Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length. Mikhail Gromov once voiced the opinion that the isoperimetric inequality was known already to the Ancient Greeks. The mythological tale of Dido, Queen of Carthage shows that problems about making a maximum area for a given perimeter were posed in a natural way, in past eras. The relation between length and area is closely related to the physical phenomenon known as surface tension, which gives a visible form to the comparable relation between surface area and volume. The familiar shapes of drops of water express minima of surface area. The purpose of this article is to explain another such relation between length and area. A space is called simply connected if every loop in the space can be contracted to a point in a continuous fashion. For example, a room with a pillar in the middle, connecting floor to ceiling, is not simply connected. In geometry, a systole is a distance which is characteristic of a compact metric space which is not simply connected. It is the length of a shortest loop in the space that cannot be contracted to a point in the space. In the room example, absent other features, the systole would be the circumference of the pillar. Systolic geometry gives lower bounds for various attributes of the space in terms of its systole. It is known that the Fubini–Study metric is the natural metric for the geometrisation of quantum mechanics. In an intriguing connection to global geometric phenomena, it turns out that the Fubini–Study metric can be characterized as the boundary case of equality in Gromov's inequality for complex projective space, involving an area quantity called the 2-systole, pointing to a possible connection to quantum mechanical phenomena. In the following, these systolic inequalities will be compared to the classical isoperimetric inequalities, which can in turn be motivated by physical phenomena observed in the behavior of a water drop. (Wikipedia).
This lecture introduces Sympy
From playlist Introduction to Pyhton for mathematical programming
From playlist Geometry: Dynamic Interactives!
An introduction to this first course in differential equations.
From playlist Differential Equations
B01 An introduction to separable variables
In this first lecture I explain the concept of using the separation of variables to solve a differential equation.
From playlist Differential Equations
Introduction to Parametric Equations
This video defines a parametric equations and shows how to graph a parametric equation by hand. http://mathispower4u.yolasite.com/
From playlist Parametric Equations
Metaphors in Systolic Geometry - Larry Guth
Larry Guth University of Toronto; Institute for Advanced Study October 18, 2010 The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the ine
From playlist Mathematics
Matthew Hastings - Building Manifolds from Error Correcting Codes - IPAM at UCLA
Recorded 02 September 2021. Matthew Hastings of Microsoft Research presents "Building Manifolds from Error Correcting Codes" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Learn more online at: http://www.ipam.ucla.edu/programs/summer-schools/graduate-summer
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter
Symplectic methods for sharp systolic inequalities - Umberto Hryniewicz
Variational Methods in Geometry Seminar Topic: Symplectic methods for sharp systolic inequalities Speaker: Umberto Hryniewicz Affiliation: Universidade Federal do Rio de Janeiro; Member, School of Mathematics Date: January 22, 2019 For more video please visit http://video.ias.edu
From playlist Variational Methods in Geometry
Stanislav Nagy: Quantiles, depth, and symmetries: Geometry in multivariate statistics
There are tools of multivariate statistics with natural counterparts in geometry. We examine these connections and outline the amount of research that has been conducted in parallel in the two fields. Advances from geometry allow us to approach problems in multivariate statistics that were
From playlist Workshop: High dimensional measures: geometric and probabilistic aspects
Lizhi Chen - Topological complexity of manifolds via systolic geometry
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Lizhi Chen, Lanzhou University Title: Topological complexity of manifolds via systolic geometry Abstract: We discuss homology and homotopy complexity of manifolds in terms of Gromov’s systolic inequality. The optimal const
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Introductory talk on series. Defining a series as a sequence of partial sums.
From playlist Advanced Calculus / Multivariable Calculus
Lizhi Chen: Triangulation Complexity of Hyperbolic Manifolds and Asymptotic Geometry
Lizhi Chen, Lanzhou University Title: Triangulation Complexity of Hyperbolic Manifolds and Asymptotic Geometry The triangulation complexity is related to volume of hyperbolic manifolds via simplicial volume. On the other hand, Gromov showed that simplicial volume is related to topological
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Examples related to Viterbo's conjectures - Michael Hutchings
IAS/PU-Montreal-Paris-Tel-Aviv Symplectic Geometry Topic: Examples related to Viterbo's conjectures Speaker: Michael Hutchings Affiliation: University of California, Berkeley Date: October 23, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Orbifolds and Systolic Inequalities - Christian Lange
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar Topic: Orbifolds and Systolic Inequalities Speaker: Christian Lange Affiliation: Mathematisches Institut der Universität München Date: January 13, 2023 In this talk, I will first discuss some instances in which orbi
From playlist Mathematics
Results on abundance of global surfaces of section - Umberto Hryniewicz
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Results on abundance of global surfaces of section Umberto Hryniewicz Date: October 15, 2021
From playlist Mathematics
C. Judge - Systoles in translation surfaces
I will discuss joint work with Hugo Parlier concerning the shortest noncontractible loops—’systoles’—in a translation surface. In particular, we provide estimates (some sharp) on the number of systoles (up to homotopy) in the strata H(2g-2) and the stratum H(1,1). We also determine the ma
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications
An introduction to surfaces | Differential Geometry 21 | NJ Wildberger
We introduce surfaces, which are the main objects of interest in differential geometry. After a brief introduction, we mention the key notion of orientability, and then discuss the division in the subject between algebraic surfaces and parametrized surfaces. It is very important to have a
From playlist Differential Geometry
Systolic inequalities - Alexey Balitskiy
Short Talks by Postdoctoral Members Topic: Systolic inequalities Speaker: Alexey Balitskiy Affiliation: Member, School of Mathematics Date: September 28, 2022
From playlist Mathematics
B02 An introduction to the Euler method
An introduction to Euler's method.
From playlist A Second Course in Differential Equations