A marked graph diagram is a link diagram possibly with marked -valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…
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The study connects periodic surface homeomorphisms to contact structures using rational open books.
problem Understanding the properties of contact structures associated with periodic surface homeomorphisms.
method Associate rational open books to marked data sets, study contact structures, and prove Stein fillability conditions.
result A class of data sets gives rise to Stein fillable contact structures under certain combinatorial conditions.
In all dimensions, we prove that the marked length spectrum of a Riemannian manifold with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…
Decision making from data involves identifying a set of attributes that contribute to effective decision making through computational intelligence. The presence of missing values greatly influences the selection of right set of attributes and this renders degradation in classification accuracies of the classifiers. As …
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
We are now witnessing the increasing availability of event stream data, i.e., a sequence of events with each event typically being denoted by the time it occurs and its mark information (e.g., event type). A fundamental problem is to model and predict such kind of marked temporal dynamics, i.e., when the next event wil…
We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set…
Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learnin…
Methodology for estimating marked Hawkes processes with neural networks.
problem Estimating conditional intensity of marked Hawkes processes.
method Proposes two models: Shallow Neural Hawkes with marks and Neural Network for Non-Linear Hawkes with Marks.
result Validation on synthetic datasets and real-world cryptocurrency order book data.
New method uses short geodesics to approximate marked length spectrum.
problem Determining a metric from its marked length spectrum.
method Recovering hypotheses from previous work using dynamical tools.
result Approximate values of MLS on a large set determine the metric.
In a wide variety of applications, humans interact with a complex environment by means of asynchronous stochastic discrete events in continuous time. Can we design online interventions that will help humans achieve certain goals in such asynchronous setting? In this paper, we address the above problem from the perspect…
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.
We show that, on an oriented compact surface, two sufficiently -close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are isometric via a diffeomorphism that fixes the boundary. We also prove that the same co…
Study proves rigidity of marked length spectra in contracting group actions.
problem Rigidity of marked length spectra in contracting group actions.
method Unified approach using the Extension Lemma and metric geometry.
result Orbit map is a rough isometry if marked length spectra match.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
Spaces with similar long paths have similar shapes.
problem Comparing shapes of Gromov hyperbolic spaces.
method Examining asymptotic marked length spectra.
result Spaces with identical spectra are roughly isometric.
Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles . With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'
Two Anosov metrics with same boundary distance are isometric.
problem Boundary rigidity for surfaces of Anosov type.
method Transfer principle linking marked length spectrum rigidity to marked boundary distance rigidity.
result Two metrics of Anosov type with the same marked boundary distance are isometric.
Yoshikawa [Yo] conjectured that a certain set of moves on marked graph diagrams generates the isotopy relation for surface links in , and this was proved by Swenton [S] and Kearton and Kurlin [KK]. In this paper, we find another proof of this fact for the case of 2-links (surface links with spherical com…
Paper introduces a neural network-based non-stationary influence kernel for complex event data.
problem Modeling complex, non-stationary, and dependent discrete event data.
method Neural Spectral Marked Point Processes (NSMPP) with a versatile non-stationary influence kernel.
result NSMPP outperforms state-of-the-art models on synthetic and real data.
Paper defines new invariants for surface-links using graph diagrams and magmas.
problem Tackles invariants for surface-links in entropic magmas.
method Uses marked graph diagrams and a generalization of Kauffman bracket magma.
result Defines new invariants for surface-links in 4-space.
We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalen…
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
We introduce a version of Khovanov homology for alternating links with marking data, , inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the page fo…
New model captures time and mark inter-dependence in TPPs.
problem Limited predictive performance of conditionally independent TPP models on entangled time and mark interactions.
method Developed a multivariate TPP that models conditional inter-dependence of time and mark, using both intensity-based and intensity-free models.
result Proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks.
This paper extends the analysis of Muni Toke and Yoshida (2020) to the case of marked point processes. We consider multiple marked point processes with intensities defined by three multiplicative components, namely a common baseline intensity, a state-dependent component specific to each process, and a state-dependent …
Proposes a new framework to disentangle event influences in MTPP.
problem Underexplored how individual events influence overall dynamics over time.
method Decoupled MTPP framework using Neural Ordinary Differential Equations (Neural ODEs).
result Significantly improves performance on real-life datasets compared to state-of-the-art methods.
New model for multivariate discrete event data with flexible interactions.
problem Modeling multivariate discrete event data with categorical interactions.
method Developed a new modeling approach with convex constraints, two estimation procedures (LS and ML).
result Proposed model can capture arbitrary shapes of historical event influence.
A new method clusters rows of a matrix of point processes.
problem Challenges in analyzing structured point process data.
method Mixture model of multi-level marked point processes, combined with ES algorithm and FPCA.
result An efficient method for clustering rows of a matrix of point processes.
We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determin…
We are interested in the 3-Calabi-Yau categories arising from quivers with potential associated to a triangulated marked surface (without punctures). We prove that the spherical twist group ST of is isomorphic to a subgroup (generated by braid twists) of the mapping class group …
We use Fox calculus to assign a marked polytope to a `nice' group presentation with two generators and one relator. Relating the marked vertices to Novikov-Sikorav homology we show that they determine the Bieri-Neumann-Strebel invariant of the group. Furthermore we show that in many cases the marked polytope is an inva…
Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.
problem Determine the -primary component of Farrell cohomology for non-orientable surfaces.
method Classify subgroups of order using topological equivalence adapted to surfaces with marked points.
result Determine the -primary component of Farrell cohomology for non-orientable surfaces.
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface wit…
PUMA augments models to remove unique data points without performance loss.
problem Preserving model performance while removing unique training data points.
method Explicitly models data influence, reweights remaining data optimally.
result PUMA effectively removes unique data points without performance degradation.
Develops a framework for modeling set-valued data in continuous-time.
problem Handling sequences where each event is associated with a set of items.
method General framework for modeling set-valued data, developed inference methods, and importance sampling techniques.
result Orders-of-magnitude improvements in efficiency for probabilistic queries over direct sampling.
Quantum duality map extended to general marked surfaces and its compatibility with skein algebras proven.
problem Generalizing quantum duality map to general marked surfaces and proving its compatibility with skein algebras.
method Generalized quantum duality map, reduced stated skein algebras, quantum trace maps, skein lifting.
result Compatibility of quantum duality map with skein algebras proven.
In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a ho…
Paper shows non-arithmetic surface with unique geometric property.
problem Non-arithmetic surfaces with unique geometric properties.
method Example of a non-arithmetic surface with marked length variety rigidity.
result Found a non-arithmetic surface with marked length variety rigidity.
On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphismmath.GT
In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary compone…
We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space o…
Framework handles both exchangeable and non-exchangeable event sequences without tuning.
problem Handling both exchangeable and non-exchangeable event sequences efficiently.
method Parametric Hawkes-process-inspired conditional probability mass function with variational inference.
result Competitive computational and predictive performance against state-of-the-art methods.
A graph helps understand Artin groups better.
problem Understanding the structure of Artin groups.
method Constructing a marking graph with transverse parabolic subgroups and defining natural moves.
result The marking graph is quasi-isometric to the group modulo its center.
New method recalibrates VaR for option books, reducing forecast errors.
problem Inaccurate VaR forecasts due to missing operational choices.
method Marking-aware sequential VaR recalibration targeting normalized book-level loss.
result Sequential VaR recalibration improves VaR performance across different markets and options.
Crimes emerge out of complex interactions of human behaviors and situations. Linkages between crime incidents are highly complex. Detecting crime linkage given a set of incidents is a highly challenging task since we only have limited information, including text descriptions, incident times, and locations. In practice,…
A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of …