• Nu S-Au Găsit Rezultate

# What is the shape of the Earth?

N/A
N/A
Protected

Share "What is the shape of the Earth?"

Copied!
21
0
0
Arată mai multe ( pagini)

Text complet

(1)

### TOPOLOGY

Ciprian Manolescu UCLA

DHC Ceremony, Babes-Bolyai University July 11, 2018

(2)

### What is the shape of the Earth?

We don’t know, so here’s an easier question:

### What is the shape of the universe?

(3)

Near each point, the Earth looks like flat space:

We say it is a two-dimensional manifold.

(4)

### Actually, the Earth is a geoid:

But in topology we do not distinguish between objects that can be deformed into each other without breaking them

### Geoid = Sphere

(5)

For topologists, a coffee mug is the same as a donut:

(6)

### However, the sphere is topologically different from the donut, and from the flat (Euclidean) space.

Classification of two dimensional manifolds:

Euclidean space

. . . . . .

. . .

(7)

### What is the shape of the universe?

Finite or infinite?

Near each point, it looks like three dimensional Euclidean space

We say it’s a three dimensional manifold.

(8)

1. Infinite (flat)

2. Poincare dodecahedral space (Luminet et al 2003)

3. Picard horn (Aurich et al 2004) etc.

The job of mathematicians is to classify all possible manifolds.

(9)

### Higher dimensions

String theory: the universe may have hidden dimensions Topology: classify manifolds of any dimension n

n=0,1,2,3,4,5,6…

A manifold of dimension n is a space such that near each point we can move in n directions (n degrees of freedom) n=0: point .

n=1: line circle

(10)

n=2:

Euclidean space

. . .

. . .

. . .

(11)

Thurston (1982) proposed a classification scheme in dimension three

. . .

(12)

### Thurston Geometrization Conjecture

The classification was showed to be true by Perelman (2003)

This implies the Poincaré conjecture, one of the seven Clay Millennium Problems (\$1 million)

(13)

### Higher dimensions

n=4, 5, 6, … Examples:

n-dimensional Euclidean space n-dimensional sphere

Theorem: One cannot classify manifolds of dimension 4 or higher.

(14)

### Triangulations

Manifolds of dimension 0, 1, 2, 3 can be triangulated:

This is not true for manifolds of dimension 4, 5, 6, … !

(15)

### Higher dimensions

Theorem: One cannot classify manifolds of dimension 4 or higher.

Instead, we can focus on simply-connected manifolds, those on which we can contract every loop:

Simply connected

Not simply connected

(16)

### Classification of n-dimensional simply- connected manifolds:

Doable in dimensions n=5,6,7,… (1960’s) Unknown in dimension n=4

Four-dimensional topology is the hardest!

(17)

### Smooth structures

If you can deform two shapes into each other without breaking them, can you deform them without making corners?

If you can’t, we say that they represent different smooth structures on the same manifold.

(18)

### Smooth structures

In dimensions n=0,1,2,3, every manifold has a unique smooth structure

The first exotic smooth structures were found by Milnor (1956) on the 7-dimensional sphere

n-dimensional Euclidean space has:

a unique smooth structure if n=0,1,2,3, 5,6,7,8,9, ….

infinitely many smooth structures if n=4

(cf. Donaldson, Gompf 1980’s)

(19)

### n-dimensional sphere

(cf. Kervaire-Milnor 1963)

dimension # structures

1 1

2 1

3 1

4 ?

5 1

6 1

7 28

8 2

9 8

10 6

dimension # structures

11 992

12 1

13 3

14 2

15 16256

16 2

17 16

18 16

19 523264

20 24

(20)

### An open question in topology

The smooth four-dimensional Poincaré conjecture:

Is there a unique smooth structure on the 4-dimensional sphere?

## ?

(21)

Earth sphere: http://www.freepik.com/free-vector/big-crystal-earth-sphere_677399.htm Tangent space: http://rqgravity.net/BasicsOfCurvature

Two-dimensional Euclidean space: http://spaceguard.iasf-roma.inaf.it/NScience/neo/dictionary/newton.htm

Geoid: Image courtesy of the University of Texas Center for Space Research and NASA.

http://celebrating200years.noaa.gov/foundations/gravity_surveys/ggm01_americas.html Coffee mug turning into doughnut: http://en.wikipedia.org/wiki/Topology

Compact two-dimensional manifolds: http://mathworld.wolfram.com/CompactManifold.html

Three-manifold three-torus: An image from inside a 3-torus, generated by Jeff Weeks' CurvedSpaces software. http://en.wikipedia.org/wiki/3- manifold#mediaviewer/File:3-Manifold_3-Torus.png

Poincare dodecahedral space: View from inside PDS along an arbitrary direction, calculated by the CurvedSpaces program, with multiple images of the Earth (from Jeff Weeks).

http://www.science20.com/news_releases/poincare_dodecahedral_space_model_gains_support_to_explain_the_shape_of_space

Grigori Perelman, solver of Poincaré conjecture, gives a lecture on his solution at New York’s Weaver Hall in 2003. Photograph: Frances M Roberts. http://www.theguardian.com/books/2011/mar/27/perfect-rigour-grigori-perelman-review

A function (blue) and a piecewise linear approximation to it (red). http://en.wikipedia.org/wiki/Piecewise_linear_function Torus: triangulated by the marching method: http://en.wikipedia.org/wiki/Surface_triangulation

A sphere is simply connected because every loop can be contracted (on the surface) to a point.

http://en.wikipedia.org/wiki/Simply_connected_space

The torus is not simply-connected: http://inperc.com/wiki/index.php?title=File:Torus.JPG

Referințe

DOCUMENTE SIMILARE

(M.O. Also, in the case of TBC patients, although the medical system provides free specific medicines they hardly access these services because of the distance

The static model of the suspension system based on 5SS axle guiding mechanism, which is shown in Figure 1, contains the bodies/parts in the suspension system (axle &amp;

T.. planning system in their structure, that is, ‘it is advisable to measure what the plan was’. Opportunity for intervention: The reporting system is useful for the

permanent tension between the possibilities of human nature and the spiritual valences of human condition, the human essence displays itself within the current political

These factors should be envisaged by Romanian society when trying to make both the minority and the majority cooperate in order to avoid the transformation of the Roma

In order to define the concept of tolerance in a manner as accessible to understanding as possible, we shall have to specify from the very beginning that we are operating in the

During the period 1992-2004, for criminal offenses with elements of abuse in the field of real estate turnover in Kosovo there were accused in total 35 persons and none

Keywords: trickster discourse, meaning, blasphemy, social change, transgression of social norms.. The Myth of the trickster and its

Dielectric measurements showed the effect of varying frequency of electric field and concentration of dysprosium on the dielectric properties like dielectric

public void doFilter(ServletRequest request, ServletResponse response, FilterChain chain) throws IOException, ServletException {.

In the latter distribution, Romanian clauses are interesting in that they allow two patterns with different underlying syntax: an agreeing variant where the embedded subject is

The evolution to globalization has been facilitated and amplified by a series of factors: capitals movements arising from the need of covering the external

The Black Sea seaside represents an important tourist and spa area not only for Romania, where there can be found the following resorts: Năvodari, Mamaia, Constanţa, Neptun,

We then go on to examine a number of prototype techniques proposed for engineering agent systems, including methodologies for agent-oriented analysis and design, formal

The best performance, considering both the train and test results, was achieved by using GLRLM features for directions {45 ◦ , 90 ◦ , 135 ◦ }, GA feature selection with DT and

Un locuitor al oglinzii (An Inhabitant of the Mirror), prose, 1994; Fascinaţia ficţiunii sau despre retorica elipsei (On the Fascination of Fiction and the Rhetoric of Ellipsis),

units, it is worth noting that the fundamental reference quantities in the British system of units (foot, pound, second) are based on the primary dimensions of length (L), force

•  A TS groups a sequence of events of one character or a stable group of characters over a period of. story Kme, which is uninterruptedly told in a span

These are ways of using signs, simpler than those that we use every day, very complicated language signs.” (Wittgenstein, 1994, p. 50-51) We are aware that there are vast stretches

The purpose of the regulation on volunteering actions is to structure a coherent approach in order to recognise the period of professional experience as well as the

The number of vacancies for the doctoral field of Medicine, Dental Medicine and Pharmacy for the academic year 2022/2023, financed from the state budget, are distributed to

The longevity of amalgam versus compomer/composite restorations in posterior primary and permanent teeth: findings From the New England Children’s Amalgam Trial..

Adrian Iftene, Faculty of Computer Science, “Alexandru Ioan Cuza” University of Iași Elena Irimia, Research Institute for Artificial Intelligence “Mihai Drăgănescu”, Romanian

Selectați limba dvs.

Site-ul web va fi tradus în limba pe care o selectați.

Limbi sugerate pentru dvs:

Alte: