Fried's theorem proven for symmetric space boundaries.
problem Characterizing manifolds with similarity structures.
method General proof for all rank one symmetric space boundary geometries.
result Closed manifolds are either complete or develop onto Heisenberg-type spaces.
Survey on twisted dynamical zeta functions and Fried's conjecture.
problem Analyzing twisted dynamical zeta functions and their relation to Fried's conjecture.
method Review of existing literature and mini-course presentation.
result Discussion and validation of Fried's conjecture.
Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.
problem Behavior of dynamical zeta functions at the origin for compact hyperbolic manifolds.
method Uses complex-valued torsion instead of Ray-Singer analytic torsion.
result Holomorphicity and value at s=0 for twisted Ruelle zeta function for arbitrary representations.
Paper proves equivariant Fried conjecture for specific flows.
problem Equivariant Fried conjecture for suspension flows.
method Analyzes suspension flows of equivariant isometries.
result Proves conjecture for various groups and cases.
We study the twisted Ruelle zeta function ζX(s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX(0). In higher dimensions, we show more generally that ζX(0) is locally constant with…
Paper shows how to transform certain flows into R-covered ones.
problem Transforming Anosov flows into R-covered ones.
method Goodman-Fried surgery along periodic points.
result Any topologically transitive Anosov flow can be transformed into an R-covered one.
We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's th…
We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovi…
Most existing feature selection methods are insufficient for analytic purposes as soon as high dimensional data or redundant sensor signals are dealt with since features can be selected due to spurious effects or correlations rather than causal effects. To support the finding of causal features in biomedical experiment…
The paper proves a conjecture linking two metrics on manifold cohomology.
problem Proving a conjecture about metrics on manifold cohomology.
method Constructing complex structures, defining metrics, and proving the conjecture.
result Ray-Singer metric equals Milnor metric, linking analytic torsion to combinatorial data.
Proves functional equation for twisted Ruelle zeta function on hyperbolic surfaces.
problem Determines the functional equation for twisted Ruelle zeta functions.
method Analyzes scattering matrix and uses topological data of hyperbolic surfaces.
result Determines the order of the divisor of R(s,χ) at s=0 and computes its Laurent expansion.
This paper uses Heegaard Floer theory to study pseudo-Anosov flows and their periodic points.
problem Understanding the differential of Heegaard Floer chain complexes associated with pseudo-Anosov flows.
method Introduces a refined grading to analyze the homology of subcomplexes representing irreducible multi-orbits.
result The homology of these subcomplexes is 1-dimensional, providing insights into periodic points of pseudo-Anosov flows.
The fully connected layers of a deep convolutional neural network typically contain over 90% of the network parameters, and consume the majority of the memory required to store the network parameters. Reducing the number of parameters while preserving essentially the same predictive performance is critically important …
Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.
problem Defining a zeta function for equivariant flows on manifolds.
method Equivariant generalization of Guillemin's trace formula.
result Computes the equivariant Ruelle zeta function in various examples.
The study examines nilpotent similarity structures on manifolds and their properties.
problem Characterizing closed manifolds with nilpotent similarity structures.
method Generalizes convexity arguments to geodesic segments in nilpotent Lie groups.
result Closed manifolds with nilpotent similarity structures are either complete or radiant.
New method identifies whether equity return predictability is due to magnitude shrinkage or directional reversal.
problem Determining the nature of equity return predictability (directional reversal vs magnitude shrinkage).
method Developed the Fourier-Residue Identity (FRI) to decompose return autocorrelation into sign and magnitude channels.
result The lag-1 autocorrelation in SPY is driven entirely by magnitude shrinkage, not directional reversal.
We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…
The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev prov…
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick…
The study finds infinitely many periodic orbits that can be used to modify Anosov flows.
problem Can surgeries on periodic orbits of Anosov flows produce equivalent flows?
method Analyzing suspension Anosov flows, the study identifies pairs of periodic orbits that can be used to modify the flow.
result For some suspension Anosov flows, there exist infinitely many pairs of periodic orbits that can be used to modify the flow.
Analytic torsion equals dynamical zeta function for certain bundles.
problem Equalities between analytic torsion and dynamical zeta functions.
method Analytic torsion and Ruelle dynamical zeta function for admissible twists.
result Generalization of previous results to admissible twists.
About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recen…
Shows Anosov flows with genus one sections, supporting a conjecture.
problem Finding genus one Birkhoff sections for Anosov flows.
method Utilizes horizontal Goodman surgery operation and correspondence with veering triangulations.
result Provides evidence for Fried and Ghys conjecture.
The paper proves a section for Anosov vector fields on compact manifolds.
problem Proving the existence of a canonical nonzero section for Anosov vector fields.
method Analyzing Anosov vector fields and flat vector bundles on compact manifolds.
result A canonical nonzero section exists and is C1 with respect to the Gauss-Manin connection. KFT improves tensor forecasting by incorporating side information.
problem Tensor factorization weaknesses in latent factors.
method Kernel Fried Tensor (KFT) with variational inference.
result Superior performance over LightGBM and FFM.
Study closed manifolds with rank one ray structures, proving completeness or covering properties.
problem Characterize closed manifolds with specific affine structures.
method Analyze the developing map and automorphism group properties.
result Closed manifolds with rank one ray structures are either complete or cover the complement of an affine subspace.
Constructs pseudo-Anosov mappings related to toral automorphisms.
problem Mapping automorphisms of tori to pseudo-Anosov mappings.
method Constructs pseudo-Anosov mappings semi-conjugate and almost-isomorphic to toral automorphisms.
result Shows stretch factors of pseudo-Anosov mappings are related to toral automorphisms.
New formula connects surface singularity zeta function to Reidemeister-Turaev torsion.
problem Calculating Reidemeister-Turaev torsion for non-unitary representations.
method Ruelle zeta function and Reidemeister-Turaev torsion for compact hyperbolic orbisurfaces.
result Value of Ruelle zeta function at 0 equals Reidemeister-Turaev torsion.
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essent…
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …
New dynamical torsion for contact Anosov flows connects to Reidemeister torsion.
problem Understanding contact Anosov flows and their properties.
method Introducing dynamical torsion and showing its properties.
result Locally constant ratio between dynamical and Turaev torsion.
Classifies Anosov flows on figure-eight knot surgeries.
problem Classifying Anosov flows on Dehn surgeries on the figure-eight knot.
method Combining Plante's classification of M(0) with Schwider's branched surfaces.
result M(r) carries a unique Anosov flow if r is an integer, and no Anosov flow if r is not an integer.
Given a 3-manifold M fibering over the circle, we investigate how the asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex vary as we vary the fibration. We formalize this problem by defining normalized asymptotic translation length functions μd for every integer d≥1 on the rational…
This paper tests LLMs in finance to assess ethical behavior.
problem Aligning AI with ethical and legal standards in finance.
method Prompted LLMs to simulate CEO behavior, analyzed with logistic regression.
result Significant heterogeneity in LLMs' unethical behavior propensity.
New flows represent Thurston norm ball faces, differing by veering mutations.
problem Dynamic representation of Thurston norm ball faces by distinct flows.
method Combining veering triangulations and mutations to represent faces by multiple flows.
result Non-fibered faces can be represented by two distinct flows differing by veering mutations.
The paper studies translation lengths on sphere complexes and related cones.
problem Understanding the translation lengths of monodromies in fibered manifolds.
method Defined the generalized fibered cone and related cones, and proved their properties.
result Proved the generalized fibered cone is a rational slice of Fried's cone, providing bounds for asymptotic translation lengths.
Introduces a massive variant of Ray-Singer Torsion to avoid zero modes in topological field theories.
problem Avoiding zero modes in the evaluation of path integrals for topological field theories.
method Introduces a massive variant of the Ray-Singer Torsion, involving determinants of the twisted Laplacian with mass but without zero modes.
result Explicitly evaluates the massive Ray-Singer Torsion on product manifolds and mapping tori.
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston …
Let M be a hyperbolic fibered 3-manifold. We study properties of sequences (Sαn,ψαn) of fibers and monodromies for primitive integral classes in the fibered cone of M. The main tool is the asymptotic translation length ℓC(ψαn) of the pseudo-Anosov monodromy ψαn on the curve …
Given a free-by-cyclic group G=FN⋊φZ determined by any outer automorphism φ∈Out(FN) which is represented by an expanding irreducible train-track map f, we construct a K(G,1) 2-complex X called the folded mapping torus of f, and equip it with a semiflow. We sh…
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.