4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Lecture notes on conifold transitions between Calabi-Yau manifolds.
problem Understanding conifold transitions in complex threefolds.
method Differential geometric approach, focusing on explicit calculations and examples.
result Necessary condition for smoothings of nodal Calabi-Yau threefolds proved.
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
4 flat 3-manifolds realized in hyperbolic 4-space.
problem Realizing flat 3-manifolds in hyperbolic 4-space.
method Using finite-volume hyperbolic 4-manifolds with cusp sections.
result 6 closed orientable flat 3-manifolds realized as cusp sections.
Study on holomorphic discs in bundles over compact surfaces, proving Fredholm regularity under certain conditions.
problem Analyzing holomorphic discs with boundary on surfaces in vector bundles over compact manifolds.
method Proves Fredholm regularity for sections with a single complex point under specific conditions.
result Holomorphic discs are Fredholm regular under certain conditions, including neutral Kähler and symplectic actions.
Theory of T-duality for transitive Courant algebroids developed.
problem Developing T-duality for transitive Courant algebroids.
method Introducing a map between sections of canonical spinor bundles and proving isomorphisms of invariant spinors and sections.
result Isomorphisms between spaces of invariant sections and spinors under T-duality.
New method constructs Birkhoff sections for pseudo-Anosov flows with controlled complexity.
problem Constructing Birkhoff sections for pseudo-Anosov flows with specific properties.
method Uses connection between pseudo-Anosov flows and veering triangulations to explicitly construct sections with controlled complexity.
result Shows that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributiona…
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functi…
The paper provides examples of geometric transitions in low dimensions.
problem Exploring geometric transitions between different types of structures in low dimensions.
method Explicit examples and computations of transitions from hyperbolic to Euclidean, spherical, and Anti-de Sitter structures.
result Details of elementary computations and techniques are provided to explain geometric transitions.
Classifies 3D loops on non-solvable reductive spaces.
problem Classifying 3D loops on non-solvable reductive spaces.
method Using Lie groups and sections σ:G/HoG for reductive homogeneous manifolds. result Classified all 3D connected strongly left alternative loops.
Shows Anosov flows with genus one sections, supporting a conjecture.
problem Finding genus one Birkhoff sections for Anosov flows.
method Utilizes horizontal Goodman surgery operation and correspondence with veering triangulations.
result Provides evidence for Fried and Ghys conjecture.
Paper shows how to transform certain flows into R-covered ones.
problem Transforming Anosov flows into R-covered ones.
method Goodman-Fried surgery along periodic points.
result Any topologically transitive Anosov flow can be transformed into an R-covered one.
The paper classifies and determines properties of specific geometric structures.
problem Classifying and understanding affine reductive homogeneous manifolds.
method Analyzing Lie groups and their homogeneous spaces, using sharply transitive sections.
result Classification of left A-loops and determination of their properties.
Paper studies shock wave transitions and completely exceptional PDEs via conformal geometry.
problem Understanding shock wave transitions and completely exceptional PDEs.
method Recasting conditions in terms of characteristics, embedding in Lagrangian Grassmannian, using BGG resolution.
result Completely exceptional PDEs, including Monge-Ampere, can be described via conformal geometry and BGG resolution.
We study properties of the cross-sectional distribution of returns. A significant anti-correlation between dispersion and cross-sectional kurtosis is found such that dispersion is high but kurtosis is low in panic times, and the opposite in normal times. The co-movement of stock returns also increases in panic times. W…
A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From thi…
We prove that the exceptional complex Lie group F4 has a transitive action on the hyperplane section of the complex Cayley plane OP2. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $\Spin(9,\C) \leq F_4$. Moreover, we identify the stabilizer of the …
A new model decomposes equity returns and volatilities into memory components.
problem Understanding long-term equity dynamics and volatility patterns.
method Proposes a multivariate generalization of the variance ratio to decompose long-horizon equity dynamics.
result Identifies a five-factor model capturing persistent, antipersistent, and multi-scale memory in returns and volatility.
Fold maps associated to geodesic random walks on curved spaces.
problem Understanding the behavior of geodesic random walks on curved surfaces.
method Analyzing mappings from the unit tangent sphere to a manifold with non-positive curvature.
result For odd powers of the unit tangent sphere, these mappings are fold maps.
This note proves equivalence between Dorfman brackets and lifts, showing universality of the Courant-Dorfman bracket.
problem Characterizing twistings and symmetries of transitive Dorfman brackets.
method Proving equivalence between Dorfman brackets and lifts, intertwining with Courant-Dorfman bracket.
result Universality of the Courant-Dorfman bracket and characterization of Dorfman brackets via lifts.
New method shows pseudo-Anosov flows on graph manifolds can be simplified.
problem Understanding pseudo-Anosov flows on graph manifolds.
method Constructing a partial Birkhoff section with genus one components that misses finitely many closed orbits.
result Every pseudo-Anosov flow on a graph manifold is almost equivalent to a totally periodic flow or a suspension Anosov flow.
Novel unsupervised feature selection method using multi-step Markov transition probability.
problem Neglected relationships between non-adjacent data points in feature selection.
method MMFS (Multi-step Markov transition probability for Feature Selection) approach, employing positive and negative viewpoints.
result MMFS effectively maintains data structure in unsupervised feature selection.
Study Liouville-type results for non-integrable almost complex manifolds.
problem Establish Liouville-type results for almost complex manifolds.
method Combine techniques from Xiaokui Yang, Valentino Tosatti, and others. Discuss key differences and introduce tools.
result Present proof of main theorem.
The paper extends a theorem for complex structures on Lie groups to Courant algebroids.
problem Characterizing integrable generalized complex structures on transitive Courant algebroids.
method Analyzing skew-symmetric fields of endomorphisms and their closure under the Dorfman bracket.
result Local form of integrable generalized complex structures is determined under certain conditions.
Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M,G). We say that the action is \emph{fixed point homogeneous} if G acts transitively on a normal sphere to some component of Fix(M,G), equivalently, if Fix(M,G) has codimension…
Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set MG. We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of MG. Fixed-point homogeneous manifolds with positive sectional curvature have been c…
This paper proves a bijection between complex and symplectic categories of tori.
problem Ambiguities in transition functions cause difficulties in constructing a functor.
method SYZ construction and transform to solve ambiguities and prove bijection.
result Proves the existence of a bijection between complex and symplectic categories of tori.
Classifies invariant differential operators on a specific geometric space.
problem Identifying invariant differential operators on curved geometries.
method Classification of strongly invariant operators between vector bundles induced by semi-holonomic Verma modules.
result Classification of invariant differential operators on Gr(3,3). This paper proves rational eigenvalues for shape operators in certain spaces.
problem Understanding rational properties of shape operators in symmetric spaces.
method Analyzing normal holonomy and shape operators in singular orbits of isotropy representations.
result Shape operators have rational eigenvalues in specific normal holonomy factors.
A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept…
CAST predicts distribution-valued time series by stabilizing and transporting simplex-supported successors.
problem Forecasting distribution-valued time series with structural failure modes.
method CAST (Causal Anchored Simplex Transport) uses successors retrieved from causal context, stabilized with a persistence anchor, and locally transported on ordered supports.
result CAST outperforms baselines on eleven public and simulated benchmarks, achieving best average rank on both one-step KL and autoregressive rollout JSD.
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through th…
This paper presents a rank rigidity result for negatively curved spaces. Let M be a compact manifold with negative sectional curvature and suppose that along every geodesic in M there is a parallel vector field making curvature −a2 with the geodesic direction. We prove that M has constant curvature equal to $-…
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
Develops methods to simulate rare transitions in molecular systems.
problem Rare transitions between metastable states in molecular systems are difficult to study due to limited data.
method Two novel methods: chain-based and midpoint-based approaches.
result Demonstrates effectiveness of methods in both data-rich and data-scarce scenarios.
Short-term trend-following has stopped delivering profits since 2009, especially on smaller market ticks.
problem The profitability of short-term trend-following has declined since 2009.
method Cross-sectional analysis of 100 liquid futures contracts from 1995-2025, evaluating four explanations.
result The decline in short-term trend-following profits is linked to smaller market ticks, not asset class or liquidity.
The paper models market crashes as phase transitions, finding dynamic transitions offer better predictions.
problem Understanding and predicting extreme financial events like market crashes.
method Employing phase transition theory, focusing on endogenous crashes, and comparing DPT, CPT, and SPT.
result Dynamic phase transitions provide more accurate predictions of market crashes compared to critical and stochastic models.
A geometric account explains why 'The Dress' is ambiguous, predicting observable signatures in image processing.
problem Understanding and predicting ambiguity in image processing, particularly in intrinsic image decomposition.
method Geometric analysis of intrinsic image decomposition, focusing on the discontinuous switch in prior-mode sections.
result Predicted signatures in albedo Jacobian and Fernet curvature can be observed in various models and datasets.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
problem Understanding and characterizing transitive double Lie algebroids.
method Using core diagrams and equivalence of transitive core diagrams with transitive double Lie groupoids.
result Transitive double Lie algebroids are completely determined by their core diagrams.
This paper shows how pseudo-Anosov flows represent stable Hamiltonian classes and limits the ways 3-manifolds can be obtained from knots.
problem Understanding the canonical representatives of stable Hamiltonian classes and their implications for 3-manifolds.
method Explains the analogy between pseudo-Anosov flows and stable Hamiltonian classes and generalizes an argument to limit the ways 3-manifolds can be obtained from knots.
result There are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere, and any 3-manifold can be obtained in at most finitely many ways as p/q surgery on a fibered hyperbolic knot in S3. The Wythoff construction takes a d-dimensional polytope P, a subset S of {0,...,d} and returns another d-dimensional polytope P(S). If P is a regular polytope, then P(S) is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want …
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Classifies transitive and periodic links in 3-manifolds.
problem Classifying transitive and periodic links in 3-manifolds.
method Generalized periodic links to transitive links, studied using link polynomials and factor links.
result Complete classification theorem of transitive links in a 3-dimensional sphere.
Two-dimensional transition rates improve life insurance reserve calculations.
problem Calculating life insurance reserves with Markov assumptions.
method Introducing two-dimensional forward and backward transition rates.
result Two-dimensional transition rates enable more accurate reserve calculations.
Urban economies follow universal scaling laws over time.
problem Identify a common economic pathway for cities.
method Mathematical framework integrating cross-sectional data into longitudinal evolution.
result Urban economies recapitulate observed cross-sectional regularity over time.
Detects phase transitions in collective behavior using manifold curvature.
problem Identifying phase transitions in multi-agent systems.
method Uses curvature and singular value ratios to detect phase transitions on manifolds.
result Phase transitions can be detected and separated into distinct sub-manifolds.