Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Researchers prove a conjecture about a specific type of 3D space.
problem Guilloux's conjecture about the Borel function on a hyperbolic 3-manifold.
method Proved Guilloux's conjecture for a particular reflection group.
result The Borel function is rigid at infinity for the tetrahedral reflection lattice.
Quantum dilogarithm function proven from a linear difference equation.
problem Proving Faddeev's quantum dilogarithm from a linear difference equation.
method Proved Faddeev's quantum dilogarithm using Borel summation of a formal power series solution of a linear difference equation.
result Borel summation of a formal power series solution produces Faddeev's quantum dilogarithm.
The paper develops axioms for uniquely decomposing functions with real arguments.
problem Decomposing functions with real arguments while preserving their overall structure.
method Developing axioms to uniquely decompose Borel measurable functions.
result Unique decompositions for all Borel measurable functions are achieved.
Extends holomorphic functions on complex manifolds to larger spaces.
problem Extending holomorphic functions on complex manifolds.
method Proving the existence of a larger space B(S,X) for continuous maps that allows holomorphic continuation. result Bounded holomorphic functions on C(S,X) can be extended to holomorphic functions on B(S,X). The study shows how certain ODEs and integrals are regular under Borel summation.
problem Analyzing the regularity of solutions to ODEs and integration problems.
method Using geometric perspective on Laplace and Borel transforms, the study examines level 1 ODEs and exponential period integrals over Lefschetz thimbles.
result Solutions of certain ODEs and integration problems are Borel regular.
Paper solves quantum differential equations for projective bundles using Borel multitransforms.
problem Integration of quantum differential equations for P1-bundles. method Introduced Borel (α,β)-multitransforms to reconstruct solutions. result Quantum analog of Leray-Hirsch theorem for quantum cohomology of P1-bundles. New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
Develops measures for non-Borel Anosov groups on Furstenberg boundary.
problem Measuring non-Borel Anosov groups on the Furstenberg boundary.
method Theory of Patterson--Sullivan measures, strict convexity, entropy rigidity.
result Existence, uniqueness, and ergodicity of measures on Furstenberg boundary.
Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.
problem Understanding non-perturbative topological string theory.
method Borel summation of Gromov-Witten potential and analysis of Stokes phenomena.
result Stokes phenomena encode Donaldson-Thomas invariants of the resolved conifold.
Extends Borel invariant to measurable cocycles of 3-manifold groups.
problem Defining and analyzing Borel invariant for measurable cocycles.
method Introducing pullback along measurable cocycles and extending Borel invariant.
result Maximal cocycles are trivializable to irreducible representations.
Study shows Roller compactification's median graph has limited asymptotic dimension.
problem Understanding the asymptotic dimension of Roller compactifications.
method Proved using finite dimensional CAT(0) cube complexes and Borel median graph.
result Borel asymptotic dimension is bounded by the complex's dimension.
We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…
In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
Proves Borel Conjecture for certain 3D spaces.
problem Characterizing fundamental groups of 3D Alexandrov spaces.
method Analyzes properties of Alexandrov 3-spaces.
result Proves Borel Conjecture for specific types of spaces.
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for eve…
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
We perform a resurgence analysis of the SU(2) Chern-Simons partition function on a Brieksorn homology sphere Σ(2,5,7). Starting from an exact Chern-Simons partition function, we study the Borel resummation of its perturbative expansion.
Study proves hyperbolic groups have specific subgroup properties.
problem Characterizing subgroups of word hyperbolic groups.
method Analyzes Anosov representations into PGL(4q+2,R). result Affirmative answer to Sambarino's question for Borel Anosov representations.
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.
Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.
problem Non-perturbative quantum geometry of open and closed topological string on the resolved conifold.
method Finite difference equations, resurgence analysis, exact WKB techniques.
result Identify 5d BPS states and relate spectral problems to quantum integrable systems.
Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.
problem Characterizing Borel Anosov subgroups in SL(d,R).
method Analysis of antipodal subsets and quasi-isometric embeddings.
result Borel Anosov subgroups are virtually free or hyperbolic surface groups.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
This research analyzes the error convergence rate of GAN models.
problem Understanding the error convergence rate of GAN models.
method Applying Talagrand inequality and Borel-Cantelli lemma to establish a tight convergence rate.
result Established a tight convergence rate for the error of GAN models.
Let X be an algebraic variety over C. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over C, every holomorphic map San→Xan is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every s…
New findings on null measurability in symmetrization interface of VC learning.
problem Null measurability issues in symmetrization interface of VC learning.
method Formalized in Lean 4, using Choquet capacitability and patching properties.
result Null-measurable bad event not Borel measurable, separating regularity levels.
Study shows how nonstandard hulls can contain metric completions.
problem Characterizing the Heine-Borel property in metric spaces.
method Using nonstandard analysis and metric completions.
result Characterization of the Heine-Borel property in terms of inapproachable finite points.
Extends equivariant contact structure results to mod p L-spaces.
problem Equivariant contact structures on minimal L-spaces.
method Uses Serre spectral sequence of Borel Floer cohomology.
result Introduces two new numerical invariants.
Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.
problem Classifying manifolds and discrete subgroups of Lie groups.
method Descriptive set theory and Borel complexity computations.
result Complexity of homeomorphism problems for manifolds and conjugacy relations for subgroups.
Let Γ be the fundamental group of a complete hyperbolic 3-manifold M with toric cusps. We define the ω-Borel invariant βnω(ρω) associated to a representation ρω:Γ→SL(n,Cω), where Cω is a field which can be constructed as a quotient of a suitable subset of $\mathbb{C}^\m…
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.
In this note we prove that the Borel class of representations of 3-manifold groups to PGL(n,C) is preserved under Cartan involution up to sign. For representations to PGL(3,C) this is implied by a more general result of E. Falbel and Q. Wang, however our proof appears to be much shorter for that special case.
New representations of surface groups into higher-dimensional PSL generalize pleated surfaces.
problem Generalizing pleated surfaces to higher-dimensional PSL groups.
method λ-Borel Anosov representations of surface groups into PSL_d(C).
result Holomorphic parametrization of space of (λ,d)-pleated surfaces.
Complex equivalence classes found in graph homotopy.
problem Complexity of proper homotopy equivalence in graphs.
method Demonstrated Borel completeness and comeager equivalence classes.
result Complex equivalence classes exist in infinite graphs.
It is shown that bootstrap approximations of an estimator which is based on a continuous operator from the set of Borel probability measures defined on a compact metric space into a complete separable metric space is stable in the sense of qualitative robustness. Support vector machines based on shifted loss functions …
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…
Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.
problem Understanding the spectral properties of Laplace-Beltrami operators on Riemannian symmetric spaces.
method Using symplectic geometry and geometric quantization, associating flag manifolds to symmetric spaces and relating their Satake diagrams.
result Harmonic polynomials on flag manifolds induce all eigenfunctions on symmetric spaces.
We give an overview about finiteness properties of soluble S-arithmetic groups. Both, the number field case and the function field case are covered. The main result is: If B is a Borel subgroup in a Chevalley group and R is an S-arithmetic ring, then the group B(R) has finiteness length |S|-1 in the function field case…
New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.
We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed PGL2(C)-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\math…
Survey of recent measures of association, including a new coefficient.
problem Exploring new measures of association in statistics.
method Survey and introduction of a new correlation coefficient.
result Proposed a new extension of the correlation coefficient to standard Borel spaces.
Heat kernel resurgent structure from Picard-Lefschetz theory
problem Short-time heat kernel asymptotics
method Picard-Lefschetz theory
result 1-Gevrey small-time expansion
Let G0 be a connected, simply connected real simple Lie group. Suppose that G0 has a compact Cartan subgroup T0, so it has discrete series representations. Relative to T0 there is a distinguished positive root system Δ+ for which there is a unique noncompact simple root ν, the "Borel -- de Siebenthal s…