Knots and dynamics
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Formal Metadata
Title 
Knots and dynamics

Title of Series  
Number of Parts 
33

Author 

License 
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Instituto de Ciencias Matemáticas (ICMAT)

Release Date 
2006

Language 
English

Content Metadata
Subject Area  
Abstract 
The trajectories of a vector field in 3space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to wellknown arithmetical functions.

Keywords 
lowdimensional dynamical systems
periodic orbits
knots and links

00:00
Knot theory
Aerodynamics
Genetic programming
Position operator
01:03
Ocean current
Point (geometry)
Group action
Dynamical system
Presentation of a group
State of matter
Length
Knot theory
Logistic distribution
Multiplication sign
Water vapor
Mereology
Rule of inference
Frequency
Mathematics
Natural number
Circle
Field (mathematics)
Physical system
Coalition
Process (computing)
Mass flow rate
Moment (mathematics)
Model theory
Projective plane
Basis (linear algebra)
Ordinary differential equation
Limit (category theory)
Quadratic form
Orbit
Flow separation
Chaos theory
Vector field
Aerodynamics
Equation
Differential equation
Game theory
Object (grammar)
Fundamental theorem of algebra
Resultant
08:09
Point (geometry)
Complex (psychology)
Dynamical system
Mereology
Number
Frequency
Fiber (mathematics)
Process (computing)
Spacetime
Forcing (mathematics)
Model theory
Volume (thermodynamics)
Measurement
Bilinear form
Orbit
Voting
Spring (hydrology)
Order (biology)
Aerodynamics
Noise
Normal (geometry)
Theory of everything
Cycle (graph theory)
Object (grammar)
Figurate number
Thermal conductivity
Resultant
14:35
Axiom of choice
Dynamical system
Rotation
Multiplication sign
Orientation (vector space)
Direction (geometry)
Range (statistics)
Kontraktion <Mathematik>
Insertion loss
Explosion
Mechanism design
Sign (mathematics)
Coefficient of determination
Manysorted logic
Chromosomal crossover
Cuboid
Physical system
Area
Spacetime
Product (category theory)
Process (computing)
Concentric
Mass flow rate
Sampling (statistics)
Basis (linear algebra)
Measurement
Time domain
Sequence
Bilinear form
Functional (mathematics)
Degree (graph theory)
Category of being
Vector space
Angle
Helix
Hausdorff dimension
Duality (mathematics)
Equation
Figurate number
Mathematician
Identical particles
Bounded variation
Point (geometry)
Suspension (chemistry)
Free group
Normal subgroup
Rule of inference
Theory
2 (number)
Number
Frequency
Natural number
Field (mathematics)
Dot product
Numerical digit
Forcing (mathematics)
Projective plane
Total S.A.
Incidence algebra
Limit (category theory)
Approximation
Vector field
Network topology
Aerodynamics
Homomorphismus
Game theory
Pressure
Film editing
33:02
Rotation
Musical ensemble
Dynamical system
State of matter
Multiplication sign
Direction (geometry)
Explosion
Coefficient of determination
Matrix (mathematics)
Manysorted logic
Square number
Cuboid
Position operator
Physical system
Area
Product (category theory)
Spacetime
Basis (linear algebra)
Hyperbolische Gruppe
Functional (mathematics)
Bilinear form
Measurement
Connected space
Graph coloring
Lattice (order)
Hausdorff dimension
Order (biology)
Phase transition
Module (mathematics)
Right angle
Identical particles
Point (geometry)
Suspension (chemistry)
Observational study
Presentation of a group
Knot theory
Manifold
Number
Scheduling (computing)
Goodness of fit
Causality
Lie group
Acoustic shadow
Field (mathematics)
Condition number
Series (mathematics)
Forcing (mathematics)
Chemical equation
Model theory
Element (mathematics)
Line (geometry)
Incidence algebra
Quadratic form
Complex number
Computer animation
Aerodynamics
Homomorphismus
Object (grammar)
51:28
Logical constant
Dynamical system
Buckling
Multiplication sign
Direction (geometry)
Equaliser (mathematics)
Sheaf (mathematics)
Survival analysis
Kontraktion <Mathematik>
Water vapor
Parameter (computer programming)
Mereology
Proper map
Vibration
Explosion
Mathematics
Matrix (mathematics)
Manysorted logic
Square number
Saddle point
Position operator
Social class
Physical system
Fundamentalbereich
Spacetime
Product (category theory)
Process (computing)
Theory of relativity
Computability
Hyperbolische Gruppe
Functional (mathematics)
Bilinear form
Orbit
Metric tensor
Connected space
Degree (graph theory)
Flow separation
Lattice (order)
Hausdorff dimension
Module (mathematics)
Equation
Quantum
Right angle
Figurate number
Mathematician
Arithmetic progression
Resultant
Point (geometry)
Link (knot theory)
Knot theory
Real number
Similarity (geometry)
Heat transfer
Distance
Rule of inference
Theory
Fraktalgeometrie
Number
Pentagon
Power (physics)
Frequency
Goodness of fit
Thermodynamisches System
Term (mathematics)
Diagonal matrix
Analytic continuation
Field (mathematics)
Condition number
Prisoner's dilemma
Forcing (mathematics)
Element (mathematics)
Exponentiation
Physical law
Model theory
Total S.A.
Complex number
Voting
Number theory
Network topology
Combinatory logic
Aerodynamics
Theory of everything
Object (grammar)
Pressure
Kongruenzklassengruppe
Coefficient
Freezing
00:00
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00:24
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00:40
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01:04
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01:27
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01:44
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08:15
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14:43
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23:48
On duty by the suspension of the because not all roles have the public is also what dating ways in the same direction how well the category of frozen did my suspensions is is already rather big categories uh that is true of studying and I think you understand that I want to tell you this question of Congo center is a case of flu which Obtained by suspension will build on what point with this Procedure is that you good to game instead of looking for flow Europe for these officers and differ on the basis of respecting the IRA is a Why wife go We can begin playing games with books not abroad So let me mention the sequence of heroes noble and then I will comment on them 1st Iran is your new Topalov torn sailed I did whites talk show Of the good of all things if dozens of this Which are Korea puzzling there which are the identity of the label of the bond In the hearings this doubt there is not your home From that little pavilion number I will come back to this Home office and later the 2nd theorem digital by now Midsize essentially notes Cannot is homomorphism is formally home office More precise in its is about the the should they don't Of have home amorphous which is normal subgroup of dish forget area this normal somewhat has normal almost on Saw the killer of cannot be is simple Brazil is a contact sheet between Colombia and had city along this irrefutable Norwood with myself The following should return to was over because we should think it's especially produces of interest in this Don't feel the has ahead this city as defined by on and the about the head this incident from the suspension of is for job This year just a simple observation lies observation which would you can use less if the next period of for pulled his or her myself and don't would go doubts the Callao a Home Office is a the logical which is that if you have Jew come off dozens of the Which are by a wholly owned Or it was over they end It is to show more officers of the have the same cannot be No. 4 if you put all this together You get doesn't booing go what the answer to our question in the special case of PROPOSED which John living by suspension each of this from office and A Many people honorable systems that live dogs Suspensions what kind of significant dual role Warburg Medical Systems think it's a good case that the origin of bridge loan or might well go ahead I cannot do you look for support for this I would not do so fire has limited the only thing I can do is to newissue Sheila They give some rough ideas white something like that could be tall and that would point is that Thanks to an effort to recover optimized definition of Get on his home and office Seoul Let me recall or tell you what is this Definition of a better start with film offers a this issue as before which is that there is no end that identity total Now I don't know that is more mobile force individual offices overblown is a contract The what's it It is connected and simply collected this means that you can go next You will more resemble the identity by some path efforts the bath starting from the identity and going to this path may all be each It is not with his homer but is even more since the war was concocted So now I realize this is your big for annual dues Asia And you look at them more shows on this point And off the ice sample paths of this drew points more and you look at the way I don't think there is moving and get some more audacious Osama look at a picture of you have True more sure of this Sorry on more of emotion that is not true Put it on warning and connecting them is what they think sometimes positively sometimes negative anomaly definition of Columbus homomorphism forfeited is about You just did a great ball for more For variation on the former list Based on the Points So if you don't know what were the only definition of cannot be be Was a little shaky little How the arranged over Points on the rotation angle my sample collected from offices of the identity Zulu That's You clearly homomorphism which is not clear is by the end of security that this is the only 1 well too Who's some of this box just say that Colombia's former officer a measure mold of work patient contained different Don't for them as well buoyed question Which I think you'll like John mother is another white questionable nature of the global War hallway on offices of the disk which are the identity of the Monterey Bay with which respect the question is is his move was simple No not issue will be priced hallway or by this the answer is no because cannot is homomorphism is a form wasn't there for the count of normal some war waiting to see who will is who use questions of war of mother would be to try to expand the definition of cut for washes away from differ with the Hollywood definitional what he makes it clear what you want to do what it wants measure the amount of work patiently 1 out The bar point is that when the function might not For your kind of lost For Mr. The question is something the 2nd guest Is it would come for normal on which has been created by bowl This is a very nice John removal Horn amorphous looms over this number but nobody knows it is almost double the figure or not this is gone some highly resumes of the disk that All Wis Take more about next south of us commanded to his door We probably discussed this kind of book let's try to your father gotten it Find more if it's really how Kellogg is final
33:07
Well about nothing more as a formal more for you usage of combined and now the lawns Budget like concept which seems to be more and more You will see full endemical system is which is a concept Polish with Liam The CDC definition is As you have in your mouth the remarks color from Myanmar to are 1 said is that it is a quasar if Cairo gum monument to child younger 1 might just how much is bonded it should be zeal is a form of bonded is resigned once it is that this is quite more for modules It satisfies Cargo gamut is and find type of gum Boost Concepts for a wonderful feeling cold bonded go home model which has been busy could lot By Michel I would really love to have oust Discussed and I cannot so that fortunately I'm much because they were you talk in Congress might be put on no I think it's next to live and uh I'm sure he will discuss connections which initially For the production of this concept but as an appetizer stole that you just mentioned tool here and that you can find as ever Crew would idea of what The main of this kind of object simple example is this if you a For you the for World or is minimal Many don't know that This concept is not useful We line officer from IBM who are thought when they are home where did order since it is a new thing as home home office so but is not the answer Or I'm wrong sort of raw however Richard Dicker food millions 1 billion For more generally a oneoff item this concept is knew we had a age and this Faisal not of formal Jin resigned was gone off hyperbole who is if a dimension so you have to Kind of opposite situation is not a million things happen having no not with office in the upper case in case a war that I should say Ozal Ohio right lattice is amazing what is your music and what some will have no choice and that did I want to discuss it For the hope that drug tested They were on hold Field All of these which are the identities of the wandering which offer a preserve it don't now that this shot on the Hyperbole who did however faces Hall modules of if dementia I forward I would just short picture the idea is just next do to the same thing as we did with Kaleb homomorphism but instead of taking 4 points that as they want points for example He was a chill using 6 . 4 0 0 On the motherboard this thinks 6 points And the want as as their warship path she evolves from zeal to 1 This export more removed is that this quarter and was soon brought 6 Show that off course Might not come back to the ordinal points however What you can do is exactly what we we did in the case of Our mode has visited in Uganda how hard it is to have the moral Still similar to the box the rights to which the . com mark of the apology was initially just segment The That's you would use of brave It's a collection of 6 shows stopping 26 points in among themselves sold With this type of construction is you have a way of constructing a right objects right and is your enough to find square like more residents on the way will be and we get a new disk with office cause To produce no and is no you them instead of on must face fails yuan or the histories of pentup but this is the wrong idea the will construct Quiz I'm on the Book of Korea preserving the amorphous souls opens got homomorphism than measure of the amount of work they should each column him from office but none of those measure the amount of writing Danger of course in the his and the we ever brave was dream a way of toast the only thing you get our continent is rotation number of number Wedgwood get what kind of a When I'm office and I was worried was on the way and the key the signature signatures of traditional classical died of raising knots and comes at the presiding officer which is Very useful in this context as no father best tried to find more ways and more Whether I could speak for hours on this topic and this not the point in this painting so Oh For some reason it seems that we buy more brought farming These days and I think anything time to describe what his new objects so I decided to choose only tourism constructions and that Was don their so The frost never want to mention the beautiful wife and both were which where the study The goal of area preserving detail offices this year And what they did is not constructive ionized more Karzai going from this past to authority out of which is presiding officer and 2nd of how wonderful report to that if my chair the film office of the studio has forgotten that has called box report contained did this For this morning's Of There are a lot less than onehalf of the forms them They got what they wanted to number of path His thing for me to come up with a number of the restriction of the disk So it's a nice way of constructing the quarries I'm office Pollack of the Mideast war has homomorphism defined or more These but I don't know I would love to go into details despicable that for a long time This It is Lies Function function of the global office in this field which is in mourning their lot Polls See hectic model before launching I would have they would form a moral duty I would but I would have to speak of your home mortgages that we did too long however what I have to do something about The construction of what they would do not fit the Prudential case they have sit on construction for more sympathetic manifolds going for his form this year to that pointed phase of mentioned for example it is not known whether or not You can find such a way that for
42:16
Any money for the 2nd exam what I wanted to mention he is a federal boat trip Deserve which is essentially the same thing except except gradually which was down by Selftest combat itself a higher bid for combat office of jeers GE deposit did Uh if a yet Doris Hydrogen sofas and the fears is and that is why we have causing more reserve on the World war funding how we don't know more wasn't Not so with the same What as before if the film half has a support continued this them is homomorphism coincides with Colombia home office work instead of going for although I do you want more and more and more Dozens and Questions I think it's time to concentrate on schedule Auburn questions directions that might be interesting for the future and mention a shield the subs would be The hope that maybe it's possible to conduct from mission shortly Ancient trade not forget confirmation And we came back to dimension which will buy because I was forced to do so You're among us that Colombia's homomorphism what kind of a weaker version of this incident and this On the suspension is cannot be be additional forces so What would be their knives used is to be able construct value After Which Did the cable suspensions would lead you wish the value and I just explain what would be even nicer is that is balance prove would be topologically died I told you I am on by the way in and there were is this construction of mental forgo wattage shows that Maybe Roger seeing the tip of the iceberg but usually not restrict that also brought mention like to walk free We had more Then there was to go higher dimensional we after June the 3 of us eclectic 100 boys and dynamics and dimension so really there need for See it off construction highdimensional why don't you I I would prefer the stop clear the presentation of the present state of affairs because that would be technical and this is the kind of things that I want to make a break Told them that I want presented 1 example Would point is that it is Fortunately for those of you walk that I'm lost Call to all that the Treasury's I wanted it want Example of a wellknown dynamical system that I want to look at it with his new wife I want to look at it with eyes of a dog launched this then dynamical system used for more demanded off On Lesotho module as is the before I destroyed the dynamics I will begin by describing Before use They use them It's Porgy but and finally By now so that this phase is to model was so issue like holding your happier when it if the caution each of the originally what to buy dormitories is the dominant 1 YBasub more of these goods from which injured were coefficient so on the basis of provisional and not if you want to get the better of his position Both aware this Is the of doctors is off to While spending spree what I a Mac is of is just storm this could sum of October which is isomorphic It's is Want so you have a picture of left and right showing 1 example of a matter of course if you again cause for me by matrix as 2 are given all letters of marijuana and this will you get Warren matters None of this based on matters of RAI want is a homogeneous Based on the reaction of isn't too far And the But that is Of the latter's Z Jew is of course has sent to Z by definition so really are marginal as it is nothing more than this Phase of lattices of Korea won the bet for Blue Note the basis for what would stop or what has been known for Manila was probably 150 this based on double space is wholly Norfolk for the compliments over 24 a lot the freezer Explain other than from a mini ways of seeing that meant the newest thanks to lead No. 2 with a 4 point of view which is easiest way explain its the shadow letters long Additionally people from members during castles 2 20 Gedo and engine afraid they are defined by either series So did to 6 5 4 of the 6 My here is 16 times the store initially or for nonzero elements On the matters of When they get to the minors for See the complex number will indeed 40 It is a It was 140 with omission only minus 6 hours or elements of the That it is and that they fear a 19th century chewing on number his Don't number bacterise More Suddenly the father is the following the not sure not for each lot is among U.S. also took gold bricks number was due to injuries don't don't doubt that This is not which is by objection each between the base of doctors is on the left lefthand side and that the pace of a complex numbers which are way a from the sole orders brokers where due to shoot would door 27 GE foryou square So this is bloodless is wholly more Compliment is His wife go for Gin would 27 inches square the summer look at a picture of This represents a simple Bloomer walks As it Gedo threequarters of the optical remained boxes His due for it is equal to 0 Can now go to concede is right Due to June is equal to 20 70 square But we cannot hold city is awful For this picture that for the national picture the Copley brew waxes is actually a prop your seat or if you prefer a quarter of all tool this is true mention them if you want to look at this base on lattices of what you have to import according 21 condition which means in practice that you have to slice this picture by UN Studio in tool which will affect the don't go which is of cult just office which will intersect don't go On the not this Not try for store since his picture was very important on understand I wish or you The film initially
51:30
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