For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.
problem Understanding spectral multiplicity and nodal sets for generic torus-invariant metrics.
method Analyzing real Δg-eigenspaces and nodal sets for generic T-invariant metrics. result For generic T-invariant metrics, real Δg-eigenspaces are irreducible and have dimension at most 2, and nodal sets are connected hypersurfaces with specific properties. Characterizes values of slice-torus invariants related to knot genus.
problem Understanding the values of slice-torus invariants for knots.
method Characterization based on stable smooth slice genus.
result Existence of slice torus invariants without explicit constructions.
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are non-trivial.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
The mirror of a projective toric manifold XΣ is given by a Landau-Ginzburg model (Y,W). We introduce a class of Lagrangian submanifolds in (Y,W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over XΣ. Through this ge…
We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for…
In this paper we study the smallest non-zero eigenvalue λ1 of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for λ1 in terms of moment polytope data. We show that this bound can only be attained for CPn endowed with the Fubini-Study metric and therefore CPn endowe…
It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height k. Then it's natural to ask whether …
We prove the existence of torus invariant almost complex structure on any positively omnioriented four dimensional primitive quasitoric orbifold. We construct pseudo-holomorphic blowdown maps for such orbifolds. We prove a version of McKay correspondence when the blowdowns are crepant.
The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow define…
Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.
problem Investigate transverse knots and symplectic surfaces via Seiberg-Witten monopole equations.
method Equivariant Seiberg-Witten theory on branched covers, introducing a novel slice-torus invariant.
result Determine the value of slice-torus invariant for certain Montesinos knots within a deviation of 2.
New invariants from divisibility of Lee classes for slice-torus.
problem Determining slice-torus knots using Lee class divisibility.
method Defined new invariants from divisibility of reduced Lee class invariants.
result New invariants coincide with Rasmussen invariant for certain cases.
In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-…
Lower bounds for a knot invariant are derived using computations and cobordism inequality.
problem Calculating the concordance invariant s# for knots. method Computation for torus knots, cobordism inequality of s#, and arguments for slice-torus invariants. result Lower bounds for s# are derived for knots. In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called…
New invariant defined for tied links in solid torus.
problem Defining an invariant for tied links in solid torus.
method Using skein relations and Jones' method over bt-algebra of type B with Markov trace.
result Recovery of invariant defined for tied links in solid torus.
Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to …
Constructs invariants from Lagrangian tori in symplectic 4-manifolds.
problem Tackles invariants of Lagrangian tori in symplectic 4-manifolds.
method Uses ECH and monopole Floer homology to construct distinguished elements.
result Repackages Gromov and Seiberg-Witten invariants of torus surgeries.
Paper develops knot invariants for long knots in a torus.
problem Understanding long knots in a torus.
method Uses picture-valued and free group valued invariants.
result Developed powerful and easy to compare knot invariants.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.
problem Constructing smooth C∗-actions on moduli spaces of super stable curves and maps of genus zero. method Using the implicit function theorem, proving smooth split atlases, and studying automorphism groups.
result Explicit descriptions of normal bundles to fixed loci in terms of spinor bundles and sections.
This study connects ReLU neural networks to toric geometry to analyze function realization.
problem Determining which continuous piecewise linear functions can be realized by ReLU neural networks.
method Established a connection between toric geometry and ReLU neural networks, defining key structures like the ReLU fan, toric variety, and Cartier divisor.
result Proved a criterion for functions realizable by unbiased shallow ReLU networks using intersection numbers.
Consider a simplicial complex that allows for an embedding into Rd. How many faces of dimension 2d or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we intr…
This thesis surveys various metrics on Riemann surface spaces.
problem Various metrics on Riemann surface spaces.
method Survey of metrics and their properties.
result Equivalence of Kähler-Einstein metric to Teichmüller metric.
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case ∥β∥α>1 is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…
We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
Introduces Finslerian convolution metrics and their properties.
problem No specific problem stated; focuses on new metric concept.
method Definition and study of Finslerian convolution metrics.
result Characterization of Finslerian convolution metrics of Riemannian, Minkowskian, and Randers types.
Survey of recent metric geometry in Kähler metrics space.
problem Understanding the metric geometry of Kähler metrics space.
method Survey and highlighting of recent results.
result Highlighting of open problems in the field.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing σ-extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics. result Existence of σ-extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds. New Finsler metrics defined by Riemannian and 1-forms are studied.
problem Characterize and study properties of (α,β,γ)-metrics. method Introduced and defined (α,β,γ)-metrics, analyzed their properties, and found conditions for local projective flatness and Douglas type. result Necessary and sufficient conditions for (α,β,γ)-metrics to be locally projectively flat and Douglas type were found. We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…
Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Study on special Finsler metrics with conditions for Riemannian and isotropic properties.
problem Characterizing Finsler metrics with specific geometric properties.
method Analyzing conditions for Riemannian and isotropic properties of AR-Finsler metrics.
result Conditions for isotropic S-curvature and mean Landsberg curvature leading to vanishing curvature. The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
Study shows convergence of Lagrangian submanifolds under certain metrics.
problem Understanding convergence of Lagrangian submanifolds under specific metrics.
method Proves convergence to an embedded Lagrangian submanifold using a monotonicity lemma applied on a carefully-chosen metric ball.
result Convergence to an embedded Lagrangian submanifold implies convergence in the Hausdorff metric for a class of metrics.
In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.
Defines a new Randers metric based on an existing one.
problem No specific problem stated; focuses on defining a new metric.
method Defines a new left-invariant Randers metric ildeF based on an existing one F. result Shows that F is of Berwald (Douglas) type if and only if ildeF is of Berwald (Douglas) type. Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…