Z_2-systolic-freedommath.GT
We give the first example of systolic freedom over torsion coefficients. The phenomenon is a bit unexpected (contrary to a conjecture of Gromov's) and more delicate than systolic freedom over the integers.
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We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential form…
We prove the -manifold $\RP^3 \# \RP^3$ is of -coefficient homology -systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define -coefficient homology -systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\…
A new systolic inequality for mod 2 systoles is established.
problem Bounding the product of mod 2 systoles in Riemannian manifolds.
method Analyzing the product of systoles of dimensions 1 and n-1 in closed manifolds with bounded local geometry.
result A systolic inequality is derived for the product of mod 2 systoles, showing a power-law relationship with volume.
In my masters thesis I prove a square root bound on the distance of homological codes that come from two dimensional surfaces, as a result of the systolic inequality. I also give a detailed version of M.H. Freedman's proof that due to systolic freedom, this bound does not hold in higher dimensions.
Given a pair of integers m and n such that 1 < m < n, we show that every n-dimensional manifold admits metrics of arbitrarily small total volume, and possessing the following property: every m-dimensional submanifold of less than unit m-volume is necessarily torsion in homology. This result is different from the case o…
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than un…
Constructs manifolds from quantum codes with novel geometric properties.
problem Creating manifolds with specific geometric constraints.
method Reverse engineering manifolds from quantum code chain complexes.
result First examples of power law systolic freedom.
In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, ins…
New quantum code breaks distance barrier with transversal non-Clifford gates.
problem Breaking the sqrt(N) distance barrier for quantum LDPC codes.
method Combining three qLDPC codes, Freedman-Hastings mapping, and triple cup product.
result Achieves Ω(N^(2/3)) distance and Θ(N^(2/3)) dimension, enabling fault-tolerant magic state preparation.
The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.
problem Optimal systolic inequalities for Möbius strip and Klein bottle.
method Alternative proof using -distance of conformal factor.
result Estimates on systolic defect for Möbius strip and Klein bottle.
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
A new systolic inequality with a remainder for the real projective plane.
problem Proving a stronger systolic inequality for the real projective plane.
method Developing a new systolic inequality with a remainder term.
result A stronger systolic inequality with a remainder for the real projective plane.
We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our…
In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…
In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic rati…
Max systoles on spheres with punctures are counted.
problem Finding the maximum number of systoles on spheres with punctures.
method Analyzing complete Riemannian metrics on spheres with punctures.
result Determined the maximal number of systoles.
Complexity in Bolza surfacemath.DG
Study on complexity of systolic geodesics on Bolza surface.
problem Complexity of systolic geodesics on the Bolza surface.
method Analysis of intersections and triangulation of geodesics.
result There are 12 second systolic geodesics forming a triangulation of the surface.
New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
Upper bound found for systolic geometry on manifolds with positive scalar curvature.
problem Relationship between systolic geometry and positive scalar curvature.
method Spinorial methods combined with geometric measure theory and curvature estimates.
result Upper bound for the two-dimensional stable systole on certain manifolds.
New manifolds with small systoles not quasi-arithmetic.
problem Finding manifolds with small systoles not quasi-arithmetic.
method Hybrid construction of known manifolds.
result Exhibited manifolds with arbitrarily small systoles.
Sharp inequalities for Kähler manifolds' systolic invariants are established.
problem Understanding systolic invariants of Kähler manifolds.
method Analyzing metrics with positive scalar curvature on Kähler manifolds and their products.
result Bounds for systolic invariants attain equality for specific manifolds.
Systoles of hyperbolic manifolds are dense and related to Salem numbers.
problem Density of systoles in hyperbolic manifolds.
method Analyzing systoles and arithmetic hyperbolic manifolds.
result Systoles of closed arithmetic hyperbolic manifolds are dense in .
Random surfaces with long systoles created from graph theory ideas.
problem Finding surfaces with long systoles.
method Two constructions inspired by graph theory.
result Proved a new lower bound on systole length.
An upperbound for the number of critical points of the systole function on the surface moduli spacemath.GT
Upper bounds found for systole function critical points on surface moduli space.
problem Finding upper bounds for systole function critical points.
method Analyzing the systole function on the surface moduli space.
result Upper bounds for critical points of systole function and their systole values.
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
Extremal length systole is maximized at the Bolza surface.
problem Finding the surface with the maximum extremal length systole.
method Analyzing the Bolza surface and comparing its extremal length systole to others.
result The extremal length systole of the Bolza surface is and is a strict local maximum.
For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.
New systolic inequality for 3D contact forms on Seifert bundles.
problem Bounding the shortest Reeb orbit period in terms of contact volume.
method Proved a general systolic inequality for S1-invariant contact forms on Seifert bundles.
result Validated systolic inequality on Seifert bundles with non-zero Euler number.
The aim of this text is to present the concept of systole of a compact riemannian manifold and to give an overview of systolic geometry. I will also present the "regularization technique", which leads to major results in systolic geometry. I will detail how this technique allows to link the systolic volume of some clos…
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
New Finsler metric on sphere disproves systolic ratio conjecture.
problem Proving the maximal systolic ratio on 2-sphere.
method Inspired by Cossarini-Sabourau, constructs a Finsler metric.
result Systolic ratio of new Finsler metric is .
The study finds a unique systole maximum in non-hyperelliptic surfaces.
problem Understanding systole functions on translation surfaces.
method Analyzing local and global maxima of systole functions.
result Local maxima are not global in non-hyperelliptic components.
Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
Study finds bounds for systole length on arithmetic punctured spheres.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for .
The study finds large systoles in translation surfaces and hyperelliptic ones.
problem Investigating large systoles in translation surfaces.
method Analyzing surfaces of increasing genus and hyperelliptic surfaces to find large systoles.
result An infinite sequence of translation surfaces with large systoles.
Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2math.SG
Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.
problem Finding the shortest period of closed Reeb orbits on invariant tight contact forms.
method Proving a sharp systolic inequality based on the Euler class of the bundle.
result A behavior of the systolic inequality depends on the Euler class of the bundle.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
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 for large genus.
New links in 3-manifolds have large systole.
problem Geometric constraints on hyperbolic links in 3-manifolds.
method Constructing hyperbolic filling links with large essential systole.
result Links have large essential systole, contrasting earlier constraints.
Paper proves optimal systolic inequality for manifolds with positive triRic curvature.
problem Optimal systolic inequality for manifolds with positive triRic curvature.
method Stable weighted -slicing, volume comparison theorem, and metric deformation.
result Proves an optimal systolic inequality and characterizes the equality case.
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.
problem Understanding the relationship between systoles and diameters on hyperbolic surfaces.
method Exploring the inequality between systoles and diameters, deducing an upper bound for their ratio.
result The ratio of systoles and diameters has a genus-dependent upper bound.
We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic -index and using it to introduce generalized Zoll convex bodies.
result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…
We show that for closed orientable manifolds the -dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…
A new method using mod n covering improves systolic inequalities.
problem Stable systolic inequalities in Riemannian geometry.
method Mod n covering approach to force nonzero cup products or indices.
result Improved stable two systolic bounds for various manifolds.