For large genus, spectral gaps on hyperbolic surfaces approach a limit.
problem Understanding spectral gaps on hyperbolic surfaces of large genus.
method Analyzing the maximum of λk−λk−1 over thick parts of moduli spaces. result The maximum of λk−λk−1 approaches 41 for large genus. The paper calculates large genus limits for quadratic differential volumes and constants.
problem Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials.
method Combining recursive relations (Virasoro constraints) and asymmetric simple random walk jump probabilities.
result Confirm predictions about Masur-Veech volumes and area Siegel-Veech constants.
Study approximates lengths of random hyperbolic surfaces' geodesics.
problem Approximating lengths of geodesics on random hyperbolic surfaces.
method Poisson approximation for the bottom part of the length spectrum.
result Large genus limit of the expected systole computed.
The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.
problem Understanding fluctuations in Laplace eigenvalues of random hyperbolic surfaces.
method Analyzing fluctuations of linear statistics of Laplace eigenvalues over moduli space of surfaces of large genus.
result The distribution of linear statistics tends to a Gaussian as the genus of surfaces increases.
GOE statistics emerge from surface moduli space averages.
problem Understanding spectral statistics on hyperbolic surfaces.
method Defined a smooth linear statistic, averaged over moduli space, and analyzed variance.
result GOE statistics are recovered in the large genus and high energy limits.
Maps with a single face converge to hyperbolic surfaces in large genus.
problem Understanding geometric properties of high genus maps.
method Analyzing uniformly random maps and their convergence to hyperbolic surfaces.
result Lengths of simple cycles converge to a Poisson process.
Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.
problem Understanding length statistics of geodesics on random hyperbolic surfaces with cusps.
method Recursion formula for tight Weil-Petersson volumes and generalization of Mirzakhani's integration formula.
result Recovery of Poisson point process in large genus limit for length statistics of tight geodesics.
Minimal surfaces in spherical space forms have limited genus.
problem Understanding minimal surfaces in spherical space forms.
method Analyzing orientable index one minimal surfaces with large fundamental groups.
result Surfaces have genus at most two, confirming a conjecture.
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
Study on spectral gaps of hyperbolic surfaces as genus increases.
problem Understanding differences in eigenvalues for large genus hyperbolic surfaces.
method Analysis of the Laplacian on degenerating hyperbolic surfaces, min-max principle.
result Supremum of spectral gaps has infimum limit of at least 1/4 as genus increases.
Quasimodular forms help calculate large genus limits of Siegel-Veech constants.
problem Counting torus coverings and their large genus limits.
method Using quasimodular forms and generating functions, connecting geometric definitions with combinatorial counting.
result Proved conjectures on large genus limits of Masur-Veech volumes and Siegel-Veech constants.
The paper calculates volumes of Abelian differential strata in large genus asymptotics.
problem Calculating volumes of Abelian differential strata in large genus asymptotics.
method Combinatorial analysis of Eskin-Okounkov's algorithm to evaluate Masur-Veech volumes.
result The volume of a stratum indexed by a partition is (4 + o(1)) * prod(m_i + 1)^(-1) as 2g - 2 = sum(m_i) tends to infinity.
Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Viewing surfaces as random points in moduli space, studying weighted counting function.
result Variance in large genus limit is asymptotic to X log X, with exceptions.
Existence proof of noncompact self-shrinkers with arbitrary genus.
problem Existence of noncompact self-shrinkers with arbitrary genus.
method Employing min-max techniques to rigorously prove existence.
result Confirmation of one asymptotically conical end for large genus self-shrinkers.
The paper calculates large genus limits for two types of Siegel-Veech constants.
problem Large genus asymptotics for Siegel-Veech constants in Abelian differentials.
method Combining combinatorial analysis and large genus asymptotics of Masur-Veech volumes.
result The saddle connection and area Siegel-Veech constants converge to specific values as genus grows large.
Study calculates volumes and constants from intersection theory on abelian differential strata.
problem Calculating volumes and constants from intersection theory on abelian differential strata.
method Intersection numbers on strata with prescribed zeros orders.
result Evaluation of large genus limits and saddle connection Siegel-Veech constants for all strata.
Finite genus embedded minimal surfaces have limited limit ends.
problem Characterizing limit ends of embedded minimal surfaces.
method Proving properties of limit ends and using them to deduce surface characteristics.
result Embedded minimal surfaces with finite genus have at most two limit ends.
Study shows energy levels on hyperbolic surfaces follow GOE fluctuations.
problem Understanding energy level fluctuations on hyperbolic surfaces.
method Analysis of Laplace eigenvalues on hyperbolic surfaces, using GOE random matrix theory.
result Energy variance on typical hyperbolic surfaces closely matches GOE fluctuations.
No compact surfaces with specific curvature can exist near singular limits.
problem Existence of surfaces with prescribed mean curvature near singular limits.
method Analyzing mappings and Delaunay tori in Euclidean 3-space.
result No parametric surface with the specified curvature exists near singular limits.
Study of lengths of cycles in large genus random maps converging to Poisson process.
problem Understanding the distribution of cycle lengths in large genus random maps.
method Teichmüller theory approach for uniformly random metric maps (ribbon graphs).
result The length spectrum converges to a Poisson point process with an explicit intensity as genus tends to infinity.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. Study shows optimal spectral gaps diminish in large genus surfaces.
problem Optimizing spectral gaps in large genus surfaces.
method Analysis of Weil-Petersson probability and eigenvalues of Laplacian.
result Probability of optimal spectral gaps vanishes as genus increases.
Study on frequencies of non-simple curves in surfaces of large genus.
problem Frequency of non-simple curves in surfaces of large genus.
method Expression for frequency, large genus asymptotics, comparison with previous work.
result Identify most common types of non-simple curves with K intersections.
We study the modularity of the genus zero open Gromov-Witten potentials and its generating matrix factorizations for elliptic orbifolds. These objects constructed by Lagrangian Floer theory are a priori well-defined only around the large volume limit. It follows from modularity that they can be analytically continued o…
Small knots have large Heegaard genus, with crossing numbers bounded.
problem Finding knots with large Heegaard genus.
method Constructing hyperbolic knots with specific properties and using crossing number bounds.
result First known examples of small knots with large Heegaard genus.
Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…
Study systole of large genus minimal surfaces in positive Ricci curvature.
problem Understanding the systole of minimal surfaces in manifolds with positive Ricci curvature.
method Colding--Minicozzi lamination theory.
result Results on the systole of large genus minimal surfaces.
The study predicts large genus behavior of quadratic differential volumes and constants.
problem Predicting large genus behavior of quadratic differential volumes and constants.
method Analyzing conjectures on asymptotic behavior of Masur-Veech volumes and area Siegel-Veech constants.
result Conjectures on large genus asymptotics of quadratic differential volumes and constants.
Constructing new minimal surfaces of higher genus.
problem Creating minimal surfaces of higher genus.
method Weierstrass data construction and numerical solutions.
result Multiple new examples of higher genus minimal surfaces.
Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.
problem Length statistics of random multicurves on large genus hyperbolic surfaces.
method Analytical proof of convergence to Poisson-Dirichlet distribution as genus tends to infinity.
result Mean lengths of the three longest components converge to specific percentages of total length as genus increases.
For large genus hyperbolic surfaces, this paper proves eigenvalue conditions and diameter bounds.
problem Eigenvalue and diameter bounds for large genus hyperbolic surfaces.
method Analysis of moduli space of hyperbolic surfaces with Weil-Petersson metric.
result Generic hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - ε.
Large PL surfaces in homology balls can have arbitrarily high genus.
problem Finding the minimum genus of PL surfaces in homology balls.
method Utilizes Heegaard Floer homology.
result The minimum genus can be arbitrarily large.
The paper calculates super Weil-Petersson volumes for large genus.
problem Calculating super Weil-Petersson volumes for large genus.
method Analyzes super intersection numbers, proves coefficients are polynomials, and provides an algorithm to compute them.
result Proves existence of a complete asymptotic expansion of super Weil-Petersson volumes.
We describe a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume.
Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus-g bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumpti…
Study of large-n asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.
problem Understanding the geometry and spectral properties of random hyperbolic surfaces with many cusps.
method Large-n asymptotic analysis, spectral theory, and moduli space volumes. result Linear number of small Laplacian eigenvalues and relative frequency of simple vs. non-simple closed geodesics.
Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
Optimizes the first eigenvalues of Riemann surfaces for large genus.
problem Finding optimal lower bounds for first eigenvalues of Riemann surfaces.
method Analyzing shortest multi-closed curves to establish a new lower bound.
result The first eigenvalue of a Riemann surface is greater than a specific formula involving the genus and a constant.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann sur…
Study of flat bundles from Calabi-Yau theory converging to Riemann-Hilbert solutions.
problem Understanding flat bundles and their limits in Calabi-Yau theory.
method Analysis of variations of BPS structures and convergence to Riemann-Hilbert problems.
result Expression for Gromov-Witten partition function in terms of confluent hypergeometric equations.
The smallest eigenvalues of Riemann surfaces grow like 1/g^2 for large genus.
problem Understanding the behavior of small eigenvalues of Riemann surfaces as genus increases.
method Constructing specific Riemann surfaces with controlled boundary curve lengths and systole counts.
result The smallest eigenvalues of Riemann surfaces are uniformly comparable to 1/g^2 for large genus.
Study describes frequencies of geodesics on hyperbolic surfaces as genus grows.
problem Large genus asymptotic behaviors of geodesic frequencies on hyperbolic surfaces.
method Proof of conjecture involving separating and nonseparating geodesics.
result Explicit function $f( rac{n}{g})$ for frequency ratio given.
Rare Teichmüller disks converge to small limit sets.
problem Understanding limit sets of Teichmüller disks.
method Analyzing Thurston boundary of Teichmüller space.
result Teichmüller disks with smallest limit sets are exceptional.
We find the asymptotic expansion of Masur-Veech volumes for large genus.
problem The asymptotic behavior of Masur-Veech volumes as genus increases.
method Combination of combinatorial and algebro-geometric approaches.
result Existence and computation of a complete asymptotic expansion.
Study shows knots can have large genus difference from concordance.
problem Understanding genus differences in knots and surfaces.
method Analyzes the topological 4-genus and minimal genus of bounded surfaces.
result Arbitrarily large genus difference between knots and their concordance.
The study of second eigenvalues of hyperbolic surfaces improves bounds and investigates their behavior for large genus.
problem Investigating the second eigenvalues of closed hyperbolic surfaces for large genus.
method Analyzing the shortest length of separating multi-geodesics and investigating the ratio of eigenvalues to this length.
result For large genus, the second eigenvalue of a hyperbolic surface is uniformly comparable to 1/ln(g).