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Of all publications in the section: 484
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Working paper
Fedor Bogomolov, Tschinkel Y. arxiv.org. math. Cornell University, 2014
We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.
Working paper
Cruz Morales J. A., Galkin S. arxiv.org. math. Cornell University, 2013. No. 1301.4541.
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].
Working paper
Finkelberg M. V., Kubrak D. arxiv.org. math. Cornell University, 2012
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2019. No. 1901.07030.
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.
Working paper
Feigin E., Makhlin I. arxiv.org. math. Cornell University, 2016. No. arXiv:1604.08844 .
FFLV polytopes describe monomial bases in irreducible representations of sln. We study various sets of vertices of FFLV polytopes. First, we prove the locality of set of vertices with respect to the type A Dynkin diagram. Second, we describe all the permutation vertices. Third, we describe all the simple vertices and prove that their number is equal to the large Scr\"oder number. Finally, we derive analogous results for symplectic algebras.
Working paper
Kelbert M., Konakov V., Menozzi S. arxiv.org. math. Cornell University, 2015. No. 1505.04610.
We provide sharp error bounds for the difference between the transition densities of some multidimensional Continuous Time Markov Chains (CTMC) and the fundamental solutions of some fractional in time Partial (Integro) Differential Equations (P(I)DEs). Namely, we consider equations involving a time fractional derivative of Caputo type   and a spatial operator corresponding to the generator of  a non degenerate  Brownian or stable driven Stochastic Differential Equation (SDE).
Working paper
Konakov V., Menozzi S. arxiv.org. math. Cornell University, 2016. No. 1604.00771v2.
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Holder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.
Working paper
Positselski L. arxiv.org. math. Cornell University, 2012. No. 1202.2697.
We define and study the derived categories of the first kind for curved DG and A-infinity algebras complete over a pro-Artinian local ring with the curvature elements divisible by the maximal ideal of the local ring. We develop the Koszul duality theory in this setting and deduce the generalizations of the conventional results about A-infinity modules to the weakly curved case. The formalism of contramodules and comodules over pro-Artinian topological rings is used throughout the paper. Our motivation comes from the Floer-Fukaya theory.
Working paper
Ivan Cheltsov, Constantin Shramov. arxiv.org. math. Cornell University, 2011
We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.
Working paper
Kolesnikov A. arxiv.org. math. Cornell University, 2009. No. 0904.1852.
Given two probability measures $\mu$ and $\nu$ we consider a mass transportation mapping $T$ satisfying 1) $T$ sends $\mu$ to $\nu$, 2) $T$ has the form $T = \phi \frac{\nabla \phi}{|\nabla \phi|}$, where $\phi$ is a function with convex sublevel sets. We prove a change of variables formula for $T$. We also establish Sobolev estimates for $\phi$, and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.
Working paper
Ornea L., Verbitsky M., Vuletescu V. arxiv.org. math. Cornell University, 2015
A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system L, called the conformal weight bundle. The L-valued cohomology of M is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the ddc-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds. Counterexamples to the Morse-Novikov ddc-lemma on Vaisman manifolds were found by R. Goto. We prove that ddc-lemma is true with coefficients in a sufficiently general power La of L on any LCK manifold with potential (this includes Vaisman manifolds). We also prove vanishing of Dolbeault and Bott-Chern cohomology with coefficients in La. The same arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral sequence with coefficients in any power of L.
Working paper
Feigin E., Makhlin I., Fourier G. et al. arxiv.org. math. Cornell University, 2017. No. 1711.00751.
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.
Working paper
Braverman A., Finkelberg M. V. arxiv.org. math. Cornell University, 2012
Working paper
Markarian N. arxiv.org. math. Cornell University, 2015. No.  arXiv:1504.01931 [.
We introduce Weyl n-algebras and show how their factorization homology may be used to define invariants of manifolds. In the appendix we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
Working paper
Markaryan N. S. arxiv.org. math. Cornell University, 2015
We introduce Weyl n-algebras and show how their factorization homology may be used to define invariants of manifolds. In the appendix we heuristically explain why these invariants must be perturbative Chern–Simons invariants.
Working paper
Makhlin I. arxiv.org. math. Cornell University, 2014
We exploit the idea that the character of an irreducible finite dimensional gln-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits of Gelfand-Tsetlin polytopes. Namely, we show that under the relevant substitution the integer point transforms of all but n! vertices vanish, the remaining ones being the summands in Weyl's formula.
Working paper
Ivan Cheltsov. arxiv.org. math. Cornell University, 2014
I prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d\geqslant 3$. I describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. I prove that every reduced plane curve of degree $d\geqslant 4$ whose log canonical threshold is smaller than $\frac{5}{2d}$ is GIT-unstable for the action of the group $\mathrm{PGL}_3(\mathbb{C})$, and I describe GIT-semistable reduced plane curves with log canonical thresholds $\frac{5}{2d}$. I prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest values of the $\alpha$-invariant of Tian of smooth surfaces in $\mathbb{P}^3$ of degree $d\geqslant 3$.
Working paper
Bezrukavnikov R., Finkelberg M. V. arxiv.org. math. Cornell University, 2012. No. 1208.3696.
Working paper
Levin A., Olshanetsky M., Zotov A. arxiv.org. math. Cornell University, 2015
We consider Yang-Baxter equations arising from its associative analog and study corresponding exchange relations. They generate finite-dimensional quantum algebras which have form of coupled GL(N) Sklyanin elliptic algebras. Then we proceed to a natural generalization of the Baxter-Belavin quantum R-matrix to the case Mat(N, C) ⊗ Mat(N, C) ⊗Mat(M, C) ⊗ Mat(M, C) . It can be viewed as symmetric form of GL(NM) R-matrix in the sense that the Planck constant and the spectral parameter enter (almost) symmetrically. Such type (symmetric) R-matrices are also shown to satisfy the Yang-Baxter like quadratic and cubic equations.