Searcharxiv⌕ Search

arXiv subjects

Andras Szenes

Publications and source records attributed to Andras Szenes.

At least 19 recordsLinked to original sources
arXiv2025

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves

The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980's. In this paper, we give a complete geometric proof of a recursive formula, which reduces the calculation of the intersection Betti numbers of the moduli spaces of semistable bundles on Riemann surfaces in degree-0 and arbitrary rank to the known formulas of the Betti numbers of the smooth, degree-1 moduli spaces. Our formula was motivated by the work of Mozgovoy and Reineke from 2015, and appears as a consequence of the Decomposition Theorem applied to the forgetful map from a parabolic moduli space. We give a detailed description of the topology of this map, and that of the relevant local systems. Our work is self-contained, geometric and focuses on using the multiplicative structures of the theory.

math.AG↗
arXiv2023

Intersection cohomology of type-A toric varieties

Type-A toric varieties may be obtained as GIT quotients with respect to a torus action with weights corresponding to roots of the group $SL(k)$ for some $k>1$. These varieties appear in various important applications, in particular, as normal cones to strata in moduli spaces of vector bundles. In this paper, we describe the intersection Betti numbers of these varieties, and those of some associated projective varieties. We present an elegant combinatorial model for these numbers, and, using the work of Hausel and Sturmfels, we show that the relevant intersection cohomology groups are endowed with a canonical product structure.

math.AG↗
arXiv2022

The parabolic Verlinde formula: iterated residues and wall-crossings

We give a new proof for the parabolic Verlinde formula in all ranks based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residue functionals. On the way, we develop a tautological variant of Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces, and present a new, transparent, local approach to the rho-shift problem of the theory.

math.AG↗
arXiv2020

An Enumerative Approach to $P=W$

The $P = W$ conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank $2$ case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers on the Higgs moduli, and then verify the top perversity part of our enumerative $P = W$ statement for even tautological classes. A key in this calculation is the existence of polynomial solutions to the Discrete Heat Equation satisfying particular vanishing properties. For odd classes, we derive a determinantal criterion for the enumerative $P = W$.

math.AG↗
arXiv2019

Superradiant diamond color center arrays coupled to concave plasmonic nanoresonators

Different types of concave plasmonic nanoresonators have been optimized to achieve superradiantly enhanced emission of SiV color centers in diamond. Comparative study has been performed to consider advantages of different N number of SiV color centers, different diamond-silver (bare) and diamond-silver-diamond (coated) core-shell nanoresonator types, as well as of spherical and ellipsoidal geometry. The complete fluorescence enhancement (qualified by Px factor) monitoring and the cQE corrected quantum efficiency weighted PxcQE objective function optimization promotes to design bad-cavities for plasmonic Dicke effect. The switching into a collective Dicke state via optimized nanoresonators results in a radiated power proportional to N^2, which manifest itself in an enhancement proportional to N both of the excitation and emission rates. Accordingly, enhancement proportional to N^2 of the Px factor and PxcQE has been reached both via four and six SiV color centers arranged in symmetrical square and hexagonal patterns inside all types of inspected nanoresonators. Coated spherical and bare ellipsoidal nanoresonators result in stronger non-cooperative fluorescence enhancement, while superradiance is better achieved via bare spherical nanoresonators independently of SiV color centers number, and via coated (bare) ellipsoidal nanoresonators seeded by four (six) SiV color centers. Indistinguishable superradiant state of four color centers and line-width narrowing is achieved via bare nanoresonators. Six color centers seeded bare spherical (ellipsoidal) nanoresonators result in larger fluorescence enhancement and more significantly overridden superradiance thresholds, while having slightly more (less) pronounced bad-cavity characteristics. Both phenomena are simultaneously optimized in ellipsoidal bare nanoresonators embedding six color centers with a slightly larger detuning.

physics.optics↗
arXiv2018

Residues, Grothendieck polynomials and K-theoretic Thom polynomials

Grothendieck polynomials were introduced by Lascoux and Schützenberger, and they play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $A_2$ singularities. We present the expansion in two versions: one displays its expected stabilization, while the other displays its expected finiteness property.

math.AT↗
arXiv2017

Improved emission of SiV diamond color centers embedded into concave plasmonic core-shell nanoresonators

Configuration of three different concave silver core-shell nanoresonators was numerically optimized to enhance the excitation and emission of embedded silicon vacancy (SiV) diamond color centers simultaneously. According to the tradeoff between the radiative rate enhancement and quantum efficiency (QE) conditional optimization was performed to ensure ~2-3-4 and 5-fold apparent cQE enhancement of SiV color centers with ~10% intrinsic QE. The enhancement spectra, as well as the near-field and charge distribution were inspected to uncover the physics underlying behind the optical responses. The conditionally optimized coupled systems were qualified by the product of the radiative rate enhancements at the excitation and emission, which is nominated as Px factor. The optimized spherical core-shell nanoresonator containing a centralized emitter is capable of enhancing considerably the emission via bonding dipolar resonance. The Px factor is 529-fold with 49.7% cQE at the emission. Decentralization of the emitter leads to appearance of higher order multipolar modes, which is not advantageous caused by their nonradiative nature. Transversal and longitudinal dipolar resonances of the optimized ellipsoidal core-shell resonator were tuned to the excitation and emission, respectively. The simultaneous enhancements result in 6.2x10^5 Px factor with 50.6% cQE at the emission. Rod-shaped concave core-shell nanoresonators exploit similarly transversal and longitudinal dipolar resonances, moreover they enhance the fluorescence more significantly due to their antenna-like geometry. Px factor of 8.34x10^5 enhancement is achievable while the cQE is 50.3% at the emission. The enhancement can result in 2.03x10^6-fold Px factor, when the criterion regarding the minimum QE is set to 20%.

physics.optics↗
arXiv2016

Enhancing diamond color center fluorescence via optimized plasmonic nanorod configuration

A novel numerical methodology has been developed, which makes possible to optimize arbitrary emitting dipole and plasmonic nano-resonator configuration with an arbitrary objective function. By selecting quantum efficiency as the objective function that has to be maximized at preselected Purcell factor criteria, optimization of plasmonic nanorod based configurations has been realized to enhance fluorescence of NV and SiV color centers in diamond. Gold and silver nanorod based configurations have been optimized to enhance excitation and emission separately, as well as both processes simultaneously, and the underlying nanophotonical phenomena have been inspected comparatively. It has been shown that considerable excitation enhancement is achieved by silver nanorods, while nanorods made of both metals are appropriate to enhance emission. More significant improvement can be achieved via silver nanorods at both wavelengths of both color centers. It has been proven that theoretical limits originating from metal dielectric properties can be approached by simultaneous optimization, which results in configurations determined by preferences corresponding to the emission. Larger emission enhancement is achieved via both metals in case of SiV center compared to the NV center. Gold and silver nanorod based configurations making possible to improve SiV centers quantum efficiency by factors of 1.18 and 5.25 are proposed, which have potential applications in quantum information processing.

physics.optics↗
arXiv2015

Optimized Superconducting Nanowire Single Photon Detectors to Maximize Absorptance

Dispersion characteristics of four types of superconducting nanowire single photon detectors, nano-cavity-array- (NCA-), nano-cavity-deflector-array- (NCDA-), nano-cavity-double-deflector-array- (NCDDA-) and nano-cavity-trench-array- (NCTA-) integrated (I-A-SNSPDs) devices was optimized in three periodicity intervals commensurate with half-, three-quarter- and one SPP wavelength. The optimal configurations capable of maximizing NbN absorptance correspond to periodicity dependent tilting in S-orientation (90° azimuthal orientation). In NCAI-A-SNSPDs absorptance maxima are reached at the plasmonic Brewster angle (PBA) due to light tunneling. The absorptance maximum is attained in a wide plasmonic-pass-band in NCDAI_1/2*lambda-A, inside a flat-plasmonic-pass-band in NCDAI_3/4*lambda-A and inside a narrow plasmonic-band in NCDAI_lambda-A. In NCDDAI_1/2*lambda-A bands of strongly-coupled cavity and plasmonic modes cross, in NCDDAI_3/4*lambda-A an inverted-plasmonic-band-gap develops, while in NCDDAI_lambda-A a narrow plasmonic-pass-band appears inside an inverted-minigap. The absorptance maximum is achieved in NCTAI_1/2*lambda-A inside a plasmonic-pass-band, in NCTAI_3/4*lambda-A at inverted-plasmonic-band-gap center, while in NCTAI_lambda-A inside an inverted-minigap. The highest 95.05% absorptance is attained at perpendicular incidence onto NCTAI_lambda-A. Quarter-wavelength type cavity modes contribute to the near-field enhancement around NbN segments except in NCDAI_lambda-A and NCDDAI_3/4*lambda-A. The polarization contrast is moderate in NCAI-A-SNSPDs (~10^2), NCDAI- and NCDDAI-A-SNSPDs make possible to attain considerably large polarization contrast (~10^2-10^3 and ~10^3-10^4), while NCTAI-A-SNSPDs exhibit a weak polarization selectivity (~10-10^2).

physics.optics↗
arXiv2014

A symplectic proof of the Horn inequalities

In this paper, we give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. Our method is a combination of tropical calculus for matrix eigenvalues, combinatorics of planar networks, and estimates for the Liouville volume. As a corollary, we give a tropical description of the Duistermaat-Heckman measure on the Horn polytope.

math.SG↗
arXiv2014

Plasmonic structure integrated single-photon detector configurations to improve absorptance and polarization contrast

Configurations capable of maximizing both absorptance and polarization contrast were determined for 1550 nm polarized light illumination of different plasmonic structure integrated superconducting nanowire single-photon detectors (SNSPDs) consisting of p=264 nm and P=792 nm periodic niobium-nitride (NbN) patterns on silica substrate. Global NbN absorptance maxima appear in case of p/s-polarized light illumination in S/P-orientation (gamma=90 azimuthal angle) and the highest polarization contrast is attained in S-orientation of all devices. Common nanophotonical origin of absorptance enhancement is collective resonance on nano-cavity-gratings with different profiles, which is promoted by coupling between localized modes in quarter wavelength MIM nano-cavities and laterally synchronized Brewster-Zenneck-type surface waves in integrated SNSPDs possessing a three-quarter-wavelength-scaled periodicity. The spectral sensitivity and dispersion characteristics reveal that device design specific optimal configurations exist.

physics.optics↗
arXiv2014

[Q,R]=0 and Kostant partition functions

On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree k. A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on k. This statement was proved by Meinrenken and Sjamaar under positivity conditions. In this paper, we give a new proof of this polynomiality property. The proof is based on a study of the Atiyah-Bott fixed point formula from the point of view of the theory of partition functions, and a technique for localizing positivity.

math.SG↗
arXiv2012

The Horn problem and planar networks

The problem of determining the set of possible eigenvalues of 3 Hermitian matrices that sum up to zero is known as the Horn problem. The answer is a polyhedral cone, which, following Knutson and Tao, can be described as the projection of a simpler cone in the space of triangular tableaux (or hives) to the boundary nodes of the tableau. In this paper, we introduce a combinatorial problem defined in terms of certain weighted planar graphs giving rise to exactly the same polyhedral cone. In our framework, the values at the inner nodes of the triangular tableaux receive a natural interpretation. Other problems of linear algebra fit into the same scheme, among them the Gelfand--Zeitlin problem. Our approach is motivated by the works of Fomin and Zelevinsky on total positivity and by the ideas of tropicalization.

math.CO↗
arXiv2008

Thom polynomials of Morin singularities

We prove a formula for Thom polynomials of Morin (or A_d) singularities in any codimension. We use a combination of the test-curve method of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and they are computationally efficient for d less than 7.

math.AT↗
arXiv2007

Exceptional points for Lebesgue's density theorem on the real line

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither zero nor one. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities are closer to 1/2 than to zero or one. The method of proof uses a combinatorial restatement of the problem.

math.CA↗
arXiv2006

Periodicity of Y-systems and flat connections

We give a proof of the periodicity of Zamolodchikov's Y-systems in the AxA case using a novel interpretation of the system as a condition of flatness of a connection on a certain graph.

math.RT↗