.
Whereas the normal forms for the variable-free formulas of RC correspond in a unique way to ordinals below ϵ0, the normal forms of RC∇ are more general. It turns out that they are related in a canonical way to the collections of proof-theoretic ordinals of (bounded) arithmetical theories for each complexity level Π0n+1.
Finally, we present an algebraic universal model for the variable-free fragment of RC∇ based on Ignatiev's Kripke frame. Our main theorem states the isomorphism of several natural representations of this algebra.

Added: Dec 26, 2017

Working paper

Let m(n, r) denote the minimal number of edges in an n-uniform hypergraph which is not r-colorable. It is known that for a fixed n one has c_n r^n < m(n, r) < C_n r^n . We prove that for any fixed n the sequence a_r := m(n, r)/r^n has a limit, which was conjectured by Alon. We also prove the list colorings analogue of this statement.

Added: Oct 22, 2019

Working paper

We give a nontrivial lower bound for global dimension of a spherical fusion category.

Added: Dec 6, 2018

Working paper

We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to minimum of a finite number n of affine functions li satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of n products Ii×Ji, where {Ii}, {Ji} are partitions of the line into unions of disjoint connected sets. The families of sets {Ii},{Ji} admit the following properties: 1) c=li on Ii×Ji, 2) {Ii},{Ji} is a couple of partitions solving an auxiliary n-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.

Added: Feb 23, 2016

Working paper

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types A_n and D_n, i.e., flag varieties Fl(1,n;n+1) and isotropic orthogonal Grassmannians OG(2,2n); in particular we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For OG(2,2n) this is the first exceptional collection proved to be full.

Added: Aug 19, 2020

Working paper

Given a probability measure \mu supported on a convex subset \Omega of Euclidean space (\mathbb{R}^d,g_0), we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on (\Omega,g_0,\mu). To this end, we change the metric g_0 to a more general Riemannian one g, adapted in a certain sense to \mu, and perform our analysis on (\Omega,g,\mu). The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when \mu is unconditional, i.e. invariant under reflection with respect to the principle hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on (\Omega,g,\mu) tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--\'Emery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on(\Omega,g_0,\mu): refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincar\'e and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--\'Emery generalized Ricci curvature tensor, and the convexity of the manifold(\Omega,g,\mu). In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

Added: Feb 23, 2016

Working paper

We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifold. We separate out a special category F_0 of leaf manifolds containing the orbifold category as a complete subcategory. Objects of F_0 may be non-Hausdorff unlike orbifolds. The topology of some objects of F_0 do not satisfies the separation axiom T_0. It is shown that for every object N of F_0 a rigid geometry on N admits a desingularization. Moreover, for every such N we prove the existence and the uniqueness of a finite-dimensional Lie group structure on the group of all automorphisms of the rigid geometry on N.

Added: Apr 14, 2017

Working paper

Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M x M there exists a rank 2n-2 reflexive hyperholomorphic sheaf E_M, whose fiber over a non-diagonal point (F,G) is Ext^1(F,G). The sheaf E_M can be deformed along some twistor path to a sheaf E_X over the cartesian square of every Kahler manifold X deformation equivalent to M. We prove that E_X is infinitesimally rigid, and the isomorphism class of the Azumaya algebra End(E_X) is independent of the twistor path chosen. This verifies conjectures in arXiv:1310.5782 and arXiv:1507.03108 on non-commutative deformations of K3 surfaces and renders the results of these two papers unconditional.

Added: Oct 10, 2017

Working paper

Mendelson S.,

math. arxive. Cornell University, 2018
Robust covariance estimation under L_4-L_2 norm equivalence

Added: Oct 8, 2018

Working paper

Izmailov P.,

, Kroptov D. math. arxive. Cornell University, 2017
We propose a method (TT-GP) for approximate inference in Gaussian Process (GP) models. We build on previous scalable GP research including stochastic variational inference based on inducing inputs, kernel interpolation, and structure exploiting algebra. The key idea of our method is to use Tensor Train decomposition for variational parameters, which allows us to train GPs with billions of inducing inputs and achieve state-of-the-art results on several benchmarks. Further, our approach allows for training kernels based on deep neural networks without any modifications to the underlying GP model. A neural network learns a multidimensional embedding for the data, which is used by the GP to make the final prediction. We train GP and neural network parameters end-to-end without pretraining, through maximization of GP marginal likelihood. We show the efficiency of the proposed approach on several regression and classification benchmark datasets including MNIST, CIFAR-10, and Airline.

Added: Oct 20, 2017

Working paper

A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the general framework and survey particular realizations of this approach in types A, B and C.

Added: Oct 15, 2019

Working paper

Cerulli Irelli G.,

, Reineke M. math. arxive. Cornell University, 2015. No. 1508.00264.
Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincar\'e polynomials of these quiver Grassmannians.

Added: Sep 15, 2015

Working paper

Added: Oct 30, 2015

Working paper

Let U be the tautological subbundle on the Grassmannian Gr(k,n). There is a natural morphism Tot(U)→𝔸^n. Using it, we give a semiorthogonal decomposition for the bounded derived category D^b_coh(Tot(U)) into several exceptional objects and several copies of D^b_coh(𝔸n). We also prove a global version of this result: given a vector bundle E with a regular section s, consider a subvariety of the relative Grassmannian Gr(k,E) of those subspaces which contain the value of s. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of s. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case k=1.

Added: Dec 6, 2018