The paper provides conditions for factoring equivariant spectral triples in unbounded KK-theory.
problem Factoring equivariant spectral triples in unbounded KK-theory.
method Sufficient conditions for factorization of equivariant spectral triples as a Kasparov product.
result Equivariant Dirac-type spectral triples on torus principal bundles always factorize.
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.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
Researchers factorize Dirac operators on Riemannian submersions.
problem Understanding factorization of Dirac operators on submersions.
method Factorization of Dirac operators using Riemannian submersions and unbounded KK-theory.
result The Dirac operator on the total space is unitarily equivalent to a tensor sum of operators on the base and fibers.
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. Study the Kasparov product on submersions of open manifolds.
problem Analyzing the Kasparov product on submersions of open manifolds.
method Showed that the tensor sum of a regular vertically elliptic operator and an elliptic operator on the base space represents the Kasparov product in KK-theory.
result Obtained a factorisation of the fundamental class of a Riemannian submersion in unbounded KK-theory.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C ∗ C^{*} C ∗ -modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
Constructs unbounded KK-cycles for Riemannian embeddings in codimension one.
problem Understanding Riemannian embeddings in codimension one.
method Constructs unbounded K K KK K K -cycles from C ( X ) C(X) C ( X ) to C 0 ( Y ) C_0(Y) C 0 ( Y ) , each with a connection, representing the shriek class. result The unbounded product of ı ! ε \imath_!^ε ! ε with the Dirac operator D Y D_Y D Y represents the K K KK K K -theoretic factorization of the fundamental class [ X ] = ı ! ⊗ [ Y ] [X] = \imath_! \otimes [Y] [ X ] = ! ⊗ [ Y ] . Reinterprets quantization commutes with reduction using KK-theory.
problem Quantization commutes with reduction in geometric quantization.
method Uses KK-theory and recent formalism by Kasparov to simplify and clarify the index theoretic parts.
result Shows conceptual simplifications and clearer relationship to Ma-Tian-Zhang approach.
The paper defines a ring structure in twisted equivariant K K KK K K -theory for Lie groups.
problem Defining a ring structure in twisted equivariant K K KK K K -theory for noncompact Lie groups. method Geometric description of representatives and use of equivariant correspondences.
result Established a ring structure in $KK^{ullet}_{G}(G/K, τ_G^G)$ .
Constructs a new class for foliations to recover a secondary characteristic class.
problem Recovering the Godbillon-Vey invariant in equivariant K K KK K K -theory. method Groupoid equivariant Kasparov class for transversely oriented foliations.
result Chern character recovers the Connes-Moscovici cyclic cocycle for the Godbillon-Vey class.
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using K K KK K K -theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C ∗ C^* C ∗ -algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a K K KK K K -theoretic context. Researchers compute spectral flow of Toeplitz operators on manifolds and domains.
problem Computing spectral flow for families of Toeplitz operators.
method Using Callias-type operators and cohomological formulas, the spectral flow is related to the index of these operators.
result A cohomological formula for spectral flow on even-dimensional manifolds.
Study perturbed Dirac operators on non-compact manifolds using KK-theory.
problem Index theory of perturbed Dirac operators on non-compact manifolds.
method KK-theory approach to factorize index classes.
result Index of perturbed Dirac operators factors as a KK-product of classes defined by the operator and the Clifford multiplication.
Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.
problem Analyzing semi-characteristics on certain manifolds.
method Combines assembly maps and Hodge theorem perspectives.
result Proves an Atiyah type vanishing theorem.
Extends fixed point theorem to noncompact manifolds using KK-theory.
problem Generalizing fixed point theorem to noncompact manifolds.
method Using KK-theory, extends equivariant index to noncompact setting.
result Obtains fixed point formula for noncompact manifolds.
Develops index theory for infinite-dimensional manifolds with loop group actions using KK-theory.
problem Establish an index theory for infinite-dimensional manifolds acted upon by loop groups.
method Constructs objects in KK-theory and defines new cycles and indices.
result Defines a KK-theoretical index and compares it with the analytic index.
We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an …
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S 1 S^1 S 1 -actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
This paper explores topological aspects of index theory for infinite-dimensional manifolds.
problem Formulating index theory for infinite-dimensional manifolds with LT-actions.
method Introducing RKK-theory and constructing assembly maps for proper LT-spaces.
result Formulation of infinite-dimensional Poincaré duality and assembly maps.
Study index theory for infinite-dimensional manifolds with LT actions.
problem Index theory for infinite-dimensional manifolds with LT actions.
method Introduce LT-equivariant KK-theory and construct three KK-elements: index, Clifford symbol, and Dirac elements.
result Satisfy a relation called the (KK-theoretical) index theorem or KK-theoretical Poincaré duality.
These notes are based on a lecture course given by the first author in the Sedano Winter School on K-theory held in Sedano, Spain, on January 22-27th of 2007. They aim at introducing K-theory of C^*-algebras, equivariant K-homology and KK-theory in the context of the Baum-Connes conjecture.
In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic …
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "offici…
Let M M M be a compact manifold. and D D D a Dirac type differential operator on M M M . Let A A A be a C ∗ C^* C ∗ -algebra. Given a bundle W W W of A A A -modules over M M M (with connection), the operator D D D can be twisted with this bundle. One can then use a trace on A A A to define numerical indices of this twisted operator. We prove an …
Defines transverse symbols for foliated manifolds and proves their K-homology class.
problem Transverse index theory for foliated manifolds.
method Using filtrations of tangent bundles, defining transverse symbols, and constructing equivariant KK-classes.
result Transversally Rockland operators yield a K-homology class and there is a Poincare duality result.
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.
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 study finds equivalent properties for CD inequalities with unbounded Laplacians.
problem Implying gradient estimates for laplace operator on graphs with unbounded Laplacians.
method Investigates equivalent properties of CD(K,∞) and CD(K,n) inequalities with unbounded Laplacians.
result Concludes equivalent properties of CD(K,∞) and CD(K,n) inequalities.
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 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 finds conditions for certain graphs to be slices.
problem Conditions for entire constant mean curvature Killing graphs to be slices.
method Analyzing conditions for entire unbounded constant mean curvature Killing graphs.
result Conditions for entire constant mean curvature Killing graphs to be slices.
Study on minimizing perimeter in unbounded convex bodies without boundary regularity.
problem Minimizing perimeter under volume constraint in unbounded convex bodies.
method Introducing uniform geometry, asymptotic cylinders, approximation, and approximation argument.
result Existence of isoperimetric regions in generalized sense and strict concavity of isoperimetric profile.
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.