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Of all publications in the section: 323
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Working paper
Akhmedova V., Takebe T., Zabrodin A. math. arxive. Cornell University, 2020. No. 2010.02277.
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.
Working paper
Prokhorov Y. math. arxive. Cornell University, 2017
We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.
Working paper
Gritsenko V., Nikulin V. V. math. arxive. Cornell University, 2016
We describe a new large class of Lorentzian Kac–Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac–Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.
Working paper
Maximov Y. math. arxive. Cornell University, 2015
Working paper
Belomestny D., Panov V., Woerner J. math. arxive. Cornell University, 2016. No. 1607.00896.
In this paper we study the problem of statistical inference for a continuous time moving average L\'evy process $$Z$$ observed at low-frequency. We construct a consistent estimator for the L\'evy triplet of $$Z$$, derive its convergence rates and prove their optimality. The performance of our estimation procedure is illustrated by numerical example.
Working paper
Huang L., Menozzi S., Priola E. math. arxive. Cornell University, 2016. No. 1607.08718.
Working paper
Menozzi S. math. arxive. Cornell University, 2016. No. 1404.0461.
Working paper
Kolesnikov A. math. arxive. Cornell University, 2018
We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩. We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that K satisfies the following analog of the Gaussian transportation inequality for the uniform measure σ on Sn−1: 1nEnt(ν)≥K(σ,ν). We show that there exists a remarkable similarity between our results and the theory of the K{\"a}hler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.
Working paper
Loginov K., Moraga J. math. arxive. Cornell University, 2020
We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition.
Working paper
Amerik E., Verbitsky M. math. arxive. Cornell University, 2018
An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation space has minimal possible dimension 2n-2, where 2n is the complex dimension of M. We study the MBM loci, defined as the subvarieties covered by deformations of an MBM curve within M. When M is projective, MBM loci are centers of birational contractions. For each MBM class z, we consider the Teichmuller space Teich^min_z of all deformations of M such that z^⊥ contains a face of the Kahler cone. We prove that for all I,J∈Teich^min_z, the MBM loci of (M, I) and (M,J) are homeomorphic under a homeomorphism preserving the MBM curves, unless possibly the Picard number of I or J is maximal.
Working paper
Jabir J. M., Šiška D., Szpruch L. math. arxive. Cornell University, 2020
We develop a framework for the analysis of deep neural networks and neural ODE models that are trained with stochastic gradient algorithms. We do that by identifying the connections between control theory, deep learning and theory of statistical sampling. We derive Pontryagin’s optimality principle and study the corresponding gradient flow in the form of Mean-Field (overdamped) Langevin dynamics (MFLD) for solving relaxed data-driven control problems. Subsequently, we study uniform-in-time propagation of chaos of time-discretised MFLD. We derive explicit convergence rate in terms of the learning rate, the number of particles/model parameters and the number of iterations of the gradient algorithm. In addition, we study the error arising when using a finite training data set and thus provide quantitive bounds on the generalisation error. Crucially, the obtained rates are dimensionindependent. This is possible by exploiting the regularity of the model with respect to the measure over the parameter space.
Working paper
Trepalin A. math. arxive. Cornell University, 2017
Let X be a minimal del Pezzo surface of degree 2 over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q/𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup of the Weyl group W(E_7). There are 60 conjugacy classes of cyclic subgroups in W(E_7) and 18 of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo surfaces of degree 2 can be achieved for given q.
Working paper
Finkelberg M., Braverman A., Ginzburg V. et al. math. arxive. Cornell University, 2019
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of GL(N−1,ℂ[[t]])-equivariant perverse sheaves on the affine Grassmannian of GL_N. We explain how our equivalences fit into a more general conjectural framework proposed by D. Gaiotto.
Working paper
Timorin V., Ross P., Lex O. et al. math. arxive. Cornell University, 2017
Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.
Working paper
Pavlov A., Buchweitz R. math. arxive. Cornell University, 2015. No. 1511.05502.
We give normal forms of determinantal representations of a smooth projective plane cubic in terms of Moore matrices. Building on this, we exhibit matrix factorizations for all indecomposable vector bundles of rank 2 and degree 0 without nonzero sections, also called Ulrich bundles, on such curves.
Working paper
Pochinka O., Morozov A. math. arxive. Cornell University, 2019
In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.
Working paper
Pushkar P. E. math. arxive. Cornell University, 2019. No. arXiv:1912.06437.
Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper.
Working paper
Verbitsky M., Sampaio J. E., Birbrair L. et al. math. arxive. Cornell University, 2018
It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture, constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.
We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}\ldots s_1) of the longest element in the Weyl group, and the Schubert subvarieties correspond to the terminal subwords in this decomposition. We prove that the resulting Newton--Okounkov polytopes for semiample line bundles satisfy the additivity property with respect to the Minkowski sum. In particular, they are Minkowski sums of Newton--Okounkov polytopes of line bundles on the complete flag varieties for GL_2,... , GL_{n}.