Study mapping class group actions on infinite-type surfaces.
problem When does the mapping class group of an infinite-type surface act on a graph with unbounded orbits?
method Introduced a topological invariant to determine the existence of such actions.
result Many big mapping class groups have nontrivial coarse geometry.
Researchers factorize Dirac operators on toric noncommutative manifolds, finding curvature terms.
problem Factorizing Dirac operators on toric noncommutative manifolds.
method Using unbounded KK-theory and Kasparov modules, they show tensor sums of operators coincide with the Dirac operator on the manifold.
result There is a curvature term that arises as an obstruction for tensor sum decomposition in unbounded KK-theory.
Free groups' automorphisms have bounded orbits.
problem Understanding automorphisms of infinite rank free groups.
method Proved coarsely bounded automorphism groups via metric space actions.
result Free groups' automorphism groups have quasi-isometry type of a point.
Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
problem Determining boundedness of spectral metric on Lagrangian orbit spaces.
method Utilized wrapped Floer cohomology to define spectral invariant and pseudo-metric.
result Proved infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
We consider the G G G -invariant spectrum of the Laplacian on an orbit space M / G M/G M / G where M M M is a compact Riemannian manifold and G G G acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the G G G -invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric…
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
Study horocycle and geodesic orbits on hyperbolic surfaces.
problem Understanding the relationship between horocyclic and geodesic orbits on geometrically infinite surfaces.
method Analyzing the topological dynamics of the horocycle flow on hyperbolic surfaces, focusing on specific vectors and their orbits.
result The closure of the horocycle flow of a non-periodic vector intersects geodesic orbits along an unbounded sequence of points.
We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a f…
Every lattice H in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if H acts on a contractible manifold W and if either 1) the action is properly discontinuous, or 2) W is equipped with a complete Riemann…
The Riemann sphere of a C*-algebra is a geometric structure derived from a specific projector.
problem Understanding the geometric properties of C*-algebras through their unitary orbits.
method Developed a Riemannian manifold structure on the unitary orbit of a specific projector in a C*-algebra.
result The Riemann sphere is a homogeneous reductive C-infinity manifold with a differential geometry.
The systolic ratio of a contact form α α α on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where T min ( α ) T_{\min}(α) T m i n ( α ) is the minimal period of closed Reeb orbits on ( S 3 , α ) (S^3,α) ( S 3 , α ) . A Zoll contact form is a contact form such that all the orbits of the corresponding R…
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
The pants graph of a free group is constructed and studied.
problem Understanding the structure of free groups through graph theory.
method Developed a pants graph and studied its properties.
result The pants graph of a free group is connected and unbounded.
Study sequences of solutions to Taubes's Seiberg-Witten equations with unbounded energy.
problem Behavior of solutions with unbounded energy and their limiting nodal sets.
method Novel maximum principle for unbounded energy solutions, connection to vector field dynamics.
result Limiting nodal set converges to invariant set of vector field X X X for slow energy growth. We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the …
Two new algorithms improve performance in adversarial bandits with unbounded losses.
problem Adversarial Multi-Armed Bandits with unbounded losses.
method Developed UMAB-NN and UMAB-G for non-negative and general unbounded losses respectively.
result UMAB-NN achieves the first adaptive and scale-free regret bound for non-negative unbounded losses.
Constructs unbounded Kasparov product for sphere embeddings into Euclidean space.
problem Embedding spheres into Euclidean space and their associated Kasparov cycles.
method Constructs unbounded Kasparov cycles, equips with connections, computes unbounded Kasparov product with Dirac operator, identifies index cycles.
result Spectral triple for algebra C ( S n ) C(\mathbb S^n) C ( S n ) differs from round sphere Dirac operator by index cycle. Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
Study utility maximization in financial markets with bounded and unbounded payoffs.
problem Utility maximization in financial markets with constraints and unbounded payoffs.
method Combines quadratic backward stochastic differential equations and convex duality.
result Established utility indifference valuation, regime switching, and consumption-investment problems in unbounded markets.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Paper derives Li-Yau inequality for unbounded Laplacian on graphs.
problem Deriving Li-Yau inequality for unbounded Laplacian on graphs.
method Assumption of curvature-dimension inequality C D E ′ ( n , K ) CDE'(n,K) C D E ′ ( n , K ) and derivation of Li-Yau inequality. result First results on Li-Yau inequality for unbounded Laplacian on graphs.
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
New manifolds show Riesz transform unbounded for p > 2.
problem Understanding Riesz transform behavior on manifolds.
method Constructing Riemannian manifolds with specific properties.
result Riesz transform unbounded on L p ( M ) L^p(M) L p ( M ) for all p > 2 p > 2 p > 2 . New inequalities for unbounded functions improve denoising score matching.
problem Statistical error bounds for denoising score matching with unbounded objective functions.
method Derive new concentration inequalities using McDiarmid's inequality and Rademacher complexity bounds.
result Improved statistical error bounds for denoising score matching.
It is shown that the compactly supported identity component of the diffeomorphism group of the 2-dimensional punctured torus T p 2 \mathbb T^2_p T p 2 is an unbounded group. It follows that the fragmentation norm of T p 2 \mathbb T^2_p T p 2 is unbounded.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
Study measures complexity of surfaces using a new graph to prove group properties.
problem Understanding the complexity and structure of mapping class groups.
method Introduces a non-peripheral curve graph and uses it to analyze the structure of mapping class groups.
result Proves properties of the mapping class group based on the complexity measure.
Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.
problem Understanding ancient solutions on graphs with unbounded Laplacians.
method Generalizing Colding and Minicozzi's theorem and Hua's result to graphs with unbounded Laplacians.
result The dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth.
Study on surfaces with unbounded mean curvature and singularities.
problem Understanding surfaces with unbounded mean curvature and singularities.
method Analyzing surfaces given by Kenmotsu-type formula.
result Characterization of singularities in surfaces with unbounded mean curvature.
The study proves properties of spectral selectors for contact manifolds and applies them to contact big fibers and geodesics.
problem Properties of spectral selectors for contact manifolds.
method Algebraic properties of spectral selectors for strongly orderable contact manifolds.
result Established contact big fiber theorem and constructed norms on contactomorphism group universal cover.
Paper tackles online control of linear systems with unbounded noise.
problem Online control of linear systems under unbounded noise with unknown convex cost functions.
method Developed an algorithm achieving i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) high-probability regret under unbounded noise, and established O ( m p o l y ( log T ) ) O({
m poly} (\log T)) O ( m p o l y ( log T )) regret bound for strongly convex costs and sub-Gaussian noise. result Achieved i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) high-probability regret under unbounded noise, and O ( m p o l y ( log T ) ) O({
m poly} (\log T)) O ( m p o l y ( log T )) regret bound for specific noise and cost conditions. Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
Study solves wealth maximization problem with unbounded mean and volatility.
problem Maximizing terminal wealth with unbounded mean and volatility under Knightian uncertainty.
method Solves utility maximization problem explicitly with Ornstein-Uhlenbeck and GARCH(1) processes.
result First work on unbounded mean and volatility with Knightian uncertainty and nondominated priors.
We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
New approach finds solutions to games with unbounded controls.
problem Existence of equilibrium in mean-field games with unbounded controls.
method Weak formulation and new existence/stability results for quadratic-growth generalized McKean-Vlasov BSDEs.
result Existence of equilibrium result for non-Markovian mean-field games with unbounded control space.
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
New aggregation strategy handles unbounded losses with regret bounds.
problem Online optimization with unbounded loss functions.
method Follow The Regularized Leader (FTRL) with φ-divergence.
result Worst regret bound for unbounded losses with alternative divergences.
Gradient estimates for unbounded graph Laplacians under Bakry-Emery curvature.
problem Gradient estimates for unbounded graph Laplacians.
method Proving gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians with ellipticity assumption.
result Gradient estimates and applications to completeness and finiteness of stochastically complete graphs.
Gradient descent on Hadamard manifolds converges to boundary points, solving optimization problems.
problem Optimization on Hadamard manifolds with unbounded convex functions.
method Gradient descent, duality theorem, moment-weight inequality.
result Gradient descent converges to boundary points, solving optimization problems.
Improved sampling from Gaussian distributions with privacy constraints.
problem Sampling from unbounded Gaussian distributions with differential privacy.
method First $\widetilde{\mathcal{O}}\left(d
ight)$ -sample algorithm for unbounded Gaussians under $\left(\varepsilon, δ
ight)$ -differential privacy.
result A quadratic improvement over previous results, settling an open question.
New PAC-Bayes bounds for unbounded loss functions.
problem Generalization bounds for learning problems with unbounded loss functions.
method Introducing HYPE, a new notion for loss range, and deriving a novel PAC-Bayesian generalization bound.
result PAC-Bayes framework extended to unbounded loss functions.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.
Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
problem Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
method Assumption of conformally equivalent initial metric to a complete background metric with bounded scalar curvature and positive Yamabe invariant.
result Global existence of Yamabe flow without requiring initial curvature bounds.
Unbounded output networks improve classification performance.
problem Improving classification accuracy in neural networks.
method Introducing UnBounded output network (UBnet) with unbounded output units and a modified mean-squared error objective.
result UBnets achieve high classification performance on MNIST, CIFAR-10, and CIFAR-100 datasets.
We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).
We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.