Metric graphs have subgraphs with entropy at least λ.
problem Finding subgraphs with high entropy in metric graphs.
method Proving existence of subgraphs with entropy at least λ for graphs of rank r with entropy 1.
result Metric graphs have subgraphs with entropy at least λ.
This is a commentary on Teichm{ü}ller's paper Unter-suchungen{ü}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost …
We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent of the basepoint, and the action of the mapping class group on the Teichmüller space …
We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
The paper discovers new ways Riemann surfaces can degenerate.
problem Understanding degeneration of infinite-type Riemann surfaces.
method Constructing a concrete example to prove degeneration phenomena.
result Existence of degenerations in Bers boundary of infinite-type surfaces.
This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with n cusps has a pants decomposition with all of its geodesics of length bounded by a constant r…
The Bers-Greenberg theorem tells that the Teichmüller space of a Riemann surface with branch points (orbifold) depends only on the genus and the number of special points, but not on the particular ramification values. On the other hand, the Maskit embedding provides a mapping from the Teichmüller space of an orbifold, …
Improves bounds on surface decompositions.
problem Surface decomposition bounds
method Improved theorem of Bers for surfaces with boundary and closed surfaces.
result Slight improvement on best known bound for closed surfaces.
Holomorphic solutions vary in Sobolev spaces for Beltrami equations.
problem Analyzing the behavior of solutions to Beltrami equations in Sobolev spaces.
method Holomorphic dependence of solutions in Sobolev spaces of varying Beltrami differentials.
result Solutions vary holomorphically in Sobolev spaces of higher order.
A new metric model for quasi-Fuchsian space defined by Bers metrics.
problem Understanding the quasi-Fuchsian space of a surface.
method Introducing Bers metrics and studying their properties to model QF(S).
result New integral representations of the Goldman symplectic form and holomorphic extension of the Weil-Petersson metric.
Let S be a closed Riemann surface of genus g(≧2) and set S˙=S∖{z^0}. Then we have the composed map φ∘r of a map r:T(S)×U→F(S) and the Bers isomorphism φ:F(S)→T(S˙), where F(S) is the Bers fiber space of S, T(X) is the …
Develops analogs of character varieties for algebraic correspondences, proving boundedness and compactifications.
problem Characterizing algebraic correspondences and their degenerations.
method Introducing new character varieties and studying degeneration of algebraic correspondences on trees of Riemann spheres.
result Boundedness and natural homeomorphism of compactifications of Teichmüller spaces for the four times punctured sphere.
Study bends 2D surfaces in 3D space using special equations.
problem Investigate infinitesimal bendings of 2D surfaces in 3D space.
method Use Bers-Vekua type equations and systems of differential equations with periodic coefficients.
result Construct bending fields for specific classes of 2D surfaces.
Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.
problem Characterizing dense points in the Bers boundary of Teichmüller space for infinite-type surfaces.
method Quasiconformal deformations and analytic conditions.
result Maximal cusps cannot approach certain points on the Bers boundary.
The Bers embebbing realizes the Teichmüller space of a Fuchsian group G as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for G. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the di…
Unified and generalized mating frameworks for Kleinian groups and rational maps.
problem Combining two frameworks for mating Kleinian groups with rational maps.
method Extended mating framework to genus zero hyperbolic orbifolds, constructed correspondences, defined parameter space.
result Explicit description and construction of conformal matings and correspondences.
We consider the weighted belief-propagation (WBP) decoder recently proposed by Nachmani et al. where different weights are introduced for each Tanner graph edge and optimized using machine learning techniques. Our focus is on simple-scaling models that use the same weights across certain edges to reduce the storage and…
Using Lipschitz distance on Outer space we give another proof of the train track theorem.
The Basilica Julia set is universally equivalent to other complex dynamics sets.
problem Establishing the universality of the Basilica Julia set.
method Quasiconformal equivalence and geometric finiteness.
result The Basilica Julia set is quasiconformally equivalent to other complex dynamics sets.
Develops a real-analytic embedding for diffeomorphisms of the line, linking to Fisher-Rao geometry.
problem Embedding diffeomorphisms of the line in a geometric framework.
method Real-analytic embedding, Lp Fisher-Rao geometry, Schwarzian curvature. result Establishes a connection between diffeomorphisms and Fisher-Rao geometry, providing explicit geodesics and connections.
In emerging Internet-of-Nano-Thing (IoNT), information will be embedded and conveyed in the form of molecules through complex and diffusive medias. One main challenge lies in the long-tail nature of the channel response causing inter-symbol-interference (ISI), which deteriorates the detection performance. If the channe…
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
We trained three Binarized Convolutional Neural Network architectures (LeNet-4, Network-In-Network, AlexNet) on a variety of datasets (MNIST, CIFAR-10, CIFAR-100, extended SVHN, ImageNet) using error-prone activations and tested them without errors to study the resilience of the training process. With the exception of …
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space a…
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
We consider time-domain digital backpropagation with chromatic dispersion filters jointly optimized and quantized using machine-learning techniques. Compared to the baseline implementations, we show improved BER performance and >40% power dissipation reductions in 28-nm CMOS.
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.
problem Improving the conditions under which wave front germs imply map germs.
method Generalization of Zakalyukin's lemma for frontals and applications to surface singularities.
result The paper provides a more general version of Zakalyukin's lemma for map germs.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
Proves a quantitative closing lemma for negatively curved manifolds.
problem Closing lemma for negatively curved manifolds.
method Quantitative closing lemma proof.
result Study of partner and pseudo-partner orbits for self-crossing closed geodesics.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
For a symplectic manifold (M,ω), not necessarily hard Lefschetz, we prove a version of the Merkulov dδ--lemma. We also study the dδ--lemma and related cohomologies for compact symplectic solvmanifolds.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-positive representations. Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.
problem Establishing a general ∂∂̄-lemma and its applications.
method Develops a general ∂∂̄-lemma and applies it to Fujino's conjecture.
result Establishes a Kähler version of Fujino's injectivity theorem.
Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.
problem Improving Schwarz lemma for holomorphic maps between Hermitian manifolds.
method Introducing new curvature constraints on source and target manifolds, controlling by holomorphic sectional curvature.
result Significant improvements on the Wu--Yau theorem and Schwarz lemma for Gauduchon connections.