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Working paper

Let Sym denote the algebra of symmetric functions and Pμ(⋅;q,t) and Qμ(⋅;q,t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q,t)-Cauchy identity
∑μPμ(x1,x2,…;q,t)Qμ(y1,y2,…;q,t)=∏i,j=1∞(xiyjt;q)∞(xiyj;q)∞
expresses the fact that the Pμ(⋅;q,t)'s form an orthogonal basis in Sym with respect to a special scalar product ⟨⋅,⋅⟩q,t. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric functions
Iμ(x1,x2,…;q,t)=Pμ(x1,x2,…;q,t)+lower degree terms.
These functions come from the N-variate interpolation Macdonald polynomials, extensively studied in the 90's by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions Jμ(⋅;q,t) with the biorthogonality property
⟨Iμ(⋅;q,t),Jν(⋅;q,t)⟩q,t=δμν.
These new functions live in a natural completion of the algebra Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit is also described.

Added: Dec 26, 2017

Working paper

Presburger Arithmetic PrA is the true theory of natural numbers with addition. We consider linear orderings interpretable in Presburger Arithmetic and establish various necessary and sucient conditions for interpretability depending on dimension n of interpretation. We note this problem is relevant to the interpretations of Presburger Arithmetic in itself, as well as the characterization of automatic orderings. For n = 2 we obtain the complete criterion of interpretability.

Added: Nov 28, 2019

Working paper

The intransitive cycle of superiority is characterized by such binary relations between A, B, and C that A is superior to B, B is superior to C, and C is superior to A (i.e., A>B>C>A—in contrast with transitive relations A>B>C). The first part of the article presents a brief review of studies of intransitive cycles in various disciplines (mathematics, biology, sociology, logical games, decision theory, etc.), and their reflections in educational materials. The second part of the article introduces the issue of intransitivity in elementary physics. We present principles of building mechanical intransitive devices in correspondence with the structure of the Condorcet paradox, and describe five intransitive devices: intransitive gears; levers; pulleys, wheels, and axles; wedges; inclined planes. Each of the mechanisms are constructed as compositions of simple machines and show paradoxical intransitivity of relations such as “to rotate faster than”, “to lift”, “to be stronger than” in some geometrical constructions. The article is an invitation to develop teaching materials and problems advancing the understanding of transitivity and intransitivity in various areas, including physics education.

Added: Sep 12, 2018

Working paper

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching invariant spanning trees. This procedure uses finite automata associated with the iterated monodromy action.

Added: Dec 6, 2018

Working paper

We completely solve the inverse Galois problem for del Pezzo surfaces of degree 2 and 3 over all finite fields.

Added: Dec 2, 2018

Working paper

In this paper we study the topological structure of moment-angle complexes ZK. We consider two classes of simplicial complexes. The first class BΔ consists of simplicial complexes K for which ZK is homotopy equivalent to a wedge spheres. The second class WΔ consists of K∈BΔ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that BΔ=WΔ. In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere which cannot be realized by any linear combination of iterated higher Whitehead products.
On the other hand we show that class WΔ is large enough. Namely, we show that the class WΔ is closed with respect to two explicitly defined operations on simplicial complexes. Then using these operations we prove that there exists a simplicial complex that realizes any given iterated higher Whitehead product. Also we describe the smallest simplicial complex that realizes an iterated product with only two pairs of nested brackets.

Added: Feb 2, 2018

Working paper

We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3.

Added: Sep 26, 2016

Working paper

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

Added: Dec 3, 2018

Working paper

For determinantal point processes governed by self-adjoint kernels, we prove in Theorem 1.2 that conditioning
on the configuration in a subset preserves the determinantal property. In Theorem 1.3 we show the tail sigma-
algebra for our determinantal point processes is trivial, proving a conjecture by Lyons. If our self-adjoint kernel is
a projection, then, establishing a conjecture by Lyons and Peres, we show in Theorem 1.5 that reproducing kernels
corresponding to particles of almost every configuration generate the range of the projection. Our argument is
based on a new local property for conditional kernels of determinantal point processes stated in Lemma 1.7.

Added: Feb 8, 2017

Working paper

We propose an r-variable version of Kostka-Shoji polynomials K−λμ for r-multipartitions λ,μ. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of an ample line bundle over Lusztig's iterated convolution diagram for the cyclic quiver Ã r−1.

Added: May 26, 2016

Working paper

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.

Added: Apr 10, 2017

Working paper

In this paper, we derive a stability result for L1 and L∞ perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and we do not require uniform convergence of perturbed diffusions. Instead, we require a weaker convergence condition in a special metric introduced in this paper, related to the Holder norm of the diffusion matrix differences. Our approach is based on a special version of the McKean-Singer parametrix expansion.

Added: Apr 3, 2021

Working paper

In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.

Added: Dec 2, 2018

Working paper

In this paper, we prove that for any A>0 there exist infinitely many primes p for which sums of the Legendre symbol modulo p over an interval of length (ln p)^A can take large values.

Added: Oct 6, 2018

Working paper

Ornea L.,

math. arxive. Cornell University, 2016
An LCK manifold with potential is a compact quotient of a Kahler manifold X equipped with a positive Kahler potential f, such that the monodromy group acts on X by holomorphic homotheties and multiplies f by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and b1(M). Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.

Added: Feb 7, 2016

Working paper

In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these parameters.

Added: Mar 31, 2017

Working paper

Cerulli Irelli G., Fang X.,

et al. math. arxive. Cornell University, 2016. No. 1603.08395.
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A

Added: Mar 30, 2016

Working paper

Let (M,I,J,K,g) be a hyperkahler manifold. Then the complex manifold (M,I) is holomorphic symplectic. We prove that for all real x,y, with x^2+y^2=1 except countably many, any finite energy (xJ+yK)-holomorphic curve with boundary in a collection of I-holomorphic Lagrangians must be constant. By an argument based on the Lojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed in the sense of Fukaya-Oh-Ohta-Ono. Moreover, the Fukaya A_∞ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.

Added: Dec 5, 2018

Working paper

We study local versions of the log-Brunn-Minkowski and p-Brunn-Minkowski conjecture.

Added: Dec 30, 2017

Working paper

A cubic polynomial f with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call f a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials. In particular, we show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets). We prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain cannot have bounded complementary domains containing IS-captures.

Added: Dec 6, 2018

Working paper

The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric onformal maps and the other one is the theor of integrable systems.In this paper we compare the both approaches.
After recalling the derivation of Loewner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarchies (dKP, dBKP, dToda and dDKP). The one-variable reductions are described by solutions of different versions of Loewner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP.We also discuss multi-variable reductions which are given by a system of Loewner equations supplemented by a system of patial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.

Added: Oct 28, 2020