Knots and dynamics
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Title 
Knots and dynamics

Title of Series  
Number of Parts 
33

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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. 
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Release Date 
2006

Language 
English

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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 
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00:24
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01:44
They're JCS you will see in a moment is fantastic speak its engineers How for how thank you much so evidence might die our won't discuss coalition between dynamical systems especially to and bottle and send for I think that there will be a year of Is that some it is what idea was think of a battlefield in These are the things on guys not Who length would begin finish it is mud Ambition to give a talk that Could be understood by Would you would you this implies bouts I spent a fair amount of time recording some very classical which envisions borrower the with is also implies that I Kind of like that the guy so about my presentation of the current state of research would be a very pessimistic photos however In the final part of my talk my plant would do want example that I want to destroy with some detail and I hope it will shed some new lights on the system comment from No. 4 hours go that ended begin my Jolie in 60th when they make their logistics and what was studying some guys there is simplified model for the portion the utmost them that his mom produced the following differential equations and ordinary differential equation it respects their white prompted the solution for this differential equation holiday is very modest computer off this in what he sold his August picture em what is most amazing with this picture is that it is very robust issued share politicians on his budget which issue out some extra what's more to the equation we used to see The same thing warned that tried to approach some Objects but is gold produced project fucked of thought but I will not discuss the bid is actual Island of this picture of the new through with emotion but when I can say is that when you see Picture and when you realize how is new judge want go standard yearend accuse you want to put it in mathematical NewWave remarkable objects world today with us job Lobbyist Lawrence picture has recalled body not picture of chaos theory bottles instant 58 is mail Monday where young must however warn of credit simple approach would be But should take such a differential equation for his face and began or what This will be ought to be a periodical bids After according to close And the Crows growth is not so bad the idea that Maybe we could approach topological is of this objects for the organizers of the periodic orbits that I don't think does have a look at some of these periodic orbits of the flow of water which is generous preview of from problems legal point of view Ms. harvests that want Also in more obligated for the topological border which was and enjoy away have to have this red which doubt be fresh or not Sold their idea woes Many would try to understand the nature of the knots which appear on this side The ruins of what is a collection of several of periodic orbits them when this is picture you say Well maybe we wind up proxy the Romans at Procter and think of it I think this is supposed to be Collection not so with is This picture In the back of your mind because you will see the game or on tour in this book so new mother was thought to have a Start Twice to endorse which Not purely which not don't appear inside the Procter and that fooled but the not In Norway's On their journey For instance the rule but the opera by the army off I will have positive signature but probably most of you don't know what is of but will buy for it the need to show you a collection of data not has some of the 1 of Williams fearing they genuine doubts among big talk handlers and 15 notched was
08:15
Math by or it for toe that crossing Then she writes of Dole's and normal and don't nor about issue little more crossings Bound for example 16 crossings which is much has been tabulated the on number of Not 16 crossings is 1 million 700 wooden wanted Thousand 900 6 not the among Damn White Jr But my computer objects only 21 of them appear The story the especially when just shorter 1 example This figure may not Tuesday on the right part of this is the 1st example of the not that you will never see the inside the right talked over How's or how does want pool such here where the twitch New A way young Was to use some kind of conduct or your approach they deal what is called a little hands and Giving formal definitions that ensure About was thought to war bones of paper is based that 204 assumed the spring IBM You for doing this side But A branch still face this brunch is rhymes longterm for that used to the middle of the spring this Put emotional object is embedded in space gallery is a natural dynamical systems I'm going to show you now As a model for this dynamical system is a semifinal pitcher for Middlebury intercepts logical segment exactly positive albeit of the bubbling up Usual Bala tries with digital enjoyed by some number between 0 and 1 direct doughnut is just ax moved to talk Ex modules So issue for periodic bids You want understand the apology Ellison would be by periodic orbits of the Dalai so for example onethird My my by Julie's drew But is The force only once felt 1 job back once of as however this is 1 vote To barked orders so that is why I sat on It was found for back toward that burned that is What did for 50 years for 50 did shift back to 150 off onefifth who finished 4th 5th produced barked onefifth of from a book is sold What B. Williams did was Put door who it on this picture and sound of told the not But created by this procedure the fact that this picture of dust will respond to the Romans with actually forgery originally put the war or screen of beer Isn't much noise results it has been proved by Warwick Tucker in 2002 For the Hill might put before that Berlin's lots of fiber about actually Is Recent photos they wait This Kind of ideas provided crew will Your after acidification of dynamical systems 1 could use others From measure all their dynamical system the complexity Of the not so that it don't think will For example on Top form this year I You could put to want full amazing examples of Robert who bruised talked to reverse a few injuries place which has saw where do they got they represent Possible And it is not It's just 20 who will give a talk and his Congress I think a true I recommend to attend however woke up Some dynamical systems of pretty complex with complicated dynamical dynamics and have a small number of the article Even some have no more like the although amazing over 15 of them would have no So many would change would involve point of view there the idea is that Why not only concentrate on periodic orbits with Ridgewood We should also look at Norton non This is an idea which is very old idea that I would like to concentrate under the name of the principle of shots on Saturday after Romania Costs that really it is what I knew was think of volume preserving measure was over dynamical systems as some kind of arson cycle
14:43
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forward Daniel Anderson of the southern from the 2nd reason is 40 more should use his dots Still have a look at any pictures of a How we alive or even for through numerical from simulation like this for example the floor to rolled by some obstacles were like this is kind of picture of you cannot prevent yourself on Quarries or I'll fight off of comments but life is so smooth and orderly we want to put you with a value of about some newly full in which cool useful for all coatings from dynamic from proindependence already got high after purported could invite in the army not topple degrees for get so that's why I believe is the question of what was question is all However Like Cary mathematician for political pressure simply so that's cool To a simple question that await or make it simple to move from their mentioned frame prevention so let me laugh Report their possible construction dynamical systems which is called suspension it is a way to move from dimension version free Suppose how much did more residents of the things that Edward F. might trigger a similar product over this into your back is based And you know The true face is by using the film office in this way you produce the foreign theory is that Who is wrong you have a field which is what they do your Ronald Toronto there which is such that old on this vector field sector with prawns this exactly like all bits of the film office So it is a way to produce from Editha won't affect called the suspension of amorphous now did not
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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