Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology g…
Study calculates integral cohomology of non-orientable infinite type surfaces.
problem Computing the first integral cohomology group of non-orientable infinite type surfaces.
method Alexander method, isomorphism to automorphism group, topological rigidity of curve graph, semi-direct product structure.
result First integral cohomology group computed for non-orientable infinite type surfaces.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
New mechanism for pure differential privacy on functional summaries using Laplace-like process.
problem Challenges in achieving differential privacy for complex, structured functional summaries.
method Independent Component Laplace Process (ICLP) mechanism for infinite-dimensional Hilbert space.
result Effective enhancement of utility of private summaries through oversmoothing.
Classifies surfaces for pure mapping class groups with automatic continuity.
problem Determining surfaces for which pure mapping class groups have automatic continuity.
method Completely classified orientable infinite-type surfaces and specific cases of surfaces with finite ends.
result Classification of surfaces for automatic continuity of pure mapping class groups.
New curves share invariant up to any fixed order.
problem Finite-order rondle invariants and lower central series.
method Formulation and construction of infinite sequences of curves.
result Infinitely many curves share invariants up to any fixed order.
Classifies pure mapping class groups based on surface properties.
problem Classifying pure mapping class groups based on surface properties.
method Complete classification without additional assumptions.
result PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface.
We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups Pn(RP2) of the projective plane. The maximal finite subgroups of Pn(RP2) are isomorphic to the quaternion group of order 8 if n=3, and to Z4 if n≥4. Further, for all n≥3, up to isomorphism, the foll…
We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to Z factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.
Paper proves knots satisfy a conjecture using Jones polynomial.
problem Proving infinite families of knots satisfy the Cosmetic Surgery Conjecture.
method Computed Jones polynomial and invariants for two knot families.
result Two infinite families of knots satisfy the Purely Cosmetic Surgery Conjecture.
Optimal algorithm for selecting high-quality arms from infinite bandit arms.
problem Efficiently choosing the best arm from an infinite set of options.
method Developed algorithms for both fixed confidence and fixed budget settings, achieving optimal or near-optimal sample complexities.
result Optimal sample complexity results for both fixed confidence and fixed budget settings, resolving open questions in the field.
We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we …
Two criteria for a closed connected definite 4-manifold with infinite cyclic fundamental group to be TOP-split are given. One criterion extends a sufficient condition made in a previous paper. The result is equivalent to a purely algebraic result on the question asking when a positive definite Hermitian form over the r…
Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.
problem Disprove conjectures about chirally cosmetic surgeries and purely cosmetic surgeries.
method Construct infinite families of chirally cosmetic surgeries and purely cosmetic surgeries on specific geometric objects.
result Found chirally cosmetic surgeries and purely cosmetic surgeries that contradict previous conjectures.
Study boundary actions of CAT(0) spaces and their C∗-algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated C∗-algebras. method Topological dynamics and C∗-algebras, focusing on actions of specific groups and their properties. result Established (strongly) pure infiniteness results for reduced crossed product C∗-algebras of boundary actions. Study shows compact mapping class groups of infinite type surfaces are never perfect.
problem Characterizing the perfection of mapping class groups of infinite type surfaces.
method Analyzing the closure of compactly supported mapping class groups and Torelli groups, examining their abelianizations.
result The abelianization of the closure of compactly supported mapping class groups contains uncountable direct sums of rationals.
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
Study bounds and asymptotic behavior of largest purely non-free actions on compact Riemann surfaces.
problem Understanding the largest purely non-free actions on compact Riemann surfaces.
method Analyzing continuous actions of finite groups on closed orientable surfaces, proving bounds and asymptotic behavior.
result The largest order of a purely non-free action on a surface of even genus is bounded below by 8g and sharp for infinitely many even g.
Study pure exploration in high-dimensional feature spaces using adaptive embeddings.
problem Overcoming the curse of dimensionality in pure exploration bandits.
method Adaptive embedding of feature representations into lower-dimensional spaces, carefully dealing with model misspecification.
result Sample complexity guarantees that depend on the effective dimension of feature spaces in kernel or neural representations.
Solves infinite family of cubic polynomial problems.
problem Infinite family of twisted polynomial problems.
method Using Dehn twists and 9-adic expansions.
result Result of twisting depends on 9-adic expansion.
Generalized Stacey-Roberts lemma for Banach manifolds.
problem Constructing Lie groupoids of smooth mappings in infinite-dimensional geometry.
method Generalization of the Stacey-Roberts lemma to Banach manifolds with smooth partitions of unity.
result Remedied an error in the original proof for finite-dimensional setting.
We propose infinite mixture prototypes to adaptively represent both simple and complex data distributions for few-shot learning. Our infinite mixture prototypes represent each class by a set of clusters, unlike existing prototypical methods that represent each class by a single cluster. By inferring the number of clust…
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.
problem Understanding the finiteness properties and commensurability of mapping class groups of infinite graphs.
method Investigation of asymptotically rigid mapping class groups, construction of explicit presentations, and analysis of algebraic and geometric properties.
result Graph Houghton groups are not commensurable with other known Houghton-type groups, defining a new class of groups.
In this note we propose that D-brane charges, in the presence of a topologically non-trivial B-field, are classified by the K-theory of an infinite dimensional C^*-algebra. In the case of B-fields whose curvature is pure torsion our description is shown to coincide with that of Witten.
Paper studies pure virtual twin groups and their automorphisms.
problem Understanding the structure and automorphisms of pure virtual twin groups.
method Analyzes stable isotopy classes of immersed circles on surfaces, extending classical knot theory.
result Proves PVTn is an irreducible right-angled Artin group with trivial center and gives its precise presentation. We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
We prove that an analog of the exterior differential acts on the space of arbitrary Lagrangians of multidimensional paths on any manifold or supermanifold, thus making this space into a cochain complex. An analog of the Stokes' formula holds. The construction and the proofs are purely geometrical, in terms of the varia…
We consider the problem of near-optimal arm identification in the fixed confidence setting of the infinitely armed bandit problem when nothing is known about the arm reservoir distribution. We (1) introduce a PAC-like framework within which to derive and cast results; (2) derive a sample complexity lower bound for near…
Optimizes privacy-preserving optimization for heavy-tailed data.
problem Privacy-preserving optimization with heavy-tailed gradients.
method Pure ε-differential privacy framework for Lipschitz extensions.
result Minimax optimal excess-risk rate for pure ε-DP heavy-tailed SCO.
The paper extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.
problem Classifying bi-invariant 2-forms on infinite-dimensional Lie groups.
method Generalized the classification from compact Lie groups to arbitrary finite-dimensional Lie groups and then to Milnor regular infinite-dimensional Lie groups.
result The classification of bi-invariant 2-forms extends to all Milnor regular infinite-dimensional Lie groups.
Study pricing options on forward contracts using infinite-dimensional affine models.
problem Pricing European-style options on forward contracts in complex stochastic volatility models.
method Model forward price curves using stochastic partial differential equations modulated by stochastic volatility processes. Analyze two classes of affine stochastic volatility models: Gaussian and pure-jump. Derive conditions for existence of exponential moments and develop semi-closed pricing formulas.
result Developed semi-closed Fourier-based pricing formulas for vanilla call and put options in infinite-dimensional affine models.
How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its width --- namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers --- is allowed to increase to infinity? Such questions have come to the forefr…
In this article, we consider a Markov process X, starting from x and solving a stochastic differential equation, which is driven by a Brownian motion and an independent pure jump component exhibiting state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability …
The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.
problem Solving the conjugacy problem in the Artin braid group quotient Bn/[Pn,Pn]. method Using systems of equations over the integers derived from the action of Bn/[Pn,Pn] on the abelianization of Pn/[Pn,Pn]. Also, explicitly realizing infinite virtually cyclic subgroups. result Explicitly realized infinite virtually cyclic subgroups in Bn/[Pn,Pn]. New examples of surface bundles found over surfaces.
problem Finding compact atoroidal surface bundles.
method Type-preserving homomorphism from knot complement to mapping class group.
result Infinitely many commensurability classes of surface subgroups.
This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete…
The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by…
The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive…
The article constructs stochastic integration in Riemannian manifolds.
problem No specific problem stated; focuses on the construction of stochastic integration.
method Functional-analytic approach to stochastic integration in Riemannian manifolds.
result There are infinitely many stochastic integrals, and they are related by a simple formula.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
The paper shows uncountable integral homology for specific mapping class groups.
problem Integral homology of mapping class groups for infinite-type surfaces.
method Analyzing compactly-supported mapping class groups and Torelli groups.
result Integral homology is uncountable in all positive degrees for specific infinite-type surfaces.
This paper shows how infinitely wide Tensor Networks converge to Gaussian Processes.
problem Understanding the relationship between Tensor Networks and Gaussian Processes.
method Analyzing the infinite-width limit of Tensor Networks and comparing them to Gaussian Processes.
result Infinitely wide Tensor Networks converge to Gaussian Processes, proving their equivalence.