Domain Adaptation in 6G wireless networks: When is it green?
problem Energy consumption of Domain Adaptation (UDA) compared to single-task training in 6G wireless networks.
method Investigate energy consumption and propose a method to determine the minimum number of target domains for UDA to be more energy-efficient than retraining.
result Proposed a method to determine the minimum number of target domains for UDA to be more energy-efficient than retraining.
In this article, we provide bounds on systoles associated to a holomorphic 1-form ω on a Riemann surface X. In particular, we show that if X has genus two, then, up to homotopy, there are at most 10 systolic loops on (X,ω) and, moreover, that this bound is realized by a unique translation surface up to homo…
This paper tackles URLLC in 6G networks with deep learning.
problem Stringent requirements on end-to-end delay and reliability for mission-critical applications.
method Develops a multi-level architecture combining theoretical models and real-world data, using deep transfer learning and federated learning.
result Demonstrates improved performance in URLLC for mission-critical applications.
We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.
It is shown, that the mapping class group of a surface of the genus g > 1 admits a faithful representation into the matrix group GL (6g-6, Z). The proof is based on a categorical correspondence between the Riemann surfaces and the so-called toric AF-algebras.
We show that if S is a finite type orientable surface of genus g and p punctures where 3g+p > 4, then EL(S) is (n-1)-connected and (n-1)-locally connected where dim(PML(S))=2n+1=6g+2p-7. Furthermore, if g=0, then EL(S) is homeomorphic to the p-4 dimensional Nobeling space.
We consider (local) parametrizations of Teichmuller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6g−6+3n geodesics. We find a large family of suitable sets of 6g−6+3n geodesics, each set forming a special structure called "admissible double pants decomposition". For …
Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to cL6g−6+2r. We obtain basic estimates for a Monge-Ampère equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmü…
It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using…
Teichmüller space and hyperelliptic surfaces parametrized by angles.
problem Parametrizing Teichmüller space and hyperelliptic surfaces using angles.
method Proved parametrization using 6g-5 and 4g-2 angle parameters for Teichmüller space and hyperelliptic surfaces respectively.
result Proved parametrization of Teichmüller space and hyperelliptic surfaces by angle parameters.
The space of measured laminations ML(Σ) associated to a topological surface Σ of genus g with n punctures is an integral piecewise linear manifold of real dimension 6g−6+2n. There is also a natural symplectic structure on ML(Σ) defined by Thurston. The integral and symplectic structures …
The study finds all trace field degrees for Torelli group mappings.
problem Identifying all possible trace field degrees for Torelli group mappings.
method Using Thurston-Veech construction of pseudo-Anosov maps, and providing examples of stretch factors with specific algebraic degrees.
result All integers 1≤d≤3g−3 are trace field degrees for g≥2. For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^{6g-6+3b} onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization…
We show that the number of square-tiled surfaces of genus g, with n marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most L squares, is asymptotic to L6g−6+2n times a product of constants appearing in Mirzakhani's count of …
The paper proves a linear diameter bound for hyperbolic knot complexes.
problem Understanding the diameter of Kakimizu complexes for hyperbolic knots.
method Defined a complex ISℓ(K) to study incompressible Seifert surfaces and proved its diameter has a linear upper bound. result The diameter of the Kakimizu complex for hyperbolic knots grows linearly with genus, confirming a conjecture.
Currents with corners help count triangulations on surfaces.
problem Counting triangulations on surfaces with weighted vertices.
method Introduced currents with corners, studied their properties, and applied them to triangulation counting.
result The number of triangulations grows polynomially of degree 6g-6.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). Let Σ_g be a closed orientable surface of genus g \geq 2 and τa graph on Σ_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space \mathcal{C}_τassociated with τ, which can be identified with the set of all pairs of a projective structure and a circle …
The Dehornoy order on braid groups is derived from a cluster algebra.
problem Deriving the Dehornoy order on braid groups from a cluster algebra.
method Using a tracial state on a cluster C∗-algebra associated to a surface. result The Dehornoy order on B2g+n is recovered from the cluster algebra. Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.
problem Counting simple closed geodesics on hyperbolic surfaces.
method Inspired by lattice point counting, uses principles of homogeneous dynamics.
result The number of simple closed geodesics of length ≤ L is asymptotic to L^(6g-6) times a constant.
Study surface subgroups acting on projective space, finding bending laminations and spheres.
problem Surface subgroups acting on RP3 with coaffine representations. method Stratification of convex core boundary, bending laminations, and analysis of holonomy.
result Projectivization of bending data space is a sphere of dimension 6g−7. Let X be a bundle over S1 with fiber a 3--manifold M and with monodromy φ. Gay and Kirby showed that if φ fixes a genus g Heegaard splitting of M then X has a genus 6g+1 trisection. Genus 3g+1 trisections have been found in certain special cases, such as the case where φ is triv…
The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g=0, 1 or \ge 2 such that the surface with this metric admits a…
Let γ0 be a curve on a surface Σ of genus g and with r boundary components and let π1(Σ)↷X be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves γ of type γ0 with translation length at most L on X. For example, as an applic…
Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.
problem Determining the volume of conformal metrics on planar domains with circular boundaries.
method Extending Epstein maps to conformal metrics, defining W-volume, using Schottky uniformization and Loewner energy.
result Shows a bound on the renormalized volume of Schottky uniformization and provides a realization of Loewner energy.
Super efficient geodesics have a unique vertex in the complex of curves.
problem Finding the unique vertex in the complex of curves for efficient geodesics.
method Intersection growth inequality and analysis of dot graph.
result Super efficient geodesics have a unique vertex in the complex of curves, independent of distance.
Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.
problem Understanding the space of complex projective structures on surfaces with circle patterns.
method Analyzing ideal polyhedral surfaces in hyperbolic ends, proving manifold properties and Lagrangian immersions.
result The space of complex projective structures on surfaces with circle patterns is a manifold of dimension 6g-6.
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
This is a mathematical commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teich…
Study on embeddings of surfaces in 4-manifolds and their mapping classes.
problem Characterizing and understanding embeddings of surfaces in 4-manifolds and their mapping classes.
method Analyzing smooth proper embeddings and mapping classes induced by diffeomorphisms of 4-manifolds.
result Most surfaces do not admit flexible embeddings in 4-manifolds with specific homology types.
We say a graph has property Pg,p when it is an induced subgraph of the curve graph of a surface of genus g with p punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to Pg,p. We introduce a new invariant of a graph, the 'nested complex…
Computes volumes of metric maps on surfaces, linking to Weil-Petersson volumes.
problem Computing volumes of specific metric maps on surfaces.
method Using recent results on discrete maps with irreducibility constraints, computes volumes as homogeneous polynomials.
result Identifies volumes as homogeneous polynomials and satisfies string and dilaton equations.
Impact of projective curvature tensor in f(R,G), f(R,T) and f(R,Lm)-gravitygr-qc The study characterizes spacetime and modified gravity models using projective curvature tensor.
problem Characterizing spacetime and modified gravity models with projective curvature tensor.
method Analyzing $f\left(R,G
ight)$, $f\left(R,T
ight)$, and $f\left(R,L_{m}
ight)$-gravity models.
result Projectively flat perfect fluid spacetimes represent dark energy era and are locally isometric to Minkowski or de-Sitter spacetimes.
Study of random multicurves and square-tiled surfaces on large genus surfaces.
problem Understanding the geometry and combinatorial properties of random multicurves and square-tiled surfaces on surfaces of large genus.
method Combination of combinatorial and geometric analysis, including large genus asymptotic analysis of moduli space volumes and intersection numbers.
result Random multicurves and square-tiled surfaces have well-approximated properties by random permutations, with specific expected values.
Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.
problem Existence and non-uniqueness of cone spherical metrics with prescribed singularities.
method Utilizing polystable extensions of line bundles, the study establishes three primary results concerning these metrics.
result Existence of multiple irreducible and reducible cone spherical metrics for certain effective divisors.
Graphs on surfaces have limits for complete walks, impacting ergodicity.
problem Graphs embedded in surfaces have limits for complete leftward walks.
method Analyzes graphs embedded in surfaces, proving limits on valence for complete walks.
result The valence of graphs embedded in surfaces is bounded for complete walks.