Unified approach to conformal and modular invariants on surfaces.
problem Constructing a general family of conformal invariants on surfaces.
method Using an identification of Teichmüller space and rigged moduli space, and analytic work on harmonic functions.
result Unified conformal and modular invariants can be viewed as generalized modular invariants and functions on the rigged moduli space.
RIG extends IG to Riemannian manifolds for explainable AI.
problem Lack of explainability in AI models.
method Extension of Integrated Gradients to Riemannian manifolds.
result RIG restricts to IG in Euclidean space.
New ML method detects incomplete bid-rigging cartels.
problem Detecting incomplete bid-rigging cartels in competitive bidding.
method Combines statistical screens with machine learning.
result Algorithm outperforms existing methods in incomplete cartels.
Enhances Koopman operator estimation with intrinsic observables in RKHS.
problem Accurate estimation of Koopman operator and its spectrum.
method Jet Extended Dynamic Mode Decomposition (JetEDMD) leveraging RKHS jets.
result Proves JetEDMD's superiority with error bounds and convergence rate.
Let $x:M\to\Bm$ be the canonical injection of a Null Hypersurface (M,g) in a semi-Riemannian manifold (M,gˉ). A rigging for M is a vector field L defined on some open set of M containing M such that Lp∈/TpM for each p∈M. Such a vector field induces a null rigging N.…
Rigging technique introduced in \cite{bi0} is a convenient way to address the study of null hypersurfaces. It offers in addition the extra benefit of inducing a Riemannian structure on the null hypersurface which is used to study geometric and topological properties on it. In this paper we develop this technique showin…
Deep learning detects bid-rigging cartels with high accuracy.
problem Detecting bid-rigging cartels using pairwise bidding interactions.
method Convolutional neural networks applied to graphs of normalized bid values.
result Convolutional neural networks achieve around 90% accuracy in classifying collusive and competitive bidding interactions.
Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.
problem Properties of Lorentzian manifolds influenced by totally geodesic null hypersurfaces.
method Coupling rigging technique with null foliation existence to prove Riemann flow structure.
result Proves curvature conditions restrict causal structure of spacetime.
We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.…
The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special rig…
Using the classification of transitive groups we classify indecomposable quandles of size <36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all indecomposable quandles of size <36 not of type D is computed.
This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical …
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
Constructs moduli spaces for complex affine and dilation surfaces.
problem Classifying and understanding moduli spaces of complex surfaces.
method Using Veech's ideas, constructs holomorphic affine bundles and covering spaces.
result Moduli spaces of dilation surfaces are orbifold K(G,1) where G is the framed mapping class group.
New Poisson structures found on Higgs bundle moduli spaces.
problem Constructing Poisson structures on moduli spaces of Higgs bundles.
method Via Lie algebroids on stacky curves, focusing on parabolic Higgs bundles.
result Provides new examples of Poisson structures on moduli spaces.
Study special Lagrangian moduli spaces with boundary.
problem Understanding geometric structures on moduli spaces of special Lagrangians.
method Investigates geometric structures and constructs special affine structures and a Hessian metric.
result Constructs a pair of special affine structures and a Hessian metric on the moduli space.
Constructs projective moduli spaces for Calabi-Yau pairs.
problem Creating projective moduli spaces for Calabi-Yau surface pairs.
method Constructs projective asymptotically good moduli spaces.
result Provides a wall crossing between KSBA and K-moduli spaces.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. Study of Hitchin moduli spaces over Teichmüller space.
problem Metric aspects of Hitchin moduli spaces over varying complex structures.
method Gauge theoretical approach, Kähler fibrations, moment map interpretation, symplectic reduction.
result Establishes natural complex and pseudo-Kähler structures on universal Hitchin moduli spaces.
The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.
problem Comparing constrained and decoupled moduli spaces of manifolds with embedded particles and discs.
method Generalized Bödigheimer--Tillmann's work to higher dimensions and different tangential structures.
result New results for surfaces with different tangential structures and higher dimensional manifolds.
We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama m…
Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.
problem Proving the quasi-projectivity of the moduli space of Higgs bundles.
method Uses analytic methods and the symplectic cut to construct a compactification.
result Proves the quasi-projectivity of the moduli space of Higgs bundles.
Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.
problem Constructing moduli spaces over Riemann surfaces.
method Finite-dimensional construction using holomorphic symplectic reduction.
result Moduli spaces over Riemann surfaces as stratified holomorphic symplectic spaces.
We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under B-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decompo…
Researchers compute the heterotic moduli-space metric up to α′2.
problem Computing the heterotic moduli-space metric up to a specific order.
method Using α′2 for backgrounds with a smooth α′o0 limit, focusing on the Kaehler potential and metric corrections. result Metric corrections from the deformation of the Hull connection are identified.
Homological stability fails for 4-manifold moduli spaces, detected by new class.
problem Homological instability for moduli spaces of 4-manifolds.
method Used Seiberg-Witten equations to construct a new characteristic class.
result Homological discrepancy between smooth and topological 4-manifold moduli spaces.
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
The purpose of the present work is to study (marginally) trapped submanifolds lying in a null hypersurface. Let $(M,g,N)\to\Bm(c)$ be a null hypersurface of a space-time with constant sectional curvature c, endowed with a Screen Integrable and Conformal rigging N. The (Marginally) Trapped Submanifolds we are intere…
We shall construct a moduli space of pairs of Kähler-Einstein structures and special lagrangians and obtain smoothness of the moduli space of these pairs. Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group.
The paper studies the moduli space of Higgs pairs and their geometric properties.
problem The moduli space of Higgs pairs and its geometric properties.
method Introduced τ-stability of Higgs pairs and established the Kobayashi-Hitchin correspondence. result Proved that the moduli space is a non-singular complex manifold for a suitable choice of τ. This note finds explicit representatives for moduli space of parabolic bundles.
problem Finding explicit representatives in deRham cohomology for the moduli space of parabolic bundles.
method Using results from vector bundles and computing intersection pairing of cohomology.
result Explicit representatives in deRham cohomology for the generators of the moduli space of parabolic bundles.
The study finds non-trivial elements in moduli spaces of curvature metrics.
problem Identifying non-trivial elements in moduli spaces of curvature metrics.
method Constructing elements in homotopy groups of moduli spaces of positive curvature metrics.
result Non-trivial elements found in moduli spaces of positive curvature metrics.
Study moduli spaces of solutions to Bogomolny equations on surfaces.
problem Understanding solutions to Bogomolny equations on surfaces with specific boundary conditions.
method Refined Kobayashi-Hitchin correspondence and identification with Higgs bundles.
result Identified moduli spaces as holomorphic lagrangian sub-manifolds.
Study of tautological forms on curve moduli spaces.
problem Understanding tautological forms on moduli spaces of curves.
method Defined and studied a system of tautological rings on moduli spaces of marked curves, showing certain 2-forms are tautological and rings are finite dimensional.
result Characterized the Kawazumi-Zhang invariant as a tautological form.
We construct proper good moduli spaces parametrizing K-polystable Q-Gorenstein smoothable log Fano pairs (X,cD), where X is a Fano variety and D is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as c varies. The main applicatio…
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the A∞ structure of the both …
We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Universal geometry organizes heterotic moduli spaces.
problem Understanding the structure of heterotic compactifications.
method Fibering compactification data over moduli space and analyzing universal curvatures.
result Deformations are components of universal curvatures, incorporating α′2 corrections. Same cohomology for curves with levels, proving stable range.
problem Stability of cohomology in moduli spaces with level structures.
method Proving cohomology equality through stable range analysis.
result Rational cohomology of moduli space with levels matches ordinary space.
We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
Study describes moduli spaces of flat bundles on Sasakian manifolds.
problem Understanding moduli spaces of flat bundles on Sasakian manifolds.
method Shows moduli space of simple flat bundles is a union of spaces with fixed basic structures.
result Detailed description of non-abelian Hodge correspondence on compact Sasakian manifolds.
This paper shows moduli spaces of RCD(0,2) structures are contractible.
problem Understanding moduli spaces of RCD(0,2) structures.
method Established a list of compact topological spaces admitting RCD(0,2) structures and described their associated moduli spaces.
result All moduli spaces of RCD(0,2) structures are contractible.