This paper explores when homeomorphisms between Morse boundaries of spaces induce quasi-isometries.
problem When does a homeomorphism between Morse boundaries of spaces imply a quasi-isometry?
method Investigates quasi-mobius homeomorphisms and 2-stability conditions.
result A homeomorphism between Morse boundaries of proper, cocompact spaces is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
CAT(0) spaces' Morse boundaries uniquely identify them.
problem Determining when a homeomorphism of Morse boundaries implies a quasi-isometry.
method Investigates Morse boundaries of cocompact CAT(0) spaces and their homeomorphisms.
result A homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish…
Classification of 4-manifolds with finite fundamental groups using stable homeomorphism criteria.
problem Classifying 4-manifolds with finite fundamental groups.
method Using stable homeomorphism criteria based on quadratic 2-types and Kirby-Siebenmann invariant.
result Two 4-manifolds are CP2-stably homeomorphic if and only if their quadratic 2-types are stably isomorphic and their Kirby-Siebenmann invariant agrees. Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the po…
New result classifies hierarchically hyperbolic groups based on their Morse boundaries.
problem Classifying hierarchically hyperbolic groups using Morse boundaries.
method Generalizing a result on Gromov boundaries to Morse boundaries, showing quasi-isometry if and only if there's a homeomorphism.
result Spaces are quasi-isometric if and only if there's a 2-stable, quasi-möbius homeomorphism between their Morse boundaries.
Uniformly perfect Morse boundaries characterize geometric properties of groups.
problem Characterizing geometric properties of groups using Morse boundaries.
method Introducing and geometrically characterizing uniformly perfect Morse boundaries for proper geodesic metric spaces.
result The Morse boundary of any finitely generated, non-elementary group is uniformly perfect if it is nonempty.
The study explores stable diffeomorphism groups in 4-manifolds using localisation and invariants.
problem Understanding stable diffeomorphism groups in 4-manifolds.
method Localisation of n-manifolds, inverting connected sum construction, using Bauer--Furuta invariants.
result K3-stable Bauer--Furuta invariants determine S^2xS^2-stable invariants.
Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.
problem Classifying stable diffeomorphism of spin 4-manifolds with given fundamental groups.
method Formulated conjectural relationships between algebraic invariants and obstructions, proved for specific groups.
result Proved conjectures for specific fundamental groups, providing complete algebraic stable classification.
Explicitly constructs moduli spaces of stable parabolic bundles.
problem Understanding moduli spaces of stable parabolic bundles over the Riemann sphere.
method Explicit construction and quotient of stable parabolic structures by bundle automorphisms.
result Explicit models of moduli spaces as smooth, compact complex manifolds.
The paper relaxes the stability condition to boost confidence in generalization for randomized learning algorithms.
problem The tension between uniform stability and L2-stability in generalization bounds. method Establishes in-expectation first moment generalization error bounds for L2-stable randomized learning algorithms and uses subbagging to achieve near-tight exponential bounds. result Improves generalization bounds for convex and non-convex optimization problems with SGD.
This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of rank 2 Hig…
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
The paper proves D-SGD's stability and generalization bound, highlighting the importance of communication topology.
problem The stability and generalization of decentralized stochastic gradient descent (D-SGD).
method Theoretical analysis of D-SGD's stability and generalization bound, considering spectral gap and communication topology.
result D-SGD's generalization bound is positively correlated with the spectral gap of the communication topology.
We study the landscape of Lagrangian functions for nonconvex optimization problems.
problem Understanding the stable equilibria of nonconvex optimization problems.
method We define a special class of Lagrangian functions and propose a stochastic primal-dual algorithm.
result We establish an asymptotic convergence rate and sample complexity for solving the online GEV problem.