Study on rectifiability of singular set in multiple valued maps.
problem Rectifiability of singular set in multiple valued energy minimizing maps.
method Analysis of Dirichlet-minimizing Q-valued maps from R^m into a smooth compact manifold.
result Singular set is (m−3)-rectifiable with uniform Minkowski bounds. The paper finds multiple points in maps from sphere to Euclidean space.
problem Existence of multiple points in maps from Sm to Rd. method Ideal-valued index of G-space. result Existence of multiple points with detailed positional relationships.
We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations a…
Proves existence of area-minimizing multiple-valued functions for given boundary data.
problem Existence of area-minimizing multiple-valued functions for given boundary data.
method Topological and complex analysis methods to prove existence and conformality.
result Optimal multiple-valued boundary data leading to a Dirichlet minimizing function with minimal energy.
The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
problem Integrating quasi-Poisson manifolds into a broader geometric framework.
method Develops new aspects of shifted symplectic and Poisson geometry, establishing Lie-type correspondences and systematic constructions.
result Identifies multiplicative D-valued moment maps integrating quasi-Poisson manifolds, extending known constructions.
The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We descr…
MAPS and MAPS-SE learn from multiple suboptimal experts to improve policies efficiently.
problem Learning from multiple suboptimal experts in reinforcement learning.
method Active policy improvement from multiple black-box oracles.
result MAPS and MAPS-SE achieve sample efficiency and state-wise imitation learning.
Shows natural quasi-Poisson structure on multiplicative Grothendieck-Springer space.
problem Exploring new structures in algebraic geometry.
method Reduction along Dirac realizations.
result Natural quasi-Poisson structure exists on multiplicative Grothendieck-Springer space.
Study finds minimum entropy of pseudo-Anosov maps decreases as surface genus increases.
problem Estimating the minimum entropy of pseudo-Anosov maps on surfaces with punctures.
method Analyzes the behavior of minimum entropy for a subset of (g,n) plane, proving Penner's speculation. result Minimum entropy behaves as g1 for fixed n. Let M be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible represent…
This paper studies covariant derivatives for Lie groupoids with representation-valued forms.
problem Understanding covariant derivatives for Lie groupoids with representation-valued forms.
method The paper explores two approaches: linear connections and multiplicative Ehresmann connections, both yielding geometrically richer curved double complexes.
result The horizontal exterior covariant derivative D is a key finding, generalizing the well-known operator from principal bundles. Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently …
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class W1,p, 1<p<∞. If f almost minimizes its p Dirichlet energy then f is Hölder continuous. If p=2 and f is squeeze and squash stationary then f is in VMO.
We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-f…
MFMs enable efficient reward alignment for generative models.
problem Computational bottleneck in controlling generative models.
method Meta Flow Maps (MFMs) extend consistency models and flow maps to stochastic regime for efficient value function estimation.
result MFMs enable inference-time steering and unbiased, off-policy fine-tuning to general rewards efficiently.
The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map sgr taking …
Extends a determinant identity for complex polynomials with repeated critical points.
problem Determining the determinant of a map relating critical points and values of complex polynomials.
method Stratifying complex space and generalizing a map to handle repeated critical points.
result Establishes a determinant identity closely related to Dyson's conjecture.
Proposes a new method for MIR using kernel mean embeddings.
problem Multiple instance regression (MIR) where bags contain multiple instances with a single label.
method Computes kernel mean embeddings of predicted label distributions and learns a regressor from these embeddings.
result Better results than baseline instance-MIR across all datasets, state-of-the-art on two.
Paper describes how to extend multiple conjugation quandles using maps.
problem Understanding affine extensions of multiple conjugation quandles.
method Introduces augmented MCQ Alexander pairs for affine extensions.
result Affine extensions of multiple conjugation quandles can be described by quadruples of maps.
Principled mapping from pure-DP ε to GDP μ for Gaussian differential privacy
problem Choosing the μ parameter in Gaussian differential privacy
method Matching the worst-case success of a membership inference attack
result Recommendation of μ ≈ ε/5 as a conservative general-purpose conversion
Expressing L-polynomials coefficients in terms of multiple zeta values.
problem Expressing L-polynomials coefficients in terms of multiple zeta values.
method Expressing L-polynomials coefficients in terms of multiple zeta values.
result Every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign.
The paper constructs a multi-valued inverse of quasiregular maps and develops pull-back theory for differential forms.
problem Understanding multi-valued inverses of quasiregular maps and their properties.
method Using Almgren's framework of multi-valued maps and developing pull-back theory for differential forms.
result The multi-valued inverse is a quasiregular ω-curve with respect to a natural n-form ω. Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
Develops multivalued Jacobi fields for stable submanifolds.
problem Stability of minimally immersed submanifolds in Riemannian manifolds.
method Defines and studies multiple valued Jacobi fields as minimizers of a second variation functional.
result Any Q-valued Jacobi field can be decomposed into classical Jacobi fields except on a singular set of codimension at least two.
The paper proves an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
problem Proving an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
method Using equivariant Riemann-Roch theorem and graded Todd class, the paper proves an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
result The weighted sum of multiplicity functions has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M.
Study fixed points of n-valued maps on surfaces using braid groups.
problem Deciding when n-valued maps can be deformed to free of fixed points.
method Fixed point theory of maps between X and its configuration spaces, braid groups.
result Algebraic criterion for surfaces to determine fixed point free n-valued maps.
Maps preserve surface symplectomorphisms for smooth functions.
problem Understanding symplectomorphisms of surfaces under smooth functions.
method Constructing a canonical map between sets of functions and diffeomorphisms.
result Symplectomorphism group is contractible or homotopy equivalent to a circle.
Study connects Gaussian processes and regularization for sequence-function mappings.
problem Understanding and interpreting sequence-function maps in biology.
method Relates Gaussian process priors, regularization, and gauge fixing in overparameterized weight space.
result Established the relationship between regularized regression and Gaussian processes in function space.
We consider a continuous map f:M→N between two manifolds and try to estimate its multiplicity from below, i.e. find a q-tuple of pairwise distinct points x1,...,xq∈M such that f(x1)=f(x2)=...=f(xq). We show that there are certain characteristic classes of vector bundle f∗TN−TM that guarant…
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
Paper details Hilbert-curve for high-performance data mining.
problem Efficiently mapping multi-dimensional data to one dimension.
method Defines Hilbert-curve using finite automaton and context-free grammar.
result Cache-oblivious algorithms for matrix operations and clustering.
The paper explores fixed points of n-valued maps on surfaces using configuration spaces.
problem The Wecken property and computation of Nielsen numbers for n-valued maps.
method Configuration spaces and braid groups.
result The projective plane and 2-sphere have the Wecken property for n-valued maps.
New method constructs multi-monopoles on mapping tori.
problem Understanding wall-crossing in multi-monopole counts.
method Adiabatic limit theorem to construct multi-monopoles.
result First explicit constructions of multi-monopoles in various chambers.
We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.
We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. W…
Bayesian approach for multivariate density regression of complex data.
problem Regression of multivariate density-valued responses on predictors.
method Bayesian inference using sliced Wasserstein barycenter and SW distance.
result Accurate fits and reliable predictions for complex data.
Note on minimal maps' uniqueness via singular values.
problem Uniqueness of minimal maps into \(\mathbb{R}^n\).
method Using singular values and convexity of area functional, proving local linearity of singular value vectors.
result Improved uniqueness theorem for minimal graphs.
Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.
problem Analyzing multi-valued sections of vector bundles and proving a reparametrization theorem.
method Develops properties of Q-multisections and provides a geometric proof. result Elementary and purely geometric proof of a reparametrization theorem for multi-valued graphs.
Proves continuity and singular set dimension for 2D maps with Q values.
problem Interior regularity of 2D Q-valued maps. method Strong concentration-compactness theorem for equicontinuous maps.
result 2D Q-valued maps are Hölder continuous with singular set dimension ≤1. Proposes a graph neural network for efficient multi-agent routing.
problem Routing multiple agents in complex, sparsely connected graphs with dynamic traffic.
method Value Iteration Network based on graph neural networks for online coordination.
result Significantly outperforms traditional methods in terms of cost and runtime.
This study recovers electromagnetic parameters on boundaries from impedance and admittance data.
problem Recovering anisotropic electromagnetic parameters from boundary impedance and admittance data.
method Formulated inverse boundary value problem for time-harmonic Maxwell's equations on differential 1-forms.
result Knowledge of impedance and admittance maps determines tangential entries of induced metrics at the boundary.
We discuss sharp Sobolev inequalities for vector valued maps.
New asymptotic e-values improve inference by eliminating data-dependent scaling inefficiency.
problem Data-dependent scaling inefficiency in existing asymptotic e-values.
method Drawing on Bentkus's near-optimal concentration inequalities, introduce Bentkus-type asymptotic e-values.
result Bentkus-type asymptotic e-values consistently deliver sharper inference than existing alternatives.
Introduces multiplicative differential forms on Lie groupoids with VB-groupoids values.
problem Describing multiplicative differential forms on Lie groupoids with VB-groupoids values.
method Introduces multiplicative differential forms on Lie groupoids with values in VB-groupoids, presents a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy, defines a differential complex whose 1-cocycles are multiplicative forms with values in VB-groupoids.
result Complete description of multiplicative differential forms on Lie groupoids with values in VB-groupoids.
The paper studies complex genera and related geometric applications, deriving formulas for multiple zeta values.
problem Understanding coefficients in Chern numbers for complex genera.
method Examining Chern numbers for complex genera, focusing on specific genera like Td^(1/2), Γ, and Todd.
result Unified formulas for multiple zeta values and transition matrices among symmetric functions.
Formula decomposes multiplicative forms on Poisson groupoids into two parts.
problem Decomposing multiplicative forms on Poisson groupoids.
method Provided a formula to decompose multiplicative k-forms into a 1-cocycle and a ρ-compatible form.
result Multiplicative forms on Poisson groupoids form a differential graded Lie algebra (DGLA) crossed module.
The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.
Characterizes values at infinity for real polynomial maps with 2D fibers.
problem Understanding atypical values at infinity for real polynomial maps.
method Characterization using indices of gradient vector fields on spheres.
result Analogous to two-variable case, but for maps with 2D fibers.