arXiv subjects

Gergő Pintér

Publications and source records attributed to Gergő Pintér.

At least 19 recordsLinked to original sources

arXiv2026

The boundary of the Milnor fibre and a linking invariant of finitely determined germs

The image of a finitely determined holomorphic germ $Φ$ from $\mathbb{C}^2$ to $\mathbb{C}^3$ defines a hypersurface singularity $(X,0)$, which is in general non-isolated. We show that the diffeomorphism type of the boundary of the Milnor fibre $\partial F$ of $X$ is a topological invariant of the germ $Φ$. We establish a correspondence between the gluing coefficients (so-called vertical indices) used in the construction of $\partial F$ and a linking invariant $L$ of the associated sphere immersion introduced by T. Ekholm and A. Szűcs. For this we provide a direct proof of the equivalence of the different definitions of $L$. Since $L$ can be expressed in terms of the cross cap number $C(Φ)$ and the triple point number $T(Φ)$ of a stable deformation of $Φ$, we obtain a relation between these invariants and the vertical indices. This is illustrated on several examples.

math.GT↗

arXiv2026

Public transport in the 15-minute city

The 15-minute city is a powerful planning concept to counter car-dependence by promoting active mobility to amenities and fostering inclusive urban environments. However, this policy has challenges in amenity-poor urban peripheries. Public transport remains underexplored in this discourse despite its role in distant access. Here, we propose a framework that incorporates public transport into the 15-minute city model using openly available data. By comparing Helsinki, Madrid, and Budapest, we demonstrate that multimodal mobility substantially increases access to amenities and enhances socio-spatial integration within a 15-minute reach. Although urban periphery benefit significantly from radial or high-speed public transport lines in their social mixing potential, such lines alone do not improve their access to amenities. These findings underscore the need to optimize polycentric public transport networks that can improve inclusive urban accessibility and complement active mobility in polycentric cities.

physics.soc-ph↗

arXiv2026

The geometry of the Hermitian matrix space and the Schrieffer--Wolff transformation

In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.

quant-ph↗

arXiv2025

Birth Quota of Non-Generic Degeneracy Points

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Non-generic isolated two-fold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine-tuning or protecting symmetry. The number of the resulting Weyl points is at least $|Q|$, where $Q$ is the topological charge associated to the non-generic degeneracy point. Here, we show that such a non-generic degeneracy point also has a birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. The birth quota is a local multiplicity associated to the non-generic degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for band structures of two- and three-dimensional crystals. Our work establishes a strong and powerful connection between singularity theory and topological band structures, and more broadly, parameter-dependent quantum systems.

cond-mat.mes-hall↗

arXiv2025

Multifold degeneracy points of quantum systems and singularities of matrix varieties

Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally lead to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic two-fold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail, and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multi-fold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.

math-ph↗

arXiv2024

Stability of Weyl node merging processes under symmetry constraints

Changes in the number of Weyl nodes in Weyl semimetals occur through merging processes, usually involving a pair of oppositely charged nodes. More complicated processes involving multiple Weyl nodes are also possible, but they typically require fine tuning and are thus less stable. In this work, we study how symmetries affect the allowed merging processes and their stability, focusing on the combination of a two-fold rotation and time-reversal ($C_2\mathcal{T}$) symmetry. We find that, counter-intuitively, processes involving a merging of three nodes are more generic than processes involving only two nodes. Our work suggests that multi-Weyl-merging may be observed in a large variety of quantum materials, and we discuss SrSi$_2$ and bilayer graphene as potential candidates

cond-mat.mes-hall↗

arXiv2024

Quantifying Barriers of Urban Mobility

Barriers in cities, such as administrative boundaries, natural obstacles, railways or major roads are thought to induce segregation. However, the empirical knowledge about this phenomenon is limited. Here, we present a network science framework to assess barriers to urban mobility along their hierarchy, across residential areas and visited amenities. Using GPS mobility data, we construct a network of blocks from the sequence of individual stays in a major European city. A community detection algorithm allows us to partition this network into non-overlapping areas of dense mobility clusters, in which the effect of transportation hubs can be tuned with a parameter. We apply the Symmetric Area Difference index to quantify the overlap between these mobility clusters and the polygons of urban area separated by barriers. Reducing the effect of transportation hubs results in smaller scale mobility clusters that fit better to lower rank administrative or road barriers compared to their higher rank pairs. We find that characteristic urban barriers can replace each other in dividing mobility clusters of different scales. Next, we define the Barrier Crossing Ratio, the fraction of barrier crossings that bridge mobility clusters. The decomposition of this indicator by origins and destinations suggests a significantly higher impact of barriers on those who live closer to the city center and smaller impact on visits to complex amenities. These results contribute to the ongoing discourse on urban segregation, emphasizing the importance of barriers to urban mobility in shaping interactions and mixing.

physics.soc-ph↗

arXiv2024

Revealing urban area from mobile positioning data

Researchers face the trade-off between publishing mobility data along with their papers while simultaneously protecting the privacy of the individuals. In addition to the fundamental anonymization process, other techniques, such as spatial discretization and, in certain cases, location concealing or complete removal, are applied to achieve these dual objectives. The primary research question is whether concealing the observation area is an adequate form of protection or whether human mobility patterns in urban areas are inherently revealing of location. The characteristics of the mobility data, such as the number of activity records or the number of unique users in a given spatial unit, reveal the silhouette of the urban landscape, which can be used to infer the identity of the city in question. It was demonstrated that even without disclosing the exact location, the patterns of human mobility can still reveal the urban area from which the data was collected. The presented locating method was tested on other cities using different open data sets and against coarser spatial discretization units. While publishing mobility data is essential for research, it was demonstrated that concealing the observation area is insufficient to prevent the identification of the urban area. Furthermore, using larger discretization units alone is an ineffective solution to the problem of the observation area re-identification. Instead of obscuring the observation area, noise should be added to the trajectories to prevent user identification.

cs.SI↗

arXiv2024

Signature of the Milnor fiber of parametrized surfaces

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Szűcs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.

math.AG↗

arXiv2023

Singularity theory of Weyl-point creation and annihilation

Weyl points (WP) are robust spectral degeneracies, which can not be split by small perturbations, as they are protected by their non-zero topological charge. For larger perturbations, WPs can disappear via pairwise annihilation, where two oppositely charged WPs merge, and the resulting neutral degeneracy disappears. The neutral degeneracy is unstable, meaning that it requires the fine-tuning of the perturbation. Fine-tuning of more than one parameter can lead to more exotic WP mergers. In this work, we reveal and analyze a fundamental connection of the WP mergers and singularity theory: phase boundary points of Weyl phase diagrams, i.e., control parameter values where Weyl point mergers happen, can be classified according to singularity classes of maps between manifolds of equal dimension. We demonstrate this connection on a Weyl--Josephson circuit where the merger of 4 WPs draw a swallowtail singularity, and in a random BdG Hamiltonian which reveal a rich pattern of fold lines and cusp points. Our results predict universal geometrical features of Weyl phase diagrams, and generalize naturally to creation and annihilation of Weyl points in electronic (phononic, magnonic, photonic, etc) band-structure models, where Weyl phase transitions can be triggered by control parameters such as mechanical strain.

math-ph↗

arXiv2023

Amenity complexity and urban locations of socio-economic mixing

Cities host diverse people and their mixing is the engine of prosperity. In turn, segregation and inequalities are common features of most cities and locations that enable the meeting of people with different socio-economic status are key for urban inclusion. In this study, we adopt the concept of economic complexity to quantify the sophistication of amenity supply at urban locations. We propose that neighborhood complexity and amenity complexity are connected to the ability of locations to attract diverse visitors from various socio-economic backgrounds across the city. We construct the measures of amenity complexity based on the local portfolio of diverse and non-ubiquitous amenities in Budapest, Hungary. Socio-economic mixing at visited third places is investigated by tracing the daily mobility of individuals and by characterizing their status by the real-estate price of their home locations. Results suggest that measures of ubiquity and diversity of amenities do not, but neighborhood complexity and amenity complexity are correlated with the urban centrality of locations. Urban centrality is a strong predictor of socio-economic mixing, but both neighborhood complexity and amenity complexity add further explanatory power to our models. Our work combines urban mobility data with economic complexity thinking to show that the diversity of non-ubiquitous amenities, central locations, and the potentials for socio-economic mixing are interrelated.

physics.soc-ph↗

arXiv2023

Weyl points in ball-and-spring mechanical systems

Degeneracy points of parameter-dependent Hermitian matrices play a fundamental role in quantum physics, as illustrated by the concept of Berry phase in quantum dynamics, the Weyl semimetals in condensed-matter physics, and the robust ground-state degeneracies in topologically ordered quantum systems. Here, we construct simple ball-and-spring mechanical systems, whose eigenfrequency degeneracies mimic the behaviour of degeneracy points of electronic band structures. These classical-mechanical arrangements can be viewed as de-quantized versions of Weyl Josephson circuits, i.e., superconducting nanostructures proposed recently to mimic band structure effects of Weyl semimetals. In the mechanical setups we study, we identify degeneracy patterns beyond simple Weyl points, including the chirality flip effect and a quadratic degeneracy point. Our theoretical work is a step toward simple and illustrative table-top experiments exploring topological and differential geometrical aspects of physics.

cond-mat.mes-hall↗

arXiv2023

Cross-caps, triple points and a linking invariant for finitely determined germs

It was recently proved that for finitely determined germs $ Φ: ( \mathbb{C}^2, 0) \to ( \mathbb{C}^3, 0) $ the number $C(Φ)$ of Whitney umbrella points and the number $T(Φ)$ of triple values of a stable deformation are topological invariants. The proof uses the fact that the combination $C(Φ)-3T(Φ)$ is topological since it equals the linking invariant of the associated immersion $S^3 \looparrowright S^5$ introduced by Ekholm and Szűcs. We provide a new, direct proof for this equality. We also clarify the relation between various definitions of the latter invariant.

math.AT↗

arXiv2022

Weyl-point teleportation

In this work, we describe the phenomenon of Weyl-point teleportation. Weyl points usually move continuously in the configuration parameter space of a quantum system when the control parameters are varied continuously. However, there are special transition points in the control space where the continuous motion of the Weyl points is disrupted. In such transition points, an extended nodal structure (nodal line or nodal surface) emerges, serving as a wormhole for the Weyl points, allowing their teleportation in the configuration space. A characteristic side effect of the teleportation is that the motional susceptibility of the Weyl point diverges in the vicinity of the transition point, and this divergence is characterized by a universal scaling law. We exemplify these effects via a two-spin model and a Weyl Josephson circuit model. We expect that these effects generalize to many other settings including electronic band structures of topological semimetals.

cond-mat.mes-hall↗

arXiv2022

Evaluating the Socioeconomic Status of a Large Social Event Attendees

In this study, Call Detail Records (CDRs) from downtown Budapest were analysed, focusing on a large-scale event in August 2014. The attendees of the main event of the Hungarian State Foundation Day have been analysed based on their Socioeconomic Status (SES). This paper proposes an approach to estimating SES by the price and age of the subscribers' phones, obtained by fusing a mobile phone property database with the CDRs. We have found some tendencies between the attendees based on the location, from where they watched the fireworks. However, the results do not show significant differences in this geographical granularity.

cs.SI↗

arXiv2022

Awakening City: Traces of the Circadian Rhythm within the Mobile Phone Network Data

In this study, Call Detail Records (CDR), covering Budapest, Hungary has been processed to analyze the circadian rhythm of the subscribers. An indicator, called wake-up time, is introduced to describe a behavior of a group of subscribers. It is defined as the time, when the mobile phone activity of a group rises in the morning. Its counterpart is the time, when the activity falls in the evening. Inhabitant and area-based aggregation are also presented. The former is to consider the people who live in an area, the latter uses the transit activity in an area to describe the behavior of a part of the city. The opening hours of the malls and the nightlife of the party district was used to demonstrate this application, as real-life examples. The proposed approach was also used to estimate the working hours of the workplaces. The findings are in a good agreement with practice in Hungary, and also support the workplace detection method. Negative correlation was found between wake-up time and mobility indicators (Entropy, Radius of Gyration): On workdays, people wake up earlier and travel more, on holidays it is quite the contrary. The wake-up time was evaluated in different socioeconomic classes, using housing prices and mobile phones prices, as well. It was found that lower socioeconomic groups tend to wake up earlier.

cs.SI↗

arXiv2021

Analyzing the Behavior and Financial Status of Soccer Fans from a Mobile Phone Network Perspective: Euro 2016, a Case Study

In this study, Call Detail Records (CDRs), covering Budapest, for the month of June in 2016 has been analyzed. During this observation period, the 2016 UEFA European Football Championship took place, which affected significantly the habit of the residents, despite the fact that not a single match was played in the city. We evaluated the fans' behavior in Budapest, during and after the Hungarian matches, and found that the mobile phone network activity reflects the football fans' behavior, demonstrating the potential of mobile phone network data within a social sensing system. The Call Detail Records are enriched with mobile phone properties to analyze the subscribers' devices. Applying the device information (Type Allocation Code) from the activity records, the Subscriber Identity Modules, that do not operate in cell phones are omitted from mobility analyses, allowing to focus on people. The mobile phone price is proposed and evaluated as a socioeconomic indicator, and correlation between the phone price and the mobility customs have been found. We also found that, beside the cell phone price, the subscriber age and the subscription type also have an effect on the mobility. On the other hand, these do not seem to affect the interest in football.

cs.SI↗

arXiv2021

Analyzing the Behavior of Soccer Fans from a Mobile Phone Network Perspective: Euro 2016, a Case Study

In this study, we analyzed one month of Call Detail Records, covering Budapest in June, 2016. During this observation period, the 2016 UEFA European Football Championship took place, where the Hungarian national football team did well, beyond expectations. This case study focuses on the fans' behavior in Budapest, during and after the Hungarian matches, from the perspective of the mobile phone network data. Tense rooting, joy of scoring a goal, self-forgetting celebration because of winning the group. All of these have an effect on the data, that can be simply revealed. We used descriptive statistical methods to analyze the available data set, and managed to demonstrate the potential of this kind of data at social sensing. We found that the mobile phone network activity reflects the football fans' behavior, even if the match is played in another country.

physics.soc-ph↗