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Alexander Goncharov

Publications and source records attributed to Alexander Goncharov.

At least 19 recordsLinked to original sources
arXiv2024

Quantum geometry of moduli spaces of local systems and representation theory

Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. We introduce a moduli space P(G,S) parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S)) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le. We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.

math.RT↗
arXiv2024

Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties

Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that non-commutative stacks X(m,S) of framed rank m flat R-vector bundles of on decorated surfaces S are birationally identified with the moduli spaces of flat line bundles on a spectral surface assigned to certain bipartite graphs on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry canonical non-commutative Poisson structure. The space X(m,S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We define non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry canonical non-commutative 2-form. The non-commutative stack A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group. The non-commutative cluster A-coordinates on the space A(m,S) are ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves, and use them to give an alternative proof of main results. We show that any stack of Stokes data is a stack of admissible dg-sheaves of certain type. Using this we prove that all stacks of framed Stokes data carry a cluster Poisson structure, equivariant under the wild mapping class group. Therefore they can be equivariantly quantized. Similar stacks of decorated Stokes data carry an equivariant cluster A-variety structure.

math.AG↗
arXiv2023

MORFEO enters final design phase

MORFEO (Multi-conjugate adaptive Optics Relay For ELT Observations, formerly MAORY), the MCAO system for the ELT, will provide diffraction-limited optical quality to the large field camera MICADO. MORFEO has officially passed the Preliminary Design Review and it is entering the final design phase. We present the current status of the project, with a focus on the adaptive optics system aspects and expected milestones during the next project phase.

astro-ph.IM↗
arXiv2023

The inverse spectral map for dimers

In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.

math.AG↗
arXiv2021

Lebesgue Constants For Cantor Sets

We evaluate the values of the Lebesgue constants in polynomial interpolation for three types of Cantor sets. In all cases, the sequences of Lebesgue constants are not bounded. This disproves the statement by Mergelyan.

math.NA↗
arXiv2021

MAORY: A Multi-conjugate Adaptive Optics RelaY for ELT

MAORY is the adaptive optics module for ELT providing two gravity invariant ports with the same optical quality for two different client instruments. It enable high angular resolution observations in the near infrared over a large field of view (~1 arcmin2 ) by real time compensation of the wavefront distortions due to atmospheric turbulence. Wavefront sensing is performed by laser and natural guide stars while the wavefront sensor compensation is performed by an adaptive deformable mirror in MAORY which works together with the telescope's adaptive and tip tilt mirrors M4 and M5 respectively.

astro-ph.IM↗
arXiv2020

Polymorphism of superionic ice

Water is abundant in natural environments but the form it resides in planetary interiors remains uncertain. We report combined synchrotron X-ray diffraction and optical spectroscopy measurements of H2O in the laser-heated diamond anvil cell up to 150 gigapascals (GPa) and 6500 kelvin (K) that reveal first-order transitions to ices with body-centered cubic (bcc) and face-centered cubic (fcc) oxygen lattices above 900 (1300) K and 20 (29) GPa, respectively. We assigned these structures to theoretically predicted superionic phases based on the distinct density, increased optical conductivity, and greatly decreased enthalpies of fusion. Our measurements address current discrepancies between theoretical predictions and various static/dynamic experiments on the existence and location of melting curve and superionic phase(s) in the pressure-temperature phase diagram indicating a possible presence of the conducting fcc-superionic phase in water-rich giant planets, such as Neptune and Uranus.

cond-mat.mtrl-sci↗
arXiv2019

Geometry of canonical bases and mirror symmetry

A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local systems on S. It has a positive structure. So the set of its integral tropical points is defined. We introduce a rational positive function W on A(G, S), the potential. The condition that its tropicalisation is non-negative determines its subset. For SL(2), we recover the set of integral laminations on S. We prove that when S is a disc with n special points on the boundary, this set parametrises top dimensional components of the convolution varieties. Thus, via geometric Satake correspondence, they provide a canonical basis in tensor product invariants of irreducible modules for the Langlands dual group. When G=GL(m), n=3, there is a special coordinate system on A(G,S). We show that it identifies our set with the set of with Knutson-Tao's hives. Our result generalises a theorem of Kamnitzer, who used hives to parametrise top components of convolution varieties for GL(m), n=3. For n>3, we prove Kamnitzer's conjecture. We define canonical bases in tensor products, generalizing the Mirkovic-Vilonen basis in a single representation. We prove that for any S, the set of positive integral tropical points of A(G, S) parametrise top components in a new space, surface affine Grasmannian. We view W as a potential for Landau-Ginzburg model on A(G,S). We conjecture that the pair (A(G,S), W) is the mirror dual to the moduli space of local systems on S for the Langlands dual group. In a special case, we recover Givental's description of the quantum cohomology connection for flag varieties.

math.RT↗
arXiv2018

Lateral distributions of electrons in air showers initiated by ultra-high energy gamma quanta taking into account LPM and geomagnetic field effects

Lateral distributions of electrons in air showers initiated by photons of ultra high energies ($10^{18}-10^{22}$ eV) obtained on the basis of numerical solution of adjoint cascade equations are presented. An extended analysis has been made considering separately the Landau-Pomeranchuk-Migdal (LPM) effect and the interaction of photons and electrons with the geomagnetic field (GMF) with respect to the scaling formalism for lateral distributions. It is shown that one-parametric scaling description of the lateral distribution of electrons remains valid up to the highest energies considering the LPM and GMF effects, that allow effective primary particle type discrimination using the surface detectors data of largest ground-based air shower arrays.

astro-ph.HE↗
arXiv2018

The Galois group of the category of mixed Hodge-Tate structures

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.

math.AG↗
arXiv2016

Mityagin's Extension Problem. Progress Report

Given a compact set $K\subset {\Bbb R}^d,$ let ${\mathcal E}(K)$ denote the space of Whitney jets on $K$. The compact set $K$ is said to have the extension property if there exists a continuous linear extension operator $W:{\mathcal E}(K) \longrightarrow C^{\infty}({\Bbb R}^d)$. In 1961 B. S. Mityagin posed a problem to give a characterization of the extension property in geometric terms. We show that there is no such complete description in terms of densities of Hausdorff contents or related characteristics. Also the extension property cannot be characterized in terms of growth of Markov's factors for the set.

math.FA↗
arXiv2016

Orthogonal polynomials on generalized Julia sets

We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green functions and Parreau-Widom criterion for a special family of real generalized Julia sets.

math.DS↗
arXiv2016

Asymptotic properties of Jacobi matrices for a family of fractal measures

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.

math.SP↗
arXiv2016

Donaldson-Thomas trasnsformations of moduli spaces of G-local systems

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S). We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution * of the space X(m,S) provided by dualising a local system on S. We calculate the DT-transformation of the moduli space X(m,S), with few exceptions. Namely, let C(m,S) be the transformation of the space X(m,S) given by the product of three commuting maps: the involution *, the product, over all punctures of S, of the longest element of the Weyl group action corresponding to the puncture, and the "shift of the special points on the boundary by one" map. Using a characterisation of a class of DT-transformations due to Keller, we prove that C(m,S) = DT. We prove that, burring few exceptions, the Weyl group and the involution * act by cluster transformations of the dual moduli space A(m, S). So the formula C(m,S) = DT is valid for the space A(m,S). Our main result, combined with the work of Gross, Hacking, Keel and Kontsevich, deliver a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S), with few exceptions.

math.AG↗
arXiv2015

Orthogonal polynomials for the weakly equilibrium Cantor sets

Let $K(γ)$ be the weakly equilibrium Cantor type set introduced in [10]. It is proven that the monic orthogonal polynomials $Q_{2^s}$ with respect to the equilibrium measure of $K(γ)$ coincide with the Chebyshev polynomials of the set. Procedures are suggested to find $Q_{n}$ of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below.

math.SP↗
arXiv2015

Symplectic double for moduli spaces of G-local systems on surfaces

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli space D(G,S) is the symplectic double of the Poisson moduli space of framed G-local systems on S. Its symplectic form is upgraded to a K2-symplectic structure for which the special coordinates are K2-Darboux coordinates.

math.AG↗